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©1996 CRC Press LLC
Editor-in-Chief Daniel Zwillinger Rensselaer Polytechnic Institute Troy, New York
Associate Editors Steven G. Krantz Washington University St. Louis, Missouri
©1996 CRC Press LLC
Kenneth H. Rosen AT&T Bell Laboratories Holmdel, New Jersey
Editorial Advisory Board George E. Andrews Pennsylvania State University University Park, Pennsylvania
Ben Fusaro Salisbury State University Salisbury, Maryland
William H. Beyer The University of Akron Akron, Ohio
Alan F. Karr National Institute Statistical Sciences Research Triangle Park, North Carolina
Robert Borrelli Harvey Mudd College Claremont, California
Steve Landsburg University of Rochester Rochester, New York
Michael F. Bridgland Center for Computing Sciences Bowie, Maryland
Al Marden University of Minnesota Minneapolis, Minnesota
Courtney Coleman Harvey Mudd College Claremont, California J. Douglas Faires Youngstown State University Youngstown, Ohio Gerald B. Folland University of Washington Seattle, Washington
©1996 CRC Press LLC
Cleve Moler The MathWorks Natick, Massachusetts William H. Press Harvard University Cambridge, Massachusetts
Preface It has long been the established policy of CRC Press to publish, in handbook form, the most up-to-date, authoritative, logically arranged, and readily usable reference material available. Prior to the preparation of this 30th Edition of the CRC Standard Mathematical Tables and Formulae, the content of such a book was reconsidered. Previous editions were carefully reviewed, and input obtained from practitioners in the many branches of mathematics, engineering, and the physical sciences. The content selected for this Handbook provides the basic mathematical reference materials required for each of these disciplines. While much material was retained, several topics were completely reworked, and many new topics were added. New and completely revised topics include: partial differential equations, scientific computing, integral equations, group theory, and graph theory. For each topic, old and new, the contents have been completely rewritten and retypeset. A more comprehensive index has been added. The same successful format which has characterized earlier editions of the Handbook is retained, while its presentation is updated and more consistent from page to page. Material is presented in a multi-sectional format, with each section containing a valuable collection of fundamental reference material—tabular and expository. In line with the established policy of CRC Press, the Handbook will be kept as current and timely as is possible. Revisions and anticipated uses of newer materials and tables will be introduced as the need arises. Suggestions for the inclusion of new material in subsequent editions and comments concerning the accuracy of stated information are welcomed. No book is created in a vacuum, and this one is no exception. Not only did we start with an excellent previous edition, but our editorial staff was superb, and the contributors did an amazingly good job. I wholeheartedly thank them all. There were also many proofreaders, too many to name individually; again, thank you for your efforts. Lastly, this book would not have been possible without the support of my loving wife, Janet Taylor. Daniel Zwillinger
[email protected]
©1996 CRC Press LLC
Contributors Karen Bolinger Clarion University Clarion, Pennsylvania
Ray McLenaghan University of Waterloo Waterloo, Ontario
Patrick J. Driscoll U.S. Military Academy West Point, New York
John Michaels SUNY Brockport Brockport, New York
M. Lawrence Glasser Clarkson University Potsdam, New York
William C. Rinaman LeMoyne College Syracuse, New York
Jeff Goldberg University of Arizona Tucson, Arizona
Catherine Roberts Northern Arizona University Flagstaff, Arizona
Rob Gross Boston College Chestnut Hill, Massachusetts Melvin Hausner Courant Institute (NYU) New York, New York Christopher Heil Georgia Tech Atlanta, Georgia Paul Jameson BBN Cambridge, Massachusetts Victor J. Katz MAA Washington, DC
John S. Robertson Georgia College Milledgeville, Georgia Joseph J. Rushanan MITRE Corporation Burlington, Massachusetts Neil J. A. Sloane AT&T Bell Labs Murray Hill, New Jersey Mike Sousa MITRE Corporation Burlington, Massachusetts Gary L. Stanek Youngstown State University Youngstown, Ohio
Silvio Levy Geometry Center University of Minnesota Minneapolis, Minnesota
Michael T. Strauss Zwillinger & Associates Newton, Massachusetts
Michael Mascagni Supercomputing Research Center Bowie, Maryland
Nico M. Temme CWI Amsterdam, The Netherlands
©1996 CRC Press LLC
George K. Tzanetopoulos University of Rhode Island Kingston, Rhode Island
Brad Wilson SUNY Brockport Brockport, New York
Ahmed I. Zayed University of Central Florida Orlando, Florida
©1996 CRC Press LLC
Table of Contents Chapter 1 Analysis John S. Robertson, Karen Bolinger, M. Lawrence Glasser, Neil J.A. Sloane, and Rob Gross
Chapter 2 Algebra Brad Wilson, John Michaels, Patrick J. Driscoll, and Rob Gross
Chapter 3 Discrete Mathematics George K. Tzanetopoulos, Jeff Goldberg, Joseph J. Rushanan, and Mel Hausner
Chapter 4 Geometry Ray McLenaghan and Silvio Levy
Chapter 5 Continuous Mathematics Catherine Roberts and Ray McLenaghan
Chapter 6 Special Functions Ahmed I. Zayed, Nicco M. Temme, and Paul Jameson
Chapter 7 Probability and Statistics William C. Rinaman, Christopher Heil, Michael T. Strauss, Michael Mascagni, and Mike Sousa
Chapter 8 Scientific Computing Gary Stanek
Chapter 9 Financial Analysis Daniel Zwillinger
Chapter 10 Miscellaneous Michael T. Strauss, Rob Gross, and Victor J. Katz
List of Notations
©1996 CRC Press LLC
Chapter
1
Analysis 1.1
CONSTANTS 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.1.6 1.1.7 1.1.8
1.2
SPECIAL NUMBERS 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7 1.2.8 1.2.9 1.2.10 1.2.11 1.2.12 1.2.13 1.2.14 1.2.15
1.3
Types of numbers Representation of numbers Decimal multiples and prefixes Roman numerals Decimal equivalents of common fractions Hexadecimal addition and subtraction table Hexadecimal multiplication table Hexadecimal–decimal fraction conversion table
Positive powers of 2 Negative powers of 2 Powers of 16 in decimal scale Powers of 10 in hexadecimal scale Special constants Factorials Important numbers in different bases Bernoulli polynomials and numbers Euler polynomials and numbers Fibonacci numbers Powers of integers Sums of powers of integers Negative integer powers Integer sequences de Bruijn sequences
SERIES AND PRODUCTS 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.3.6 1.3.7 1.3.8 1.3.9 1.3.10 1.3.11 1.3.12
Definitions General properties Convergence tests Types of series Summation formulae Improving convergence: Shanks transformation Summability methods Operations with series Miscellaneous sums and series Infinite series Infinite products Infinite products and infinite series
©1996 CRC Press LLC
1.4
FOURIER SERIES 1.4.1 1.4.2 1.4.3 1.4.4
1.5
COMPLEX ANALYSIS 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 1.5.6 1.5.7 1.5.8 1.5.9 1.5.10 1.5.11 1.5.12 1.5.13 1.5.14 1.5.15 1.5.16
1.6
Definitions Operations on complex numbers Powers and roots of complex numbers Functions of a complex variable Cauchy–Riemann equations Cauchy integral theorem Cauchy integral formula Taylor series expansions Laurent series expansions Zeros and singularities Residues The argument principle Transformations and mappings Bilinear transformations Table of transformations Table of conformal mappings
REAL ANALYSIS 1.6.1 1.6.2 1.6.3 1.6.4 1.6.5 1.6.6 1.6.7 1.6.8 1.6.9 1.6.10
1.7
Special cases Alternate forms Useful series Expansions of basic periodic functions
Relations Functions (mappings) Sets of real numbers Topology Metric space Convergence in R Continuity in R Convergence in Lp Convergence in L2 Asymptotic relationships
GENERALIZED FUNCTIONS
REFERENCES
©1996 CRC Press LLC
1.1 CONSTANTS
1.1.1 TYPES OF NUMBERS Natural numbers The natural numbers are customarily denoted by N. They are the set {0, 1, 2, . . . }. Many authors do not consider 0 to be a natural number.
Integers The integers are customarily denoted by Z. They are the set {0, ±1, ±2, . . . }.
Rational numbers The rational numbers are customarily denoted by Q. They are the set { pq | p, q ∈ Z, q = 0}. Two fractions pq and rs are equal if ps = qr. Addition of fractions is defined by pq + rs = ps+qr . Multiplication of fractions qs p r pr is defined by q · s = qs .
Real numbers The real numbers are customarily denoted by R. Real numbers are defined to be converging sequences of rational numbers or as decimals that might or might not repeat. Real numbers are often divided into two subsets. One subset, the algebraic numbers, are real numbers which solve a polynomial equation in one variable with integer coefficients. For example; √12 is an algebraic number because it solves the polynomial equation 2x 2 − 1 = 0, and rational numbers are algebraic. Real numbers that are not algebraic numbers are called transcendental numbers. Examples of transcendental numbers include π and e.
Complex numbers The complex numbers are customarily denoted by C. They are numbers of the form a + bi, where i 2 = −1, and a and b are real numbers. See Section 1.5. The sum of two complex numbers a + bi and c + di is a + c + (b + d)i. The product of two complex numbers a + bi and c + di is ac − bd + (ad + bc)i. The a b reciprocal of the complex number a + bi is a 2 +b 2 − a 2 +b2 i. If z = a + bi, the complex conjugate of z is z = a − bi. Properties include: z + w = z + w and zw = z w.
©1996 CRC Press LLC
1.1.2 REPRESENTATION OF NUMBERS Numerals as usually written have radix or base 10, because the numeral an an−1 . . . a2 a1 a0 represents the number an 10n +an−1 10n−1 +· · ·+a2 102 +a1 10+a0 . However, other bases can be used, particularly bases 2, 8, and 16 (called binary, octal, and hexadecimal, respectively). When another base is used, it is indicated by a subscript: 5437 = 5 × 72 + 4 × 7 + 3 = 276, 101112 = 1 × 24 + 0 × 23 + 1 × 22 + 1 × 2 + 1 = 23, A316 = 10 × 16 + 3 = 163. When writing a number in base b, the digits used can range from 0 to b −1. If b > 10, then the digit A stands for 10, B for 11, etc. The above algorithm can be used to convert a numeral from base b to base 10. To convert a numeral from base 10 to base b, divide the numeral by b, and the remainder will be the last digit. Then divide the quotient by b, using the remainder as the previous digit. Continue dividing the quotient by b until a quotient of 0 is arrived at. For example, to convert 574 to base 12, divide, yielding a remainder of 10 and a quotient of 47. Hence, the last digit of the answer is A. Divide 47 by 12, giving a remainder of 11 again and a quotient of 3. Divide 3 by 12, giving a remainder of 3 and a quotient of 0. Therefore, 57510 = 3BA12 . In general, to convert from base b to base r, it is simplest to convert to base 10 as an intermediate step. However, it is simple to convert from base b to base ba . For example, to convert 1101111012 to base 16, group the digits in fours (because 16 is 24 ), yielding 1 1011 1101, and then convert each group of 4 to base 16 directly, yielding 1BD16 .
©1996 CRC Press LLC
1.1.3 DECIMAL MULTIPLES AND PREFIXES The prefix and symbols below are taken from Conference G´en´erale des Poids et Mesures, 1991. The common names are for the U.S. Multiple 10(10 10100 1024 1021 1018 1015 1012 109 106 103 102 101 10−1 10−2 10−3 10−6 10−9 10−12 10−15 10−18 10−21 10−24
100
Prefix
Symbol
Common name
Y Z E P T G M K h da d c m µ (Greek mu) n p f a z y
googolplex googol heptillion hexillion quintillion quadrillion trillion billion million thousand hundred ten tenth hundreth thousandth millionth billionth trillionth quadrillionth quintillionth hexillionth heptillionth
)
yotta zetta exa peta tera giga mega kilo hecto deca deci centi milli micro nano pico femto atto zepto yocto
1.1.4 ROMAN NUMERALS The major symbols in Roman numerals are I = 1, V = 5, X = 10, L = 50, C = 100, D = 500, and M = 1,000. The rules for constructing Roman numerals are 1. A symbol following one of equal or greater value adds its value (for example, II = 2, XI = 11, DV = 505). 2. A symbol following one of lesser value has the lesser value subtracted from the larger value (for example, IV = 4, IX = 9, VM = 995). 3. When a symbol stands between two of greater value, its value is subtracted from the second and the result is added to the first (for example, XIV = 10+(5−1) = 14, CIX = 100 + (10 − 1) = 109, DVL = 500 + (50 − 5) = 545). 4. When two ways exist for representing a number, the one in which the symbol of larger value occurs earlier in the string is preferred (for example, 14 is represented as XIV, not as VIX).
©1996 CRC Press LLC
Decimal number Roman numeral 10 X
14 XIV
1950 MCML
50 L
1 I
2 II
3 III
4 IV
5 V
200 CC
400 CD
500 D
600 DC
1960 MCMLX
1996 MCMXCVI
1970 MCMLXX
1997 MCMXCVII
6 VI
7 VII
999 CMXCIX
1980 MCMLXXX
1998 MCMXCVIII
8 VIII
9 IX
1000 M
1990 MCMXC
1999 MCMXCIX
2000 MM
1.1.5 DECIMAL EQUIVALENTS OF COMMON FRACTIONS 1/32 1/16
2/32 3/32
1/8
4/32 5/32
3/16
6/32 7/32
1/4
8/32 9/32
5/16
10/32 11/32
3/8
12/32 13/32
7/16
14/32 15/32
1/2
16/32
©1996 CRC Press LLC
1/64 2/64 3/64 4/64 5/64 6/64 7/64 8/64 9/64 10/64 11/64 12/64 13/64 14/64 15/64 16/64 17/64 18/64 19/64 20/64 21/64 22/64 23/64 24/64 25/64 26/64 27/64 28/64 29/64 30/64 31/64 32/64
0.015625 0.03125 0.046875 0.0625 0.078125 0.09375 0.109375 0.125 0.140625 0.15625 0.171875 0.1875 0.203125 0.21875 0.234375 0.25 0.265625 0.28125 0.296875 0.3125 0.328125 0.34375 0.359375 0.375 0.390625 0.40625 0.421875 0.4375 0.453125 0.46875 0.484375 0.5
17/32 9/16
18/32 19/32
5/8
20/32 21/32
11/16
22/32 23/32
3/4
24/32 25/32
13/16
26/32 27/32
7/8
28/32 29/32
15/16
30/32 31/32
1/1
32/32
33/64 34/64 35/64 36/64 37/64 38/64 39/64 40/64 41/64 42/64 43/64 44/64 45/64 46/64 47/64 48/64 49/64 50/64 51/64 52/64 53/64 54/64 55/64 56/64 57/64 58/64 59/64 60/64 61/64 62/64 63/64 64/64
0.515625 0.53125 0.546875 0.5625 0.578125 0.59375 0.609375 0.625 0.640625 0.65625 0.671875 0.6875 0.703125 0.71875 0.734375 0.75 0.765625 0.78125 0.796875 0.8125 0.828125 0.84375 0.859375 0.875 0.890625 0.90625 0.921875 0.9375 0.953125 0.96875 0.984375 1
1.1.6 HEXADECIMAL ADDITION AND SUBTRACTION TABLE A = 10, B = 11, C = 12, D = 13, E = 14, F = 15. Example: 6 + 2 = 8; hence 8 − 6 = 2 and 8 − 2 = 6. Example: 4 + E = 12; hence 12 − 4 = E and 12 − E = 4. 1 2 3 4 5 6 7 8 9 A B C D E F
1 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10
2 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11
3 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12
4 05 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13
5 06 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14
6 07 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15
7 08 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16
8 09 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16 17
9 0A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16 17 18
A 0B 0C 0D 0E 0F 10 11 12 13 14 15 16 17 18 19
B 0C 0D 0E 0F 10 11 12 13 14 15 16 17 18 19 1A
C 0D 0E 0F 10 11 12 13 14 15 16 17 18 19 1A 1B
D 0E 0F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C
E 0F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D
F 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E
C 0C 18 24 30 3C 48 54 60 6C 78 84 90 9C A8 B4
D 0D 1A 27 34 41 4E 5B 68 75 82 8F 9C A9 B6 C3
E 0E 1C 2A 38 46 54 62 70 7E 8C 9A A8 B6 C4 D2
F 0F 1E 2D 3C 4B 5A 69 78 87 96 A5 B4 C3 D2 E1
1.1.7 HEXADECIMAL MULTIPLICATION TABLE Example: 2 × 4 = 8. Example: 2 × F = 1E. 1 2 3 4 5 6 7 8 9 A B C D E F
1 01 02 03 04 05 06 07 08 09 0A 0B 0C 0D 0E 0F
2 02 04 06 08 0A 0C 0E 10 12 14 16 18 1A 1C 1E
3 03 06 09 0C 0F 12 15 18 1B 1E 21 24 27 2A 2D
©1996 CRC Press LLC
4 04 08 0C 10 14 18 1C 20 24 28 2C 30 34 38 3C
5 05 0A 0F 14 19 1E 23 28 2D 32 37 3C 41 46 4B
6 06 0C 12 18 1E 24 2A 30 36 3C 42 48 4E 54 5A
7 07 0E 15 1C 23 2A 31 38 3F 46 4D 54 5B 62 69
8 08 10 18 20 28 30 38 40 48 50 58 60 68 70 78
9 09 12 1B 24 2D 36 3F 48 51 5A 63 6C 75 7E 87
A 0A 14 1E 28 32 3C 46 50 5A 64 6E 78 82 8C 96
B 0B 16 21 2C 37 42 4D 58 63 6E 79 84 8F 9A A5
1.1.8 HEXADECIMAL–DECIMAL FRACTION CONVERSION TABLE Hex
Decimal
Hex
Decimal
Hex
Decimal
Hex
Decimal
.00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .0A .0B .0C .0D .0E .0F
0 0.003906 0.007812 0.011718 0.015625 0.019531 0.023437 0.027343 0.031250 0.035156 0.039062 0.042968 0.046875 0.050781 0.054687 0.058593
.40 .41 .42 .43 .44 .45 .46 .47 .48 .49 .4A .4B .4C .4D .4E .4F
0.250000 0.253906 0.257812 0.261718 0.265625 0.269531 0.273437 0.277343 0.281250 0.285156 0.289062 0.292968 0.296875 0.300781 0.304687 0.308593
.80 .81 .82 .83 .84 .85 .86 .87 .88 .89 .8A .8B .8C .8D .8E .8F
0.500000 0.503906 0.507812 0.511718 0.515625 0.519531 0.523437 0.527343 0.531250 0.535156 0.539062 0.542968 0.546875 0.550781 0.554687 0.558593
.C0 .C1 .C2 .C3 .C4 .C5 .C6 .C7 .C8 .C9 .CA .CB .CC .CD .CE .CF
0.750000 0.753906 0.757812 0.761718 0.765625 0.769531 0.773437 0.777343 0.781250 0.785156 0.789062 0.792968 0.796875 0.800781 0.804687 0.808593
.10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .1A .1B .1C .1D .1E .1F
0.062500 0.066406 0.070312 0.074218 0.078125 0.082031 0.085937 0.089843 0.093750 0.097656 0.101562 0.105468 0.109375 0.113281 0.117187 0.121093
.50 .51 .52 .53 .54 .55 .56 .57 .58 .59 .5A .5B .5C .5D .5E .5F
0.312500 0.316406 0.320312 0.324218 0.328125 0.332031 0.335937 0.339843 0.343750 0.347656 0.351562 0.355468 0.359375 0.363281 0.367187 0.371093
.90 .91 .92 .93 .94 .95 .96 .97 .98 .99 .9A .9B .9C .9D .9E .9F
0.562500 0.566406 0.570312 0.574218 0.578125 0.582031 0.585937 0.589843 0.593750 0.597656 0.601562 0.605468 0.609375 0.613281 0.617187 0.621093
.D0 .D1 .D2 .D3 .D4 .D5 .D6 .D7 .D8 .D9 .DA .DB .DC .DD .DE .DF
0.812500 0.816406 0.820312 0.824218 0.828125 0.832031 0.835937 0.839843 0.843750 0.847656 0.851562 0.855468 0.859375 0.863281 0.867187 0.871093
©1996 CRC Press LLC
Hex .20 .21 .22 .23 .24 .25 .26 .27 .28 .29 .2A .2B .2C .2D .2E .2F
Decimal 0.125000 0.128906 0.132812 0.136718 0.140625 0.144531 0.148437 0.152343 0.156250 0.160156 0.164062 0.167968 0.171875 0.175781 0.179687 0.183593
Hex .60 .61 .62 .63 .64 .65 .66 .67 .68 .69 .6A .6B .6C .6D .6E .6F
Decimal 0.375000 0.378906 0.382812 0.386718 0.390625 0.394531 0.398437 0.402343 0.406250 0.410156 0.414062 0.417968 0.421875 0.425781 0.429687 0.433593
Hex .A0 .A1 .A2 .A3 .A4 .A5 .A6 .A7 .A8 .A9 .AA .AB .AC .AD .AE .AF
Decimal 0.625000 0.628906 0.632812 0.636718 0.640625 0.644531 0.648437 0.652343 0.656250 0.660156 0.664062 0.667968 0.671875 0.675781 0.679687 0.683593
Hex .E0 .E1 .E2 .E3 .E4 .E5 .E6 .E7 .E8 .E9 .EA .EB .EC .ED .EE .EF
Decimal 0.875000 0.878906 0.882812 0.886718 0.890625 0.894531 0.898437 0.902343 0.906250 0.910156 0.914062 0.917968 0.921875 0.925781 0.929687 0.933593
.30 .31 .32 .33 .34 .35 .36 .37 .38 .39 .3A .3B .3C .3D .3E .3F
0.187500 0.191406 0.195312 0.199218 0.203125 0.207031 0.210937 0.214843 0.218750 0.222656 0.226562 0.230468 0.234375 0.238281 0.242187 0.246093
.70 .71 .72 .73 .74 .75 .76 .77 .78 .79 .7A .7B .7C .7D .7E .7F
0.437500 0.441406 0.445312 0.449218 0.453125 0.457031 0.460937 0.464843 0.468750 0.472656 0.476562 0.480468 0.484375 0.488281 0.492187 0.496093
.B0 .B1 .B2 .B3 .B4 .B5 .B6 .B7 .B8 .B9 .BA .BB .BC .BD .BE .BF
0.687500 0.691406 0.695312 0.699218 0.703125 0.707031 0.710937 0.714843 0.718750 0.722656 0.726562 0.730468 0.734375 0.738281 0.742187 0.746093
.F0 .F1 .F2 .F3 .F4 .F5 .F6 .F7 .F8 .F9 .FA .FB .FC .FD .FE .FF
0.937500 0.941406 0.945312 0.949218 0.953125 0.957031 0.960937 0.964843 0.968750 0.972656 0.976562 0.980468 0.984375 0.988281 0.992187 0.996093
1.2 SPECIAL NUMBERS 1.2.1 POSITIVE POWERS OF 2 n 1 2 3 4 5
2n 2 4 8 16 32
©1996 CRC Press LLC
n 6 7 8 9 10
2n 64 128 256 512 1024
n 11 12 13 14 15
2048 4096 8192 16384 32768
2n
n 51 52 53 54 55
2n 2251799813685248 4503599627370496 9007199254740992 18014398509481984 36028797018963968
16 17 18 19 20
65536 131072 262144 524288 1048576
56 57 58 59 60
72057594037927936 144115188075855872 288230376151711744 576460752303423488 1152921504606846976
21 22 23 24 25
2097152 4194304 8388608 16777216 33554432
61 62 63 64 65
2305843009213693952 4611686018427387904 9223372036854775808 18446744073709551616 36893488147419103232
26 27 28 29 30
67108864 134217728 268435456 536870912 1073741824
66 67 68 69 70
73786976294838206464 147573952589676412928 295147905179352825856 590295810358705651712 1180591620717411303424
31 32 33 34 35
2147483648 4294967296 8589934592 17179869184 34359738368
71 72 73 74 75
2361183241434822606848 4722366482869645213696 9444732965739290427392 18889465931478580854784 37778931862957161709568
36 37 38 39 40
68719476736 137438953472 274877906944 549755813888 1099511627776
76 77 78 79 80
75557863725914323419136 151115727451828646838272 302231454903657293676544 604462909807314587353088 1208925819614629174706176
41 42 43 44 45
2199023255552 4398046511104 8796093023308 17592186044416 351843720088832
81 82 83 84 85
2417851639229258349412352 4835703278458516698824704 9671406556917033397649408 19342813113834066795298816 38685626227668133590597632
46 47 48 49 50
70368744177664 140737488355328 281474976710656 562949953421312 1125899906842624
86 87 88 89 90
77371252455336267181195264 154742504910672534362390528 309485009821345068724781056 618970019642690137449562112 1237940039285380274899124224
©1996 CRC Press LLC
1.2.2 NEGATIVE POWERS OF 2 2−n
n 1 2 3 4 5
0.5 0.25 0.125 0.0625 0.03125
6 7 8 9 10
0.015625 0.0078125 0.00390625 0.001953125 0.0009765625
11 12 13 14 15
0.00048828125 0.000244140625 0.0001220703125 0.00006103515625 0.000030517578125
16 17 18 19 20
0.0000152587890625 0.00000762939453125 0.000003814697265625 0.0000019073486328125 0.00000095367431640625
21 22 23 24 25
0.000000476837158203125 0.0000002384185791015625 0.00000011920928955078125 0.000000059604644775390625 0.0000000298023223876953125
26 27 28 29 30
0.00000001490116119384765625 0.000000007450580596923828125 0.0000000037252902984619140625 0.00000000186264514923095703125 0.000000000931322574615478515625
31 32 33 34 35
0.0000000004656612873077392578125 0.00000000023283064365386962890625 0.000000000116415321826934814453125 0.0000000000582076609134674072265625 0.00000000002910383045673370361328125
36 37 38 39 40
0.000000000014551915228366851806640625 0.0000000000072759576141834259033203125 0.00000000000363797880709171295166015625 0.000000000001818989403545856475830078125 0.0000000000009094947017729282379150390625
©1996 CRC Press LLC
1.2.3 POWERS OF 16 IN DECIMAL SCALE n= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
16n = 1 16 256 4096 65536 1048576 16777216 268435456 4294967296 68719476736 1099511627776 17592186044416 281474976710656 4503599627370496 72057594037927936 1152921504606846976 18446744073709551616 295147905179352825856 4722366482869645213696 75557863725914323419136 1208925819614629174706176
16−n = 1 0.0625 0.00390625 0.000244140625 0.0000152587890625 0.00000095367431640625 0.000000059604644775390625 0.0000000037252902984619140625 0.00000000023283064365386962890625 0.000000000014551915228366851806640625 0.0000000000009094947017729282379150390625
1.2.4 POWERS OF 10 IN HEXADECIMAL SCALE n= 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
10n =
10−n =
1 A 64 3E8 2710 186A0 F4240 989680 5F5E100 3B9ACA00 2540BE400 174876E800 E8D4A51000 9184E72A000 5AF3107A4000 38D7EA4C68000 2386F26FC10000
1 0.19999999999999999999 . . . 0.028F5C28F5C28F5C28F5 . . . 0.004189374BC6A7EF9DB2 . . . 0.00068DB8BAC710CB295E . . . 0.0000A7C5AC471B478423 . . . 0.000010C6F7A0B5ED8D36 . . . 0.000001AD7F29ABCAF485 . . . 0.0000002AF31DC4611873 . . . 0.000000044B82FA09B5A5 . . . 0.000000006DF37F675EF6 . . . 0.000000000AFEBFF0BCB2 . . . 0.000000000119799812DE . . . 0.00000000001C25C26849 . . . 0.000000000002D09370D4 . . . 0.000000000000480EBE7B . . . 0.0000000000000734ACA5 . . .
©1996 CRC Press LLC
1.2.5 SPECIAL CONSTANTS The constant π The transcedental number π is defined as the ratio of the circumference of a circle to the diameter. It is also the ratio of the area of a circle to the square of the radius (r) and appears in several other formulas from elementary geometry (see Section 4.6) C(circle) = 2πr, A(circle) = πr 2 ,
4 3 πr , 3 SA(sphere) = 4πr 2 . V (sphere) =
One method of computing π is to use the infinite series for the function tan−1 x and one of the identities 1 π = 4 tan−1 1 = 6 tan−1 √ 3 1 1 1 1 = 2 tan−1 + 2 tan−1 + 8 tan−1 − 2 tan−1 2 3 5 239 −1 1 −1 1 −1 1 = 24 tan + 8 tan + 4 tan 8 57 239 1 1 1 + 32 tan−1 − 20 tan−1 = 48 tan−1 18 57 239 There are many other identities involving π. See Section 1.4.3. For example: (−1)n π3 1 1 1 = ∞ n=0 (2n+1)3 = 1 − 27 + 125 − 343 + . . . . 32 To 200 decimal places: π ≈ 3.
14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 82148 08651 32823 06647 09384 46095 50582 23172 53594 08128 48111 74502 84102 70193 85211 05559 64462 29489 54930 38196
©1996 CRC Press LLC
To 50 decimal places: π/20 ≈ π/15 ≈ π/12 ≈ π/11 ≈ π/10 ≈ π/9 ≈ π/8 ≈ π/7 ≈ π/6 ≈ π/5 ≈ π/4 ≈ π/3 ≈ π/2 ≈ 2π/3 ≈ 3π/2 ≈ 5π/2 √ ≈ π≈
0.15707 96326 79489 66192 31321 69163 97514 42098 58469 96876 0.20943 95102 39319 54923 08428 92218 63352 56131 44626 62501 0.26179 93877 99149 43653 85536 15273 29190 70164 30783 28126 0.28559 93321 44526 65804 20584 89389 04571 67451 97218 12501 0.31415 92653 58979 32384 62643 38327 95028 84197 16939 93751 0.34906 58503 98865 91538 47381 53697 72254 26885 74377 70835 0.39269 90816 98724 15480 78304 22909 93786 05246 46174 92189 0.44879 89505 12827 60549 46633 40468 50041 20281 67057 05359 0.52359 87755 98298 87307 71072 30546 58381 40328 61566 56252 0.62831 85307 17958 64769 25286 76655 90057 68394 33879 87502 0.78539 81633 97448 30961 56608 45819 87572 10492 92349 84378 1.04719 75511 96597 74615 42144 61093 16762 80657 23133 12504 1.57079 63267 94896 61923 13216 91639 75144 20985 84699 68755 2.09439 51023 93195 49230 84289 22186 33525 61314 46266 25007 4.71238 89803 84689 85769 39650 74919 25432 62957 54099 06266 7.85398 16339 74483 09615 66084 58198 75721 04929 23498 43776 1.77245 38509 05516 02729 81674 83341 14518 27975 49456 12239
The constant e The transcedental number e is the base of natural logarithms. It is given by ∞ 1 1 n = . e = lim 1 + n→∞ n n! n=0 To 200 decimal places: e ≈ 2.
71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66967 62772 40766 30353 54759 45713 82178 52516 64274 27466 39193 20030 59921 81741 35966 29043 57290 03342 95260 59563 07381 32328 62794 34907 63233 82988 07531 95251 01901
To 50 decimal places: e/8 ≈ e/7 ≈ e/6 ≈ e/5 ≈ e/4 ≈ e/3 ≈ e/2 ≈ 2e/3 ≈
0.33978 52285 57380 65442 00359 33919 08281 22196 55886 71249 0.38832 59754 94149 31933 71839 24478 95178 53938 92441 95714 0.45304 69714 09840 87256 00479 11892 11041 62928 74515 61666 0.54365 63656 91809 04707 20574 94270 53249 95514 49418 73999 0.67957 04571 14761 30884 00718 67838 16562 44393 11773 42499 0.90609 39428 19681 74512 00958 23784 22083 25857 49031 23332 1.35914 09142 29522 61768 01437 35676 33124 88786 23546 84998 1.81218 78856 39363 49024 01916 47568 44166 51714 98062 46664
The function ex is defined by ex = formula eπi = −1.
©1996 CRC Press LLC
∞
xn n=0 n! .
The numbers e and π are related by the
To 50 decimal places: eπ ≈ πe ≈
23.14069 26327 79269 00572 90863 67948 54738 02661 06242 60021 22.45915 77183 61045 47342 71522 04543 73502 75893 15133 99669
The constant γ Euler’s constant γ is defined by γ = lim
n→∞
n 1 k=1
k
− log n .
To 200 decimal places: γ ≈ 0.
57721 56649 01532 86060 65120 90082 40243 10421 59335 93992 35988 05767 23488 48677 26777 66467 09369 47063 29174 67495 14631 44724 98070 82480 96050 40144 86542 83622 41739 97644 92353 62535 00333 74293 73377 37673 94279 25952 58247 09492
It is not known whether γ is rational or irrational.
The constant φ The golden ratio φ is defined as the positive root of the equation φ=
√ 1+ 5 . 2
φ 1
=
1+φ ; φ
that is
To 200 decimal places: φ ≈ 1.
61803 39887 49894 84820 45868 34365 63811 77203 09179 80576 28621 35448 62270 52604 62818 90244 97072 07204 18939 11374 84754 08807 53868 91752 12663 38622 23536 93179 31800 60766 72635 44333 89086 59593 95829 05638 32266 13199 28290 26788
1.2.6 FACTORIALS The factorial of n, denoted n!, is the product of all integers less than or equal to n. n! = n · (n − 1) · (n − 2) · · · 2 · 1. The double factorial of n, denoted n!!, is the product of every other integer: n!! = n · (n − 2) · (n − 4) · · · , where the last element in the product is either 2 or 1, depending on whether n is even or odd. The generalization of the factorial function is the gamma function (see Section 6.11). When n is an integer, (n) = (n − 1)!. The shifted factorial (also called the falling factorial and Pochhammer’s symbol) is denoted by (a)n (sometimes a n ) and is defined as (a)n = a · (a + 1) · (a + 2) · · · =
n terms
©1996 CRC Press LLC
(a + n − 1)! (a + n) = . (a − 1)! (a)
(1.2.1)
The q-shifted factorial is defined as (a; q)n = (1 − a)(1 − aq) . . . (1 − aq n−1 ).
(a; q)0 = 1,
(1.2.2) n
n!
log10 n!
n!!
log10 n!!
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 30 40 50 60 70 80 90 100 110 120 130 140 150 500 1000
1 1 2 6 24 120 720 5040 40320 3.6288 × 105 3.6288 × 106 3.9917 × 107 4.7900 × 108 6.2270 × 109 8.7178 × 1010 1.3077 × 1012 2.0923 × 1013 3.5569 × 1014 6.4024 × 1015 1.2165 × 1017 2.4329 × 1018 5.1091 × 1019 1.1240 × 1021 2.5852 × 1022 6.2045 × 1023 1.5511 × 1025 2.6525 × 1032 8.1592 × 1047 3.0414 × 1064 8.3210 × 1081 1.1979 × 10100 7.1569 × 10118 1.4857 × 10138 9.3326 × 10157 1.5882 × 10178 6.6895 × 10198 6.4669 × 10219 1.3462 × 10241 5.7134 × 10262 1.2201 × 101134 4.0239 × 102567
0.00000 0.00000 0.30103 0.77815 1.38021 2.07918 2.85733 3.70243 4.60552 5.55976 6.55976 7.60116 8.68034 9.79428 10.94041 12.11650 13.32062 14.55107 15.80634 17.08509 18.38612 19.70834 21.05077 22.41249 23.79271 25.19065 32.42366 47.91165 64.48307 81.92017 100.07841 118.85473 138.17194 157.97000 178.20092 198.82539 219.81069 241.12911 262.75689 1134.0864 2567.6046
1 1 2 3 8 15 48 105 384 945 3840 10395 46080 1.3514 × 105 6.4512 × 105 2.0270 × 106 1.0322 × 107 3.4459 × 107 1.8579 × 108 6.5473 × 108 3.7159 × 109 1.3749 × 1010 8.1750 × 1010 3.1623 × 1011 1.9620 × 1012 7.9059 × 1012 4.2850 × 1016 2.5511 × 1024 5.2047 × 1032 2.8481 × 1041 3.5504 × 1050 8.9711 × 1059 4.2088 × 1069 3.4243 × 1079 4.5744 × 1089 9.5934 × 1099 3.0428 × 10110 1.4142 × 10121 9.3726 × 10131 5.8490 × 10567 3.9940 × 101284
0.00000 0.00000 0.30103 0.47712 0.90309 1.17609 1.68124 2.02119 2.58433 2.97543 3.58433 4.01682 4.66351 5.13077 5.80964 6.30686 7.01376 7.53731 8.26903 8.81606 9.57006 10.13828 10.91249 11.50001 12.29270 12.89795 16.63195 24.40672 32.71640 41.45456 50.55028 59.95284 69.62416 79.53457 89.66033 99.98197 110.48328 121.15050 131.97186 567.76709 1284.6014
©1996 CRC Press LLC
1.2.7 IMPORTANT NUMBERS IN DIFFERENT BASES Base 2
π= e= √γ = 2= log 2 =
11.00100100001111110110101010001 . . . 10.10110111111000010101000101100 . . . 0.100100111100010001100111111000 . . . 1.011010100000100111100110011001 . . . 0.101100010111001000010111111101 . . .
Base 8
π= e= γ √ = 2= log 2 =
3.110375524210264302151423063050 . . . 2.557605213050535512465277342542 . . . 0.447421477067666061722321574376 . . . 1.324047463177167462204262766115 . . . 0.542710277575071736325711707316 . . .
Base 12
π= e= γ √ = 2= log 2 =
3.184809493B918664573A6211BB1515 . . . 2.8752360698219BA71971009B388AA8 . . . 0.6B15188A6760B381B754334520434A . . . 1.4B79170A07B85737704B0854868535 . . . 0.839912483369AB213742A346792537 . . .
Base 16
π= e= γ √ = 2= log 2 =
3.243F 6A8885A308D313198A2E037073 . . . 2.B7E151628AED2A6ABF 7158809CF 4F 3 . . . 0.93C467E37DB0C7A4D1BE3F 810152CB . . . 1.6A09E667F 3BCC908B2FB1366EA957D . . . 0.B17217F 7D1CF 79ABC9E3B39803F 2F 6 . . .
1.2.8 BERNOULLI POLYNOMIALS AND NUMBERS The Bernoulli polynomials Bn (x) are defined by the generating function ∞ text tn B (x) = . n et − 1 n! n=0
(1.2.3)
These polynomials can also be defined recursively by means of B0 (x) = 1, Bn (x) = 1 nBn−1 (x), and 0 Bn (x) dx = 0 for n ≥ 1. The identity Bk+1 (x + 1) − Bk+1 (x) = (k + 1)x k means that 1k + 2k + · · · + nk = n 0 1 2 3 4 5
Bk+1 (n + 1) − Bk+1 (0) . k+1
(1.2.4)
Bn (x) 1 (2x − 1)/2 (6x 2 − 6x + 1)/6 (2x 3 − 3x 2 + x)/2 4 (30x − 60x 3 + 30x 2 − 1)/30 (6x 5 − 15x 4 + 10x 3 − x)/6
The Bernoulli numbers are the Bernoulli polynomials evaluated at 0: Bn = Bn (0). A generating function for the Bernoulli numbers is
©1996 CRC Press LLC
∞
n
t t Bn n! = et −1 . In the following table each Bernoulli number is written as a fraction of integers: Bn = Nn /Dn . Note that B2m+1 = 0 for m ≥ 1. n=0
n 0 1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Nn
Dn
1 1 −1 2 1 6 −1 30 1 42 −1 30 5 66 −691 2730 7 6 −3617 510 43867 798 −174611 330 854513 138 −236364091 2730 8553103 6 −23749461029 870 8615841276005 14322 −7709321041217 510 2577687858367 6 −26315271553053477373 1919190 2929993913841559 6 −261082718496449122051 13530
Bn 1.000000000 × 100 −5.000000000 × 10−1 1.666666667 × 10−1 −3.333333333 × 10−2 2.380952381 × 10−2 −3.333333333 × 10−2 7.575757576 × 10−2 −2.531135531 × 10−1 1.166666667 × 100 −7.092156863 × 100 5.497117794 × 101 −5.291242424 × 102 6.192123188 × 103 −8.658025311 × 104 1.425517167 × 106 −2.729823107 × 107 6.015808739 × 108 −1.511631577 × 1010 4.296146431 × 1011 −1.371165521 × 1013 4.883323190 × 1014 −1.929657934 × 1016
1.2.9 EULER POLYNOMIALS AND NUMBERS The Euler polynomials En (x) are defined by the generating function ∞ 2ext tn E (x) = . n et + 1 n! n=0
n 0 1 2 3 4 5
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En (x) 1 (2x − 1)/2 x2 − x 3 (4x − 6x 2 + 1)/4 x 4 − 2x 3 + x 5 (2x − 5x 4 + 5x 2 − 1)/2
(1.2.5)
Alternating sums of powers can be computed in terms of Euler polynomials n
(−1)n−i i k = nk − (n − 1)k + · · · ∓ 2k ± 1k =
i=1
Ek (n + 1) + (−1)n Ek (0) . 2 (1.2.6)
The Euler numbers are the Euler polynomials evaluated at 1/2, and scaled: En = 2n En ( 21 ). A generating function for the Euler numbers is ∞ n=0
n 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48
En
2et tn = 2t n! e +1 En
−1 5 −61 1385 −50521 2702765 −199360981 19391512145 −2404879675441 370371188237525 −69348874393137901 15514534163557086905 −4087072509293123892361 1252259641403629865468285 −441543893249023104553682821 177519391579539289436664789665 −80723299235887898062168247453281 41222060339517702122347079671259045 −23489580527043108252017828576198947741 14851150718114980017877156781405826684425 −10364622733519612119397957304745185976310201 7947579422597592703608040510088070619519273805 −6667537516685544977435028474773748197524107684661 6096278645568542158691685742876843153976539044435185
1.2.10 FIBONACCI NUMBERS The Fibonacci numbers {Fn } are defined by the recurrence: F1 = 1,
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F2 = 1,
Fn+2 = Fn + Fn+1 .
An exact formula is available: Fn =
√1 5
√ n 1+ 5 2
−
√ φ, the golden ratio. Also, Fn ∼ φ n / 5 as n → ∞. n 1 2 3 4 5 6 7 8 9 10 11 12 13
Fn 1 1 2 3 5 8 13 21 34 55 89 144 233
n 14 15 16 17 18 19 20 21 22 23 24 25 26
Fn 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393
n 27 28 29 30 31 32 33 34 35 36 37 38 39
Fn 196418 317811 514229 832040 1346269 2178309 3524578 5702887 9227465 14930352 24157817 39088169 63245986
√ n 1− 5 . 2
n 40 41 42 43 44 45 46 47 48 49 50 51 52
Note that lim
n→∞
Fn+1 = Fn
Fn 102334155 165580141 267914296 433494437 701408733 1134903170 1836311903 2971215073 4807526976 7778742049 12586269025 20365011074 32951280099
1.2.11 POWERS OF INTEGERS n 1 2 3 4 5 6 7 8 9 10 11 12
n3 1 8 27 64 125 216 343 512 729 1000 1331 1728
n4 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736
n5 1 32 243 1024 3125 7776 16807 32768 59049 100000 161051 248832
n6 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 2985984
n7 1 128 2187 16384 78125 279936 823543 2097152 4782969 10000000 19487171 35831808
n8 1 256 6561 65536 390625 1679616 5764801 16777216 43046721 100000000 214358881 429981696
1.2.12 SUMS OF POWERS OF INTEGERS
n10 1 1024 59049 1048576 9765625 60466176 282475249 1073741824 3486784401 10000000000 25937424601 61917364224
Define sk (n) = 1k + 2k + · · · + nk = nm=1 mk . Properties include: • sk (n) = (k + 1)−1 Bk+1 (n + 1) − Bk+1 (0) (where the Bk are Bernoulli polynomials, see Section 1.2.8).
©1996 CRC Press LLC
k−m+2 • Writing sk (n) as k+1 there is the recursion formula: m=1 am n k+1 k+1 k+2 sk+1 (n) = a1 n a3 nk + ··· + k+2 k k+1 am k+1 2 + ··· + ak+1 n + 1 − (k + 1) n. 2 k+3−m m=1
s1 (n) = 1 + 2 + 3 + · · · + n =
(1.2.7)
1 n(n + 1). 2
s2 (n) = 12 + 22 + 32 + · · · + n2 1 = n(n + 1)(2n + 1). 6 s3 (n) = 13 + 23 + 33 + · · · + n3 1 = (n2 (n + 1)2 ) = [s1 (n)]2 . 4 s4 (n) = 14 + 24 + 34 + · · · + n4 1 = (3n2 + 3n − 1)s2 (n). 5 s5 (n) = 15 + 25 + 35 + · · · + n5 1 2 = n (n + 1)2 (2n2 + 2n − 1). 12 s6 (n) = 16 + 26 + 36 + · · · + n6 n (n + 1)(2n + 1)(3n4 + 6n3 − 3n + 1). = 42 s7 (n) = 17 + 27 + 37 + · · · + n7 n2 (n + 1)2 (3n4 + 6n3 − n2 − 4n + 2). 24 s8 (n) = 18 + 28 + 38 + · · · + n8 n (n + 1)(2n + 1)(5n6 + 15n5 + 5n4 − 15n3 − n2 + 9n − 3). = 90 s9 (n) = 19 + 29 + 39 + · · · + n9 =
n2 (n + 1)2 (2n6 + 6n5 + n4 − 8n3 + n2 + 6n − 3). 20 s10 (n) = 110 + 210 + 310 + · · · + n10 n = (n + 1)(2n + 1)(3n8 + 12n7 + 8n6 − 18n5 66 − 10n4 + 24n3 + 2n2 − 15n + 5). =
©1996 CRC Press LLC
n
n
k=1
k
n
k=1
k2
n
k=1
k3
n
k=1
k4
n
k=1
k5
1 2 3 4 5
1 3 6 10 15
1 5 14 30 55
1 9 36 100 225
1 17 98 354 979
1 33 276 1300 4425
6 7 8 9 10
21 28 36 45 55
91 140 204 285 385
441 784 1296 2025 3025
2275 4676 8772 15333 25333
12201 29008 61776 120825 220825
11 12 13 14 15
66 78 91 105 120
506 650 819 1015 1240
4356 6084 8281 11025 14400
39974 60710 89271 127687 178312
381876 630708 1002001 1539825 2299200
16 17 18 19 20
136 153 171 190 210
1496 1785 2109 2470 2870
18496 23409 29241 36100 44100
243848 327369 432345 562666 722666
3347776 4767633 6657201 9133300 12333300
21 22 23 24 25
231 253 276 300 325
3311 3795 4324 4900 5525
53361 64009 76176 90000 105625
917147 1151403 1431244 1763020 2153645
16417401 21571033 28007376 35970000 45735625
1.2.13 NEGATIVE INTEGER POWERS Riemann’s zeta function is ζ (n) = α(n) =
∞ (−1)k+1 k=1
kn
,
∞
β(n) =
1 k=1 k n . ∞ k=0
Related functions are
(−1)k , (2k + 1)n
γ (n) =
∞ k=0
1 . (2k + 1)n
Properties include: α(n) = (1 − 21−n )ζ (n), γ (n) = (1 − 2−n )ζ (n), (2π)2k |B2k | , 2(2k)! (π/2)2k+1 |E2k | . β(2k + 1) = 2(2k)! ζ (2k) =
(1.2.8)
The series β(1) = 1− 13 + 15 −· · · = π/4 is known as Gregory’s series. Catalan’s constant is G = β(2).
©1996 CRC Press LLC
ζ (2) = π 2 /6 ζ (4) = π 4 /90 ζ (6) = π 6 /945 ζ (8) = π 8 /9450 ζ (10) = π 10 /93555 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
ζ (n) =
∞ 1 kn k=1
∞ 1.6449340669 1.2020569032 1.0823232337 1.0369277551 1.0173430620 1.0083492774 1.0040773562 1.0020083928 1.0009945752 1.0004941886 1.0002460866 1.0001227133 1.0000612482 1.0000305882 1.0000152823 1.0000076372 1.0000038173 1.0000019082 1.0000009540
β(1) = π/4 β(3) = π 3 /32 β(5) = 5π 5 /1536 β(7) = 61π 7 /184320 β(9) = 277π 9 /8257536
∞ (−1)k+1 k=1
kn
0.6931471805 0.8224670334 0.9015426773 0.9470328294 0.9721197705 0.9855510912 0.9925938199 0.9962330018 0.9980942975 0.9990395075 0.9995171435 0.9997576851 0.9998785428 0.9999391703 0.9999695512 0.9999847642 0.9999923783 0.9999961879 0.9999980935 0.9999990466
∞ k=0
1 (2k + 1)n
∞ 1.2337005501 1.0517997903 1.0146780316 1.0045237628 1.0014470766 1.0004715487 1.0001551790 1.0000513452 1.0000170414 1.0000056661 1.0000018858 1.0000006281 1.0000002092 1.0000000697 1.0000000232 1.0000000077 1.0000000026 1.0000000009 1.0000000003
∞ k=0
(−1)k (2k + 1)n
0.7853981633 0.9159655941 0.9689461463 0.9889445517 0.9961578281 0.9986852222 0.9995545079 0.9998499902 0.9999496842 0.9999831640 0.9999943749 0.9999981224 0.9999993736 0.9999997911 0.9999999303 0.9999999768 0.9999999923 0.9999999974 0.9999999991 0.9999999997
1.2.14 INTEGER SEQUENCES These sequences are arranged in numerical order (disregarding any leading zeros or ones). Note that C(n, k) = nk . 1. 1, −1, −1, 0, −1, 1, −1, 0, 0, 1, −1, 0, −1, 1, 1, 0, −1, 0, −1, 0, 1, 1, −1, 0, 0, 1, 0, 0, −1, −1, −1, 0, 1, 1, 1, 0, −1, 1, 1, 0, −1, −1, −1, 0, 0, 1, −1, 0, 0, 0, 1, 0, −1, 0, 1, 0 M¨obius function µ(n), n ≥ 1 2. 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0 Number of ways of writing n as a sum of 2 squares, n ≥ 0 3. 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2 Number of distinct primes dividing n, n ≥ 1 4. 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2 Number of Abelian groups of order n, n ≥ 1
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5. 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267 Number of groups of order n, n ≥ 1 6. 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 3, 4, 4, 5, 4, 5, 5, 6, 1, 2, 2, 3, 2 Number of 1’s in binary expansion of n, n ≥ 0 7. 1, 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161, 2182, 4080, 7710, 14532, 27594, 52377, 99858, 190557, 364722, 698870, 1342176, 2580795, 4971008 Number of binary irreducible polynomials of degree n, or n-bead necklaces, n ≥ 0 8. 1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 8, 1, 3, 3, 8, 1, 8, 1, 8, 3, 3, 1, 20, 2, 3, 4, 8, 1, 13, 1, 16, 3, 3, 3, 26, 1, 3, 3, 20, 1, 13, 1, 8, 8, 3, 1, 48, 2, 8, 3, 8, 1, 20, 3, 20, 3, 3, 1 Number of perfect partitions of n, or ordered factorizations of n + 1, n ≥ 0 9. 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 2, 1 Thue–Morse nonrepeating sequence 10. 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 9, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 10, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 9, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 12, 1, 2, 1, 4 Hurwitz–Radon numbers 11. 1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 6, 4, 4, 4, 9, 2, 4, 4, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 4, 8, 2, 6 d(n), the number of divisors of n, n ≥ 1 12. 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16 π(n), the number of primes ≤ n, for n ≥ 1 13. 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296, 340, 390, 448, 512, 585, 668, 760, 864, 982, 1113, 1260, 1426 Number of partitions of n into distinct parts, n ≥ 1 14. 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8, 12, 10, 22, 8, 20, 12, 18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42 Euler totient function φ(n): count numbers ≤ n and prime to n, for n ≥ 1 15. 1, 1, 1, 0, 1, 1, 2, 2, 4, 5, 10, 14, 26, 42, 78, 132, 249, 445, 842, 1561, 2988, 5671, 10981, 21209, 41472, 81181, 160176, 316749, 629933, 1256070, 2515169, 5049816 Number of series-reduced trees with n unlabeled nodes, n ≥ 0 16. 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131 Prime powers 17. 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90, 96, 108, 120, 144, 168, 180, 210, 216, 240, 288, 300, 336, 360, 420, 480, 504, 540, 600, 630, 660 Highly abundant numbers: where sum-of-divisors function increases 18. 1, 2, 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, 62, 69, 72, 77, 82, 87, 97, 99, 102, 106, 114, 126, 131, 138, 145, 148, 155, 175, 177, 180, 182, 189, 197, 206, 209 Ulam numbers: next is uniquely the sum of 2 earlier terms 19. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 60, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 168, 173 Orders of simple groups 20. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181 Prime numbers
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21. 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883 Number of partitions of n, n ≥ 1 22. 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433 Mersenne primes: p such that 2p − 1 is prime 23. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269 Fibonacci numbers: F (n) = F (n − 1) + F (n − 2) 24. 1, 2, 3, 6, 10, 20, 35, 70, 126, 252, 462, 924, 1716, 3432, 6435, 12870, 24310, 48620, 92378, 184756, 352716, 705432, 1352078, 2704156, 5200300, 10400600, 20058300 Central binomial coefficients: C(n, [n/2]), n ≥ 1 25. 1, 1, 2, 3, 6, 11, 20, 40, 77, 148, 285, 570, 1120, 2200, 4323, 8498, 16996, 33707, 66844, 132568, 262936, 521549, 1043098, 2077698, 4138400, 8243093 Stern’s sequence: a(n + 1) is sum of 1 + [n/2] preceding terms, n ≥ 1 26. 1, 1, 2, 3, 6, 11, 22, 42, 84, 165, 330, 654, 1308, 2605, 5210, 10398, 20796, 41550, 83100, 166116, 332232, 664299, 1328598, 2656866, 5313732, 10626810 Narayana–Zidek–Capell numbers: a(2n) = 2a(2n − 1), a(2n + 1) = 2a(2n) − a(n) 27. 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, 207, 451, 983, 2179, 4850, 10905, 24631, 56011, 127912, 293547, 676157, 1563372, 3626149, 8436379, 19680277, 46026618, 107890609 Wedderburn–Etherington numbers: interpretations of X n , n ≥ 1 28. 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, 551, 1301, 3159, 7741, 19320, 48629, 123867, 317955, 823065, 2144505, 5623756, 14828074, 39299897, 104636890, 279793450 Number of trees with n unlabeled nodes, n ≥ 1 29. 2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 Dedekind numbers: number of monotone Boolean functions of n variables, n ≥ 0 30. 1, 1, 2, 3, 7, 16, 54, 243, 2038, 33120, 1182004, 87723296, 12886193064, 3633057074584, 1944000150734320, 1967881448329407496 Number of Euler graphs or 2-graphs with n nodes, n ≥ 1 31. 0, 0, 1, 1, 2, 3, 7, 18, 41, 123, 367, 1288, 4878 Number of alternating prime knots with n crossings, n ≥ 1 32. 0, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, 2176, 9988 Number of prime knots with n crossings, n ≥ 1 33. 1, 1, 2, 3, 8, 14, 42, 81, 262, 538, 1828, 3926, 13820, 30694, 110954, 252939, 933458, 2172830, 8152860, 19304190, 73424650, 176343390, 678390116, 1649008456 Meandric numbers: ways a river can cross a road n times, n ≥ 1 34. 0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106 Numbers that are sums of 2 squares 35. 1, 2, 4, 5, 8, 10, 14, 15, 16, 21, 22, 25, 26, 28, 33, 34, 35, 36, 38, 40, 42, 46, 48, 49, 50, 53, 57, 60, 62, 64, 65, 70, 77, 80, 81, 83, 85, 86, 90, 91, 92, 100, 104, 107 MacMahon’s prime numbers of measurement, or segmented numbers 36. 1, 2, 4, 6, 10, 14, 20, 26, 36, 46, 60, 74, 94, 114, 140, 166, 202, 238, 284, 330, 390, 450, 524, 598, 692, 786, 900, 1014, 1154, 1294, 1460, 1626, 1828, 2030, 2268, 2506 Binary partitions (partitions of 2n into powers of 2), n ≥ 0 37. 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33554432, 67108864 Powers of 2
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38. 1, 1, 2, 4, 9, 20, 48, 115, 286, 719, 1842, 4766, 12486, 32973, 87811, 235381, 634847, 1721159, 4688676, 12826228, 35221832, 97055181, 268282855, 743724984, 2067174645 Number of rooted trees with n unlabeled nodes, n ≥ 1 39. 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415 Motzkin numbers: ways to join n points on a circle by chords 40. 1, 1, 2, 4, 9, 22, 59, 167, 490, 1486, 4639, 14805, 48107, 158808, 531469, 1799659, 6157068, 21258104, 73996100, 259451116, 951695102, 3251073303 Number of different scores in n-team round-robin tournament, n ≥ 1 41. 1, 1, 2, 4, 11, 34, 156, 1044, 12346, 274668, 12005168, 1018997864, 165091172592, 50502031367952, 29054155657235488, 31426485969804308768 Number of graphs with n unlabeled nodes, n ≥ 0 42. 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428, 38613965, 93222358, 225058681, 543339720 Pell numbers: a(n) = 2a(n − 1) + a(n − 2) 43. 1, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 13079255, 50107909, 192622052, 742624232, 2870671950, 11123060678, 43191857688 Polyominoes with n cells, n ≥ 1 44. 1, 1, 2, 4, 12, 56, 456, 6880, 191536, 9733056, 903753248, 154108311168, 48542114686912, 28401423719122304, 31021002160355166848 Number of outcomes of n-team round-robin tournament, n ≥ 1 45. 1, 1, 2, 5, 14, 38, 120, 353, 1148, 3527, 11622, 36627, 121622, 389560, 1301140, 4215748, 13976335, 46235800, 155741571, 512559185, 1732007938, 5732533570 Number of ways to fold a strip of n blank stamps, n ≥ 1 46. 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020 Catalan numbers: C(2n, n)/(n + 1), n ≥ 0 47. 1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, x 5832742205057 Bell or exponential numbers: expansion of e(e −1) 48. 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, 353792, 2702765, 22368256, 199360981, 1903757312, 19391512145, 209865342976, 2404879675441, 29088885112832 Euler numbers: expansion of sec x + tan x 49. 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332 Pronic numbers: n(n + 1), n ≥ 0 50. 1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, 184756, 705432, 2704156, 10400600, 40116600, 155117520, 601080390, 2333606220, 9075135300, 35345263800 Central binomial coefficients: C(2n, n), n ≥ 0 51. 1, 1, 1, 2, 6, 21, 112, 853, 11117, 261080, 11716571, 1006700565, 164059830476, 50335907869219, 29003487462848061, 31397381142761241960 Number of connected graphs with n unlabeled nodes, n ≥ 0 52. 1, 2, 6, 22, 101, 573, 3836, 29228, 250749, 2409581, 25598186, 296643390, 3727542188, 50626553988, 738680521142 Kendall–Mann numbers: maximal inversions in permutation of n letters, n ≥ 1 53. 1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000, 355687428096000, 6402373705728000 Factorial numbers: n!, n ≥ 0
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54. 1, 2, 7, 42, 429, 7436, 218348, 10850216, 911835460, 129534272700, 31095744852375, 12611311859677500, 8639383518297652500 Robbins numbers: n−1 k=0 (3k + 1)!/(n + k)!, n ≥ 1 55. 1, 2, 8, 42, 262, 1828, 13820, 110954, 933458, 8152860, 73424650, 678390116, 6405031050, 61606881612, 602188541928, 5969806669034, 59923200729046 Closed meandric numbers: ways a loop can cross a road 2n times, n ≥ 1 56. 1, 2, 8, 48, 384, 3840, 46080, 645120, 10321920, 185794560, 3715891200, 81749606400, 1961990553600, 51011754393600, 1428329123020800, 42849873690624000 Double factorial numbers: (2n)!! = 2n n!, n ≥ 0 57. 0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664 Derangements: permutations of n elements with no fixed points, n ≥ 1 58. 1, 2, 16, 272, 7936, 353792, 22368256, 1903757312, 209865342976, 29088885112832, 4951498053124096, 1015423886506852352, 246921480190207983616 Tangent numbers: expansion of tan x 59. 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124 σ (n), sum of the divisors of n, n ≥ 1 60. 1, 3, 4, 7, 9, 12, 13, 16, 19, 21, 25, 27, 28, 31, 36, 37, 39, 43, 48, 49, 52, 57, 61, 63, 64, 67, 73, 75, 76, 79, 81, 84, 91, 93, 97, 100, 103, 108, 109, 111, 112, 117, 121, 124, 127 Numbers of the form x 2 + xy + y 2 61. 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, 15127, 24476, 39603, 64079, 103682, 167761, 271443, 439204, 710647, 1149851, 1860498 Lucas numbers: L(n) = L(n − 1) + L(n − 2) 62. 1, 1, 1, 3, 4, 12, 27, 82, 228, 733, 2282, 7528, 24834, 83898, 285357, 983244, 3412420, 11944614, 42080170, 149197152, 531883768, 1905930975, 6861221666, 24806004996 Number of ways to cut an n-sided polygon into triangles, n ≥ 1 63. 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780 Triangular numbers: n(n + 1)/2, n ≥ 1 64. 1, 3, 6, 11, 17, 25, 34, 44, 55, 72, 85, 106, 127, 151 Shortest Golomb ruler with n marks, n ≥ 2 65. 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1479, 2485, 4167, 6879, 11297, 18334, 29601, 47330, 75278, 118794, 186475, 290783, 451194, 696033, 1068745, 1632658 Number of planar partitions of n, n ≥ 1 66. 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, 169, 171, 189, 193, 195, 201, 205 Lucky numbers (defined by sieve similar to prime numbers) 67. 1, 3, 7, 19, 47, 130, 343, 951, 2615, 7318, 20491, 57903, 163898, 466199, 1328993, 3799624, 10884049, 31241170, 89814958, 258604642 Number of mappings from n unlabeled points to themselves, n ≥ 1 68. 1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, 229377, 491521, 1048577, 2228225, 4718593, 9961473, 20971521, 44040193, 92274689 Cullen numbers: n · 2n + 1, n ≥ 1 69. 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203 Powers of 3
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70. 1, 3, 9, 33, 139, 718, 4535 Number of topologies or transitive-directed graphs with n unlabeled nodes, n ≥ 1 71. 1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, 518859, 2646723, 13648869, 71039373, 372693519, 1968801519, 10463578353, 55909013009, 300159426963 Schroeder’s second problem: ways to interpret X1 X2 . . . Xn , n ≥ 1 72. 1, 3, 11, 50, 274, 1764, 13068, 109584, 1026576, 10628640, 120543840, 1486442880, 19802759040, 283465647360, 4339163001600, 70734282393600, 1223405590579200 Stirling numbers of first kind: n2 , n ≥ 2. 73. 1, 3, 13, 75, 541, 4683, 47293, 545835, 7087261, 102247563, 1622632573, 28091567595, 526858348381, 10641342970443, 230283190977853, 5315654681981355 Preferential arrangements of n things, n ≥ 1 74. 1, 3, 15, 105, 945, 10395, 135135, 2027025, 34459425, 654729075, 13749310575, 316234143225, 7905853580625, 213458046676875, 6190283353629375 Double factorial numbers: (2n + 1)!! = 1 · 3 · 5 · · · (2n + 1), n ≥ 1 75. 1, 3, 16, 125, 1296, 16807, 262144, 4782969, 100000000, 2357947691, 61917364224, 1792160394037, 56693912375296, 1946195068359375, 72057594037927936 Number of trees with n labeled nodes: nn−2 , n ≥ 2 76. 1, 3, 16, 218, 9608, 1540944, 882033440, 1793359192848, 13027956824399552, 341260431952972580352, 32522909385055886111197440 Directed graphs with n unlabeled nodes, n ≥ 1 77. 1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905, 51943281731, 2905151042481, 191329672483963, 14655626154768697, 1291885088448017715 Genocchi numbers: expansion of tan(x/2) 78. 0, 1, 4, 5, 16, 17, 20, 21, 64, 65, 68, 69, 80, 81, 84, 85, 256, 257, 260, 261, 272, 273, 276, 277, 320, 321, 324, 325, 336, 337, 340, 341, 1024, 1025, 1028, 1029, 1040, 1041 Moser–de Bruijn sequence: sums of distinct powers of 4 79. 4, 7, 8, 9, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 26, 26, 2 Chromatic number of surface of genus n, n ≥ 0 80. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296 The squares 81. 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631, 694, 760, 829, 901, 976, 1054, 1135, 1219, 1306, 1396, 1489, 1585, 1684, 1786 Centered triangular numbers: (3n2 + 3n + 2)/2, n ≥ 1 82. 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984 Tetrahedral numbers: C(n + 3, 3), n ≥ 0 83. 1, 1, 4, 26, 236, 2752, 39208, 660032, 12818912, 282137824, 6939897856, 188666182784, 5617349020544, 181790703209728, 6353726042486272 Schroeder’s fourth problem: families of subsets of an n set, n ≥ 1 84. 1, 4, 29, 355, 6942, 209527, 9535241, 642779354, 63260289423, 8977053873043, 1816846038736192, 519355571065774021 Number of transitive-directed graphs with n labeled nodes, n ≥ 1 85. 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820 Pentagonal numbers: n(3n − 1)/2, n ≥ 1
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86. 1, 5, 13, 25, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 841, 925, 1013, 1105, 1201, 1301, 1405, 1513, 1625, 1741, 1861, 1985, 2113, 2245 Centered square numbers: n2 + (n − 1)2 , n ≥ 1 87. 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201, 6930, 7714, 8555, 9455, 10416 Square pyramidal numbers: n(n + 1)(2n + 1)/6, n ≥ 1 88. 1, 5, 25, 125, 625, 3125, 15625, 78125, 390625, 1953125, 9765625, 48828125, 244140625, 1220703125, 6103515625, 30517578125, 152587890625, 762939453125, 3814697265625 Powers of 5 89. 1, 5, 52, 1522, 145984, 48464496, 56141454464, 229148550030864, 3333310786076963968, 174695272746749919580928 Number of possible relations on n unlabeled points, n ≥ 1 90. 1, 1, 5, 61, 1385, 50521, 2702765, 199360981, 19391512145, 2404879675441, 370371188237525, 69348874393137901, 15514534163557086905, 4087072509293123892361 Euler numbers: expansion of sec x 91. 1, 5, 109, 32297, 2147321017, 9223372023970362989, 170141183460469231667123699502996689125 Number of ways to cover an n set, n ≥ 1 92. 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, 276, 325, 378, 435, 496, 561, 630, 703, 780, 861, 946, 1035, 1128, 1225, 1326, 1431, 1540, 1653, 1770, 1891, 2016, 2145, 2278 Hexagonal numbers: n(2n − 1), n ≥ 1 93. 1, 6, 25, 90, 301, 966, 3025, 9330, 28501, 86526, 261625, 788970, 2375101, 7141686, 21457825, 64439010, 193448101, 580606446, 1742343625, 5228079450, 15686335501 Stirling numbers of second kind: n3 , n ≥ 3 94. 6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176 Perfect numbers: equal to the sum of their proper divisors 95. 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936, 1045, 1160, 1281, 1408, 1541, 1680, 1825, 1976, 2133, 2296, 2465, 2640, 2821, 3008, 3201 Octagonal numbers: n(3n − 2), n ≥ 1 96. 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389 The cubes 97. 1, −24, 252, −1472, 4830, −6048, −16744, 84480, −113643, −115920, 534612, −370944, −577738, 401856, 1217160, 987136, −6905934, 2727432, 10661420 Ramanujan τ function 98. 341, 561, 645, 1105, 1387, 1729, 1905, 2047, 2465, 2701, 2821, 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911, 10261, 10585, 11305, 12801, 13741, 13747 Sarrus numbers: pseudo-primes to base 2 99. 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545 Carmichael numbers 100. 1, 744, 196884, 21493760, 864299970, 20245856256, 333202640600, 4252023300096, 44656994071935, 401490886656000, 3176440229784420, 22567393309593600 Coefficients of the modular function j
For more information about all of these sequences including formulae and references, see N.J.A. Sloane and S. Plouffe, Encyclopedia of Integer Sequences, Academic Press, 1995, where over 5000 other sequences are also described.
©1996 CRC Press LLC
1.2.15 DE BRUIJN SEQUENCES A sequence of length q n over an alphabet of size q is a de Bruijn sequence if every possible n-tuple occurs in the sequence (allowing wraparound to the start of the sequence). There are de Bruijn sequences for any q and n. The table below gives some small examples. q 2 2 3 4
n 2 3 2 2
Length 4 8 9 16
Sequence 0110 01110100 001220211 0011310221203323
1.3 SERIES AND PRODUCTS
1.3.1 DEFINITIONS If {an } is a sequence of numbers or functions, then • SN = N n=1 an = a1 + a2 + ... + aN . • SN is the N th partial sum of S. • The series is said to converge if the limit exists and diverge if it does not. • For an infinite series: S = limN→∞ SN = ∞ n=1 an (when the limit exists). • If an = bn x n , where bn is independent of x, then S is called a power series. • If an = (−1)n |an |, then S is called an alternating series. • If |an | converges, then the series converges absolutely. • If S converges, but not absolutely, then it converges conditionally. For example, the harmonic series S = 1+ 21 + 13 +. . . diverges. The corresponding alternating series (called the alternating harmonic series) S = 1 − 21 + 31 + · · · + (−1)n−1 n1 + . . . converges (conditionally) to log 2.
1.3.2 GENERAL PROPERTIES 1. Adding or removing a finite number of terms does not affect the convergence or divergence of an infinite series. 2. The terms of an absolutely convergent series may be rearranged in any manner without affecting its value.
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3. A conditionally convergent series can be made to converge to any value by suitably rearranging its terms. 4. If the component series are convergent, then (αan +βbn ) = α an +β bn . ∞ ∞ ∞ 5. an bn = cn where cn = a0 bn + a1 bn−1 + · · · + an b0 . n=0
n=0
n=0
6. Summation by parts: let
an and
an bn =
where Sn is the nth partial sum of
bn converge. Then
Sn (bn − bn+1 )
an .
7. A power series may be integrated and differentiated term by term within its interval of convergence. 8. Schwarz inequality:
|an | |bn | ≤
an2
1/2
bn2
1/2
9. Holder’s inequality: |an bn | ≤ |an |1/p |bn |1/q when 1/p + 1/q = 1 and p, q > 1
10. Minkowski’s inequality:
|an + bn |p
1/p
≤
|an |p
1/p
+
|bn |p
1/p
when p ≥ 1 For example: 1. Let T be the alternating harmonic series S rearranged so that each positive term is followed by the next two negative terms. By combining each positive term of T with the succeeding negative term, we find that T3N = 21 S2N . Hence, T = 21 log 2. 2. The series 1 + 1+
1 2
−
1 3
+
1 4
+
1 5
−
1 6
+
1 7
+
1 8
−
1 9
+ . . . diverges, whereas
1 1 1 1 1 1 1 1 − + + − + ··· + + − + ... 3 2 5 7 4 4n − 3 4n − 1 2n
√ converges to log(2 2).
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1.3.3 CONVERGENCE TESTS
1. Comparison test: If |an | ≤ bn and bn converges, then an converges. 2. Limit test: If limn→∞ an = 0, then an is divergent. 3. Ratio test: Let ρ = limn→∞ | aan+1 |. If ρ < 1, the series converges absolutely. n If ρ > 1, the series diverges. 4. Cauchy root test: Let σ = limn→∞ |an |1/n . If σ < 1, the series converges. If σ > 1, it diverges. 5. Integral test: Let |an | = ∞f (n) with f (x) being monotone decreasing, and limx→∞ f (x) = 0. Then A f (x) dx and an both converge or both diverge for any A > 0. an+1 = 1 − p + An where q > 1 and the sequence {An } is 6. Gauss’s test: If an n nq bounded, then the series is absolutely convergent if and only if p > 1. 7. Alternating series test: If |an | tends monotonically to 0, then (−1)n |an | converges. For example: 1 c c n 1. For S = ∞ n=1 n x , ρ = lim n→∞ (1 + n ) x = x. Hence, using the ratio test, S converges for 0 < x < 1 and any value of c. n 5n , σ = limn→∞ ( n520 )1/n = 5. Therefore the series diverges. 2. For S = ∞ n20 n=1 −s −s 3. For S = ∞ n=1 n , let f (x) = x . Then ∞ ∞ dx 1 f (x) dx = = s x s − 1 1 1
for s > 1, and the integral diverges for s ≤ 1. Hence, S converges for s > 1. 1 4. The sum ∞ n=2 n(log n)s converges for s > 1 by the integral test. n 5. Let an = (c) = c(c+1)...(c+n−1) where c is not 0 or a negative integer. Then n! n! |an+1 /an | = 1 − (c + 1)/n + (c + 1)/n2 (1 + 1/n). By Gauss’s test, the series converges absolutely if and only if c > 0.
1.3.4 TYPES OF SERIES Bessel series 1. Fourier–Bessel series:
∞
an Jν (jν,n z)
n=0
2. Neumann series:
∞ n=0
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an Jν+n (z)
3. Kapteyn series:
∞
an Jν+n [(ν + n)z]
n=0
4. Schl¨omilch series:
∞
an Jν (nz)
n=1
For example; • • •
∞
1 n=0 n! Jν+n (2)
∞
n=1
∞
Jn (nz) = n+1
n=1 (−1)
=
1 (ν+1)
1 z 2 1−z
for 0 < z < 1
J0 (nz) =
1 2
for 0 < z < π
Dirichlet series
an These are series of the form ∞ n=1 nx . They converge for x > x0 , where x0 is the abscissa of convergence. Assuming the limits exist: If If
log |a1 + · · · + an | . n→∞ log n
an diverges, then x0 = lim
an converges, then x0 = lim
n→∞
log |an+1 + an+2 + . . . | . log n
For example: • Riemann zeta function: ζ (x) =
• •
∞
∞ 1 , x0 = 1 nx n=1
µ(n) n=1 nx
=
d(n) n=1 nx
= [ζ (x)]2 , x0 = 1 (d(n) is the number of divisors of n)
∞
1 , ζ (x)
x0 = 1 (µ(n) denotes the M¨obius function)
Fourier series If f (x) satisfies certain properties, then (see page 46 for details) f (x) =
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∞ a0 nπx nπx + + bn sin an cos 2 L L n=1
(1.3.1)
• If f (x) has the Laplace transform F (k), then ∞
1 F (k) cos(kt) = 2 k=1 ∞
1 2
F (k) sin(kt) =
k=1
∞
cos(t) − e−x f (x) dx, cosh(x) − cos(t)
∞
f (x) dx. cosh(x) − cos(t)
0
0
(1.3.2)
• Since the cosine transform of (cosh(x) − cos(t))−1 with respect to x is π csc(t) csch(πy) sinh(π − t)y, we find that ∞ k sin(kt)
k2
k=1
• • • • •
∞
+
=
y2
π sinh(π − t)y . 2 sinh(πy)
n=1
sin(2nπx) n2k+1
=
(−1)k−1 (2π)2k+1 B2k+1 (x), 2 2k+1
n=1
cos(2nπx) n2k
=
(−1)k−1 (2π)2k B (x) 2 (2k)! 2k
n=1
sin((2n+1)πx−πk/2) (2n+1)k+1
n=1
a n sin(nx) =
a sin(x) 1−2a cos(x)+a 2
for |a| < 1
n=0
a n cos(nx) =
1−a cos(x) 1−2a cos(x)+a 2
for |a| < 1
∞ ∞ ∞ ∞
=
for 0 < x <
for 0 < x <
1 2
1 2
π k+1 Ek (x) 4k!
Hypergeometric series The hypergeometric function is p Fq
a1 b1
a2 b2
... ...
ap x bq
=
∞ (a1 )n (a2 )n . . . (ap )n x n n=0
(b1 )n (b2 )n . . . (bq )n n!
where (a)n = (a + n)/ (a) is the shifted factorial. Any infinite series An with An+1 /An a rational function of n is of this type. These include series of products and quotients of binomial coefficients. For example: 2 F1
3 F2
4 F3
a, c
b
(c) (c − a − b) (Gauss) 1 = (c − a) (c − b)
−n, c,
a, 1+a+b−c−n
a, a/2,
1 + a/2, 1 + a − b,
2n2n+m+1 m m m (−1) 4n+2m+2 (3 m=0 2m 2n
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b
(c − a)n (c − b)n 1 = (Saalschutz) (c)n (c − a − b)n
b, 1 + 2b − n
−n
1 =
√ (3/4)n (5/4)n + 2 2)m = (7/8)n (9/8)n
(a − 2b)n (−b)n (Bailey) (1 + a − b)n (−2b)n
Power series
n 1. The values of x for which the power series ∞ n=0 an x converges, form an interval (interval of convergence) which may or may not include one or both endpoints. 2. A power series may be integrated and differentiated term-by-term within its interval of convergence. n −1 n 3. Note that [1 + ∞ = 1− ∞ n=1 an x ] n=1 bn x , where b1 = a1 and bn = n−1 an + k=1 bn−k ak for n ≥ 2. ∞ n n 4. Inversion of power series: If s = ∞ n=1 an x , then x = n=1 An s , where 5 3 2 A1 = 1/a1 , A2 = −a2 /a1 , A3 = (2a2 − a1 a3 )/a1 , A4 = (5a1 a2 a3 − a12 a4 − 5a23 )/a17 , A5 = (6a12 a2 a4 + 3a12 a32 + 14a24 − a13 a5 − 21a1 a22 a3 )/a19 .
Taylor series 1. Taylor series in 1 variable: f (a + x) =
N xn n=0
n!
f (n) (a) + RN .
2. Lagrange’s form of the remainder: RN =
x N+1 f (N+1) (θa), (N + 1)!
for some 0 < θ < 1.
3. Taylor series in 2 variables: f (a + x, b + y) = f (a, b) + xfx (a, b) + yfy (a, b) + 1 2 x fxx (a, b) + 2xyfxy (a, b) + y 2 fyy (a, b) + . . . . 2! 4. Taylor series for vectors: N [(x · ∇)n f ](a) f (a + x) = + RN (a) = f (a) + x · ∇f (a) + . . . . n! n=0 For example (see also page 40): • Binomial series: (x + y)ν =
∞ n=0
(ν + 1) x n y ν−n . (ν − n + 1) n!
When ν is a positive integer, the series terminates at n = ν. (2k)! k 1 = ∞ • √1−4x k=0 (k!)2 x for |x| < 1/4 ∞ xt xn • exe x −1 = n=0 Bn (t) n! ∞ xt xn • e2e x +1 = n=0 En (t) n!
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• •
x u ∞ n 1 xn n=1 (n+1)(n+3) = x 3 0 u du 0 dt 0 t 1 1 2 2 = 2x 3 x + 2 x + (1 − x ) log(1 − x)
∞ ∞
k
x k=1 k n
for |x| < 1
= Lin (x) (polylogarithm)
Telescoping series If limn→∞ F (n) = 0, then ∞ n=1
∞
n=1 [F (n)
− F (n + 1)] = F (1). For example,
∞ 1 1 1 1 = − = . (n + 1)(n + 2) n+1 n+2 2 n=1
The GWZ symboliccomputer algorithm expresses a proposed identity in the form of a telescoping series k [F (n + 1, k) − F (n, k)] = 0, then searches for a G(n, k) that satisfies F (n + 1, k) − F (n, k) = G(n, k + 1) − G(n, k) and G(n, ±∞) = 0. The search assumes that G(n, k) = R(n, k)F (n, k − 1) where R(n, k) is a rational expression in n and k. When R is found, the proposed identity is verified. For example, the Pfaff–Saalschutz identity has the following proof: ∞ (c − a + n)!(c − b + n)! (a + k)!(b + k)!(c − a − b + n − 1 − k)! = , (k + 1)!(n − k)!(c + k)! (n + 1)!(c + n)! k=−∞
R(n, k) = −
(b + k)(a + k) . (c − b + n + 1)(c − a + n + 1)
Other types of series 1. Arithmetic series: N
1 (a + nd) = Na + N(N + 1)d. 2 n=1
2. Arithmetic power series: N
(a + nb)x n =
n=1
a − (a + bN) xN+1 bx(1 − x N ) + , 1−x (1 − x)2
(x = 1).
3. Geometric series: 1 + x + x2 + x3 + · · · =
1 , 1−x
(|x| < 1).
4. Arithmetic–geometric series: a + (a + b)x + (a + 2b)x 2 + (a + 3b)x 3 + · · · =
©1996 CRC Press LLC
bx a + , 1−x (1 − x)2 (|x| < 1).
5. Combinatorial sums: x+1 = n • nk=0 x−k n−k m • k=−∞ (−1)k xk = (−1)m x−1 m n k+m m+n+1 • k=0 k = n −x m k x+m • k=−∞ (−1) k = m x y x+y • ∞ k=−∞ m+k n−k = m+n l x l+x • ∞ k=−∞ m+k n+k = l−m+n l x+k k = (−1)l+m x−m • ∞ k=−∞ (−1) m+k n n−l x l k l−k l+m x−m−l • k=−∞ (−1) m k−n = (−1) l−m−n q+k l+q+1 = m+n+1 (m ≥ q) • lk=0 l−k m n 6. Generating functions:
• • • •
7. Multiple series: √ • • • • •
∞
Jk (x)zk = exp( 21 x z z−1 ) z(z+2x) n Chebyshev polynomials: ∞ n=1 Tn (x)z = 2xz−z2 −1 Hn (x) n 2 Hermite polynomials: ∞ n=0 n! z = exp(2xz − z ) xz (α) n −α−1 Laguerre polynomials: ∞ exp[ z−1 ] n=0 Ln (x)z = (1 − z) ∞ 1 , for |x| < 1 Legendre polynomials: n=0 Pn (z)x n = √1−xz+x 2
• Bessel functions:
2
k=−∞
(−1)l+m+n
(l+1/6)2 +(m+1/6)2 +(n+1/6)2
(−1)n (n+1/2) m+n m,n=0 n! (m+n+1/2) z
∞
m2 −n2 m,n=1 (m2 +n2 )2 1 k12 k22 ...kn2
√
3 for −∞ < l, m, n < ∞ not all zero
= 4β(z)ζ (z) for −∞ < m, n < ∞ not both zero
1 (m2 +n2 )z
∞
=
=
=
π 2n (2n+1)!
=
√
π ez
erf√(√z−z) z−z
for z > 0
π 4
for 1 ≤ k1 < · · · < kn < ∞
1.3.5 SUMMATION FORMULAE 1. Euler–Maclaurin summation formula: As n → ∞, n
f (k) ∼
k=0
©1996 CRC Press LLC
1 f (n) + 2
0
n
f (x) dx + C +
∞ j =1
(−1)j +1 Bj +1
f (j ) (n) (j + 1)!
where C = lim
m→∞
m
(−1)j Bj +1
j =1
1 f (j ) (0) + f (0) (j + 1)! 2 ∞ (−1)m Bm+1 (x − [x])f (m+1) (x) dx . + (m + 1)! 0
2. Poisson summation formula: If f is continuous, ∞ ∞ ∞ ∞ 1 f (0) + f (n) = f (x) dx + 2 f (x) cos(2nπx)dx . 2 0 0 n=1 n=1
For example: n 1 B2 1 • k=1 k ∼ log n + γ + 2n − 2n2 − . . . where γ is Euler’s constant. −n2 x −π 2 n2 /x • 1+2 ∞ = πx [1 + 2 ∞ ] (Jacobi) n=1 e n=1 e
1.3.6 IMPROVING CONVERGENCE: SHANKS TRANSFORMATION Let sn be the nth partial sum. The sequences {S(sn )}, {S(S(sn ))}, . . . often converge successively more rapidly to the same limit as {sn }, where S(sn ) = For example, for sn =
n
sn+1 sn−1 − sn2 . sn+1 + sn−1 − 2sn
k k k=0 (−1) z ,
we find S(sn ) =
1 1+z
(1.3.3) for all n.
1.3.7 SUMMABILITY METHODS Unique values can be assigned to divergent series in a variety of ways which preserve the values of convergent series. ∞ n 1. Abel summation: ∞ n=0 an = lim r→1− n=0 an r . ∞ 1 +...sN 2. Cesaro (C, 1)-summation: n=0 an = limN→∞ s0 +sN+1 where sn = nm=0 am . For example: • 1 − 1 + 1 − 1 + ··· =
1 2
(in the sense of Abel summation)
• 1 − 1 + 0 + 1 − 1 + 0 + 1 − ··· =
©1996 CRC Press LLC
1 3
(in the sense of Cesaro summation)
1.3.8 OPERATIONS WITH SERIES Let y = a1 x + a2 x 2 + a3 x 3 + . . . , and let z = z(y) = b1 x + b2 x 2 + b3 x 3 + . . . . z=
b0
1 1−y √ 1+y
b1
b2
b3
1
a1
a12 + a2
a13 + 2a1 a2 + a3
1
1 a 2 1
− 18 a12 + 21 a2
1 3 a 16 1
(1 + y)−1/2
1
− 21 a1
3 2 a 8 1
− 21 a2
5 3 − 16 a1 + 43 a1 a2 − 21 a3
ey
1
a1
1 2 a 2 1
+ a2
1 3 a 6 1
log(1 + y)
0
a1
a2 − 21 a12
a3 − a1 a2 + 13 a13
sin y
0
a1
a2
− 16 a13 + a3
cos y
1
0
− 21 a12
−a1 a2
tan y
0
a1
a2
1 3 a 3 1
− 41 a1 a2 + 21 a3
+ a1 a2 + a3
+ a3
1.3.9 MISCELLANEOUS SUMS AND SERIES • • • • • • • • • •
N
n+1 k n n=1 (−1) n 1 k=1 k(k+1)(k+2)
n
kmk k=1 (k+m)!
= 21 (−1)N+1 [Ek (N + 1) + (−1)N Ek ] =
1 4
−
1 2(n+1)(n+2) n+1
1 m = (m−1)! − (m+n)! 2n k 2n −1 = 2n+1 k=0 (−1) k n+1 n mnm+n+k m+n2 = m k=0 k k m+n n−1 1 2 x 2 x 2 1 k=0 4k sech 2k = 4n csch 2n − csch x n n−1 n πk k n πk k=0 (−1) cos n = 2n−1 , k=0 sec n = 0, ∞ π 1 k=−∞ (k+a)(k+b) = b−a (cot πa − cot πb) √ √ ∞ 1 π√ sinh(aπ √2)+sin(aπ √2) 1 k=0 k 4 +a 4 = 2a 4 + 2a 3 2 cosh(aπ 2)−cos(aπ 2) ∞ 1 1 k k=1 e2πk −1 = 24 − 8π
• The series
∞ n=3
(k = n/2)
1 converges to 38.43 . . . so slowly that it ren log n(log log n)2 86
quires 103.14·10 terms to give two-decimal accuracy ∞ 1 • The series diverges, but the partial sums exceed 10 only n log n(log log n) n=3 after a googolplex of terms have appeared
©1996 CRC Press LLC
1.3.10 INFINITE SERIES Algebraic functions n n−2 2 n n−1 x y + .... (x + y) = x + x y+ 2 1 n n 2 n 3 (1 ± x)n = 1 ± x+ x ± x + ..., 1 2 3 n n+1 2 n+2 3 −n (1 ± x) = 1 ∓ x+ x ∓ x + ..., 1 2 3 √ 1 1 1 5 4 x + ..., 1 + x = 1 + x − x2 + x3 − 2 8 16 128 1 5 35 4 3 (1 + x)−1/2 = 1 − x + x 2 − x 3 + x + ..., 2 8 16 128 (1 ± x)−1 = 1 ∓ x + x 2 ∓ x 3 + x 4 ∓ x 5 + . . . , n
n
(1 ± x)−2 = 1 ∓ 2x + 3x 2 ∓ 4x 3 + 5x 4 ∓ 6x 5 + . . . ,
(x 2 < 1). (x 2 < 1). (x 2 < 1). (x 2 < 1). (x 2 < 1). (x 2 < 1).
Exponential functions 1 1 1 + + ··· + + ··· . (all real values of x) 1! 2! n! x x2 xn ex = 1 + + + ··· + + ··· , 1! 2! n! (x − a)2 (x − a)n = ea 1 + (x − a) + + ··· + + ... . 2! n! 2 n (x log a) (x log e e a) a x = 1 + x loge a + + ··· + + ··· . 2! n! e =1+
©1996 CRC Press LLC
Logarithmic functions log x =
x−1 1 + x 2
x−1 x
2 + ··· +
1 n
x−1 x
n
+ ··· ,
1 1 = (x − 1) − (x − 1)2 + (x − 1)3 − . . . , 2 3 x−1 1 x−1 3 1 x−1 5 + + ... , =2 + x+1 3 x+1 5 x+1
(x > 1/2), (2 ≥ x > 0), (x > 0).
(x − a) (x − a)2 (x − a)3 + − ..., (0 < x ≤ 2a). − 2 a 2a 3a 2 x2 x3 x4 log(1 + x) = x − + − + ..., −1 < x ≤ 1. 2 3 4 1 1 1 log(n + 1) = log(n − 1) + 2 + 5 + ... . + n 3n3 5n 3 5 x 1 1 x x log(a + x) = log a + 2 + + ... , + 2a + x 3 2a + x 5 2a + x = log a +
(a > 0, −a < x). log
1+x x3 x 2n−1 =2 x+ + ··· + + ... , 1−x 3 2n − 1
(−1 < x < 1).
Trigonometric functions x3 x5 x7 + − + .... (all real values of x). 3! 5! 7! x2 x4 x6 cos x = 1 − + − + .... (all real values of x). 2! 4! 6! x3 2x 5 (−1)n−1 22n (22n − 1)B2n 2n−1 tan x = x + + + ...+ x + ... 3 15 (2n)! sin x = x −
(x 2 < π 2 /4, Bn is the nth Bernoulli number). cot x =
1 x x3 2x 5 x7 (−1)n+1 22n B2n 2n−1 + ... − − − − − ...+ x x 3 45 945 4725 (2n)! (x 2 < π 2 , Bn is the nth Bernoulli number).
sec x = 1 +
x2 61 6 277 8 (−1)n E2n 2n 5 + x4 + x + x + ...+ x + ... 2 24 720 8064 (2n)! (x 2 < π 2 /4, En is the nth Euler number).
csc x =
1 x 7x 3 31x 5 (−1)n+1 2(22n−1 − 1)B2n 2n−1 + + + + ...+ x + ... x 6 360 15120 (2n)!
©1996 CRC Press LLC
(|x| < π, Bn is the nth Bernoulli number). x2 x4 x6 − − − ... (x 2 < π 2 ). 6 180 2835 x4 x6 17x 6 x2 − − − ... (x 2 < π 2 /4). log cos x = − − 2 12 45 2520 x2 7x 4 62x 6 log tan x = log x + + + + ... (x 2 < π 2 /4). 3 90 2835 3x 4 8x 5 3x 6 56x 7 x2 esin x = 1 + x + − − − + + .... 2! 4! 5! 6! 7! 4x 4 31x 6 x2 ecos x = e 1 − + − + ... . 2! 4! 6! x2 3x 3 9x 4 37x 5 etan x = 1 + x + + + + + ... (x 2 < π 2 /4). 2! 3! 4! 5! (x − a)2 (x − a)3 sin x = sin a + (x − a) cos a − sin a− cos a + . . . . 2! 3! log sin x = log x −
Inverse trigonometric functions sin−1 x = x +
1 3 1·3 5 1·3·5 7 x + x + x + ... 2·3 2·4·5 2·4·6·7 (x 2 < 1, − π2 < sin−1 x <
cos
−1
π ). 2
π 1·3 5 1·3·5 7 1 3 x= − x+ x + x + x + ... 2 2·3 2·4·5 2·4·6·7 (x 2 < 1, 0 < cos−1 x < π).
x3 x5 x7 + − + ... 3 5 7 π 1 1 1 1 = − + 3 − 5 + 7 − ... 2 x 3x 5x 7x π 1 1 1 1 = − − + 3 − 5 + 7 − ... 2 x 3x 5x 7x π x3 x5 x7 −1 cot x = − x + − + − ... 2 3 5 7 tan−1 x = x −
©1996 CRC Press LLC
(x 2 < 1), (x > 1), (x < −1). (x 2 < 1).
Hyperbolic functions x3 x5 x7 x (2n+1) + + + ··· + + .... 3! 5! 7! (2n + 1)! sin x 2 sin 2x 3 sin 3x 2 − 2 + 2 + ... (|x| < π). = sinh πa 2 π a + 12 a + 22 a + 32 x2 x6 x 2n x4 =1+ + + ··· + + .... + 4! 6! (2n)! 2! cos x cos 2x cos 3x 2a 1 − 2 + 2 − 2 + ... (|x| < π). = sinh πa π 2a 2 a + 12 a + 22 a + 32 1 2 22n (22n − 1)B2n 2n−1 = x − x3 + x5 − · · · + x + ... (|x| < π/2), 3 15 (2n)! (Re x > 0), = 1 − 2e−2x + 2e−4x − 2e−6x + . . . 1 1 1 = 2x 2 + 2 + 2 + ... . π 3π 5π 2 2 +x +x + x2 2 2 2
sinh x = x + sinh ax cosh x cosh ax tanh x
1 x x3 2x 5 22n B2n 2n−1 + − + + ··· + x + ... (0 < |x| < π), x 3 45 945 (2n)! (Re x > 0), = 1 + 2e−2x + 2e−4x + 2e−6x + . . . 1 1 1 1 = +2x + 2 + + + ... (Re x > 0). x π + x2 (2π)2 + x 2 (3π)2 + x 2 1 5 61 E2n 2n sech x = 1 − x 2 + x 4 − x 6 + · · · + x + ... 2! 4! 6! (2n)! coth x =
(|x| < π/2, En is the nth Euler number), (Re x > 0), = 2 e−x − e−3x + e−5x − e−7x + . . . 1 3 5 = 4π 2 − + + ... . π + 4x 2 (3π)2 + 4x 2 (5π)2 + 4x 2 1 x 7x 3 2(22n−1 − 1)B2n 2n−1 csch x = − + + ... (0 < |x| < π), + ··· + x x 6 360 (2n)! −x = 2 e + e−3x + e−5x + e−7x + . . . (Re x > 0), 1 2x 2x 2x = − 2 + − + .... x π + x2 (2π)2 + x 2 (3π)2 + x 2 (n − 2) sinh nu = sinh u (2 cosh u)n−1 − (2 cosh u)n−3 1! (n − 3)(n − 4) (2 cosh u)n−5 + 2! (n − 4)(n − 5)(n − 6) (2 cosh u)n−7 + . . . . − 3!
©1996 CRC Press LLC
cosh nu =
1 n n(n − 3) (2 cosh u)n − (2 cosh u)n−2 + (2 cosh u)n−4 2 1! 2! n(n − 4)(n − 5) (2 cosh u)n−6 + . . . . − 3!
Inverse hyperbolic functions 1 3 1·3 5 1·3·5 7 (|x| < 1), x + x − x + ... 2·3 2·4·5 2·4·6·7 1 1 1·3 1 1·3·5 1 = log(2x) + · 2 + + ... (|x| > 1). · 4+ · 2 2x 2 · 4 4x 2 · 4 · 6 6x 6 1 1 1·3 1 · = ± log(2x) − · 2 − + ... (x > 1). 2 2x 2 · 4 4x 4 1 1 1 1·3 1 1·3·5 1 = − · 3− − + ... (|x| > 1), · · x 2 3x 2 · 4 5x 5 2 · 4 · 6 7x 7 2 1 x2 1 · 3 x4 1 · 3 · 5 x6 = log + · − · + · − ... (0 < x < 1). x 2 2 2·4 4 2·4·6 6 1 · 3 x4 1 · 3 · 5 x6 2 1 x2 − · − · − ... (0 < x < 1). = log − · 2·4 4 2·4·6 6 x 2 2 x5 x7 x 2n+1 x3 + + + ··· + + ... (|x| < 1). =x+ 3 5 7 2n + 1 1 1 1 1 1 = + 3 + 5 + 7 + ··· + + ... (|x| > 1). x 3x 5x 7x (2n + 1)x 2n+1 1 E2n 1 (|x| < 1). = x − x3 + x5 + · · · + x 2n+1 + . . . 6 24 (2n + 1)!
sinh−1 x = x −
cosh−1 x csch−1 x
sech−1 x tanh−1 x coth−1 x gd x
1.3.11 INFINITE PRODUCTS
For the sequence of complex numbers {ak }, the infinite product is defined as ∞ k=1 (1+ is that limn→∞ an = 0. A necessary and ak ). A necessary condition for convergence sufficient condition for convergence is that ∞ k=1 log(1 + ak ) converges. Examples: z ∞ 1 + k1 • z! = 1 + kz k=1 ∞
z 2k k=1 ∞ z2 1− 2 • sin πz = πz k k=1 ∞ z2 • cos πz = 1− (k − 21 )2 k=1 • sin z = z
cos
©1996 CRC Press LLC
• • • •
z 1+ sin(a + z) = (sin a) a + kπ k=0,±1,±2,... ∞ 2z cos(a + z) = (cos a) 1+ 2a + kπ k=±1,±3,±5,... ∞ z2 sinh z = z 1+ 2 2 k π k=1 ∞ 4z2 cosh z = 1+ (2k + 1)2 π 2 k=0 ∞
Weierstrass theorem
w2 wm . For k = Define E(w, m) = (1 − w) exp w + + ··· + m 2 1, 2, . . . let {bk } be a sequence of complex numbers such that |bk | → ∞. ∞ z Then the infinite product P (z) = E , k is an entire function bk k=1 with zeros at bk and at these points only. The multiplicity of the root at bn is equal to the number of indices j such that bj = bn .
1.3.12 INFINITE PRODUCTS AND INFINITE SERIES 1. The Rogers–Ramanujan identities (for a = 0 or a = 1) are 1 +
∞ k=1
=
q k +ak (1 − q)(1 − q 2 ) · · · (1 − q k ) 2
∞
j =0
1 (1 −
q 5j +a+1 )(1
− q 5j −a+4 )
.
(1.3.4)
2. Jacobi’s triple product identity is ∞
q (2 ) x k = k
k=−∞
∞
(1 − q j )(1 + x −1 q j )(1 + xq j −1 ).
j =1
(1.3.5)
3. The quintuple product identity is ∞
(−1)k q (3k
2
−k)/2 3k
x (1 + xq k )
k=−∞
=
∞ j =1
(1 − q j )(1 + x −1 q j )(1 + xq j −1 )(1 + x −2 q 2j −1 )(1 + x 2 q 2j −1 ). (1.3.6)
©1996 CRC Press LLC
1.4 FOURIER SERIES If f (x) is a bounded periodic function of period 2L (that is, f (x + 2L) = f (x)) and satisfies the Dirichlet conditions, 1. In any period, f (x) is continuous, except possibly for a finite number of jump discontinuities. 2. In any period f (x) has only a finite number of maxima and minima. Then f (x) may be represented by the Fourier series,
f (x) =
∞ a0 nπx nπx an cos + + bn sin , 2 L L n=1
where {an } and {bn } are determined as follows: 1 α+2L nπx an = dx f (x) cos L α L 2L 1 nπx = dx, f (x) cos L L 0 1 L nπx = f (x) cos dx; L −L L 1 α+2L nπx bn = f (x) sin dx L α L 1 2L nπx = f (x) sin dx, L 0 L 1 L nπx = f (x) sin dx, L −L L
(1.4.1)
for n = 0, 1, 2, . . . ,
(1.4.2)
for n = 1, 2, 3, . . . ,
(1.4.3)
where α is any real number (the second and third lines of each formula represent α = 0 and α = L respectively). The series in Equation (1.4.1) will converge (in the Cesaro sense) to every point f (x + ) + f (x − ) (i.e., the average of the left hand where f (x) is continuous, and to 2 and right hand limits) at every point where f (x) has a jump discontinuity.
1.4.1 SPECIAL CASES 1. If, in addition to the Dirichlet conditions, f (x) is an even function (i.e., f (x) = f (−x)), then the Fourier series becomes f (x) =
©1996 CRC Press LLC
∞ nπx a0 + . an cos 2 L n=1
(1.4.4)
That is, every bn = 0. In this case, the {an } may be determined from
2 an = L
L
f (x) cos
0
nπx dx L
n = 0, 1, 2, . . . .
(1.4.5)
If, in addition to the above requirements, f (x) = f (L − x), then an will be zero for all even values of n. In this case the expansion becomes f (x) =
∞
a2m−1 cos
m=1
(2m − 1)πx . L
(1.4.6)
2. If, in addition to the Dirichlet conditions, f (x) is an odd function (i.e., f (x) = −f (−x)), then the Fourier series becomes f (x) =
∞
bn sin
n=1
nπx . L
(1.4.7)
That is, every an = 0. In this case, the {bn } may be determined from bn =
2 L
L
f (x) sin
0
nπx dx L
n = 1, 2, 3, . . . .
(1.4.8)
If, in addition to the above requirements, f (x) = f (L − x), then bn will be zero for all even values of n. In this case the expansion becomes f (x) =
∞
b2m−1 sin
m=1
(2m − 1)πx . L
(1.4.9)
The series in Equation (1.4.6) and Equation (1.4.9) are known as odd harmonic series, since only the odd harmonics appear. Similar rules may be stated for even harmonic series, but when a series appears in even harmonic form, it means that 2L has not been taken to be the smallest period of f (x). Since any integral multiple of a period is also a period, series obtained in this way will also work, but, in general, computation is simplified if 2L is taken as the least period. Writing the trigonometric functions in terms of complex exponentials, we obtain the complex form of the Fourier series known as the complex Fourier series or as the exponential Fourier series. It is represented by f (x) =
∞ n=−∞
where ωn =
cn eiωn x
(1.4.10)
nπ for n = 0, ±1, ±2, . . . and the {cn } are determined from L L 1 f (x)e−iωn x dx. (1.4.11) cn = 2L −L
The set of coefficients {cn } is often referred to as the Fourier spectrum.
©1996 CRC Press LLC
1.4.2 ALTERNATE FORMS The Fourier series in Equation (1.4.1) may be represented in the alternate forms: 1. When φn = tan−1 (−an /bn ), an = cn sin φn , bn = −cn cos φn , and cn = an2 + bn2 , then f (x) =
∞
nπx a0 + + φn . cn sin 2 L n=1
2. When φn = tan−1 (an /bn ), an = cn sin φn , bn = cn cos φn , and cn = then ∞
nπx a0 + + φn . cn cos f (x) = 2 L n=1
(1.4.12) an2 + bn2 ,
(1.4.13)
1.4.3 USEFUL SERIES
4 πx 1 3πx 1 5πx 1= sin + sin + sin + ... [0 < x < k]. π k 3 k 5 k 2k πx 1 2πx 1 3πx x= sin − sin + sin + ... [−k < x < k]. π k 2 k 3 k πx 4k 1 1 3πx 5πx k + 2 cos + 2 cos + ... [0 < x < k]. x = − 2 cos 2 π k 3 k 5 k 3πx π2 2k 2 4 πx π2 2πx 4 π2 x2 = 3 − sin − sin + − 3 sin π 1 1 k 2 k 3 3 k 2 2 4πx 5πx π 4 π sin − + − 3 sin ... [0 < x < k]. 4 k 5 5 k πx 4k 2 1 1 1 k2 2πx 3πx 4πx 2 − 2 cos − 2 cos + 2 cos − 2 cos + ... x = 3 π k 2 k 3 k 4 k [−k < x < k] 1− 1 22 1 1− 2 2 1 1+ 2 3 1 1 + 2 22 4 1+
©1996 CRC Press LLC
1 1 1 + − 3 5 7 1 1 + 2+ 2 3 4 1 1 + 2− 2 3 4 1 1 + 2+ 2 5 7 1 1 + 2+ 2 6 8
+ ... = + ... = + ... = + ... = + ... =
π . 4 π2 . 6 π2 . 12 π2 . 8 π2 . 24
(1.4.14)
1.4.4 EXPANSIONS OF BASIC PERIODIC FUNCTIONS f(x)
• f (x) =
4 π
n=1,3,5,...
nπx 1 sin n L
1 0
L
2L x
-1 f(x)
• f (x) =
2 π
∞ n=1
nπc nπx (−1) cos − 1 sin n L L n
c
1 0
L
-1
2L
x
2L
x
2L
x
c
f(x)
∞ (−1)n 2 nπc nπx c sin cos • f (x) = + L π n=1 n L L
2c
1 0
L
f(x) ∞ nπx 2 nπ sin(nπc/2L) sin • f (x) = sin L n=1 2 nπc/2L L
1/c
c 3 L/2
0
L
L/2
1/c
c
f(x) ∞ nπx 2 (−1)n+1 sin • f (x) = π n=1 n L
1
2L
L
0
x
-1
4 1 • f (x) = − 2 2 π
n=1,3,5,...
f(x)
nπx 1 cos n2 L
1 0
L
2L
x
f(x)
• f (x) =
8 π2
n=1,3,5,...
(n−1)/2
(−1) n2
sin
nπx L
1 0
3L/2 L/2
L
-1 ∞ 1 1 1 nπx • f (x) = − sin 2 π n=1 n L
• f (x) =
2L
x
2L
x
f(x) 1 0
∞ 1+a nπx 1 2 (−1)n cos nπa − 1 cos + 2 2 π (1 − a) n=1 n2 L
(a =
c ) 2L
f(x) c 1 0
©1996 CRC Press LLC
L
2L
x
∞ 2 (−1)n−1 sin nπa nπx • f (x) = 1+ sin π n=1 n nπ(1 − a) L
(a =
c ) 2L
f(x) c/2 1 0
L
-1
• f (x) =
4 1 − 2 π 2 (1 − 2a)
n=1,3,5,...
2L
x
c/2
nπx 1 cos nπa cos n2 L
(a =
c ) 2L
f(x) c
1 0
• f (x) =
c/2
2L - c/2
L
∞ (−1)n 1 + (−1)n nπx 2 1+ sin nπa sin π n=1 n nπ(1 − 2a) L
2L
x
(a =
c ) 2L
f(x) c/2 1 0
2L - c/2 c/2
L
-1
• f (x) =
2L
x
c/2
∞ nπ nπx 1 4 sin sin nπa sin π n=1 n 4 L
(a =
c ) 2L
f(x) c 1 7L/4 0
L/4
2L
L
x
-1 c
• f (x) =
∞ nπx 9 nπa 1 sin sin 2 π n=1 n2 3 L
(a =
c ) 2L
f(x) 1 5L/3 0
L/3
2L x
L
-1
• f (x) =
∞ nπx 32 nπa 1 sin sin 2 2 3π n=1 n 4 L
(a =
c ) 2L
f(x) 1 0 -1
©1996 CRC Press LLC
7 L / 4 2L L/4
L
x
• f (x) =
1 2 1 + sin ωt − π 2 π
n=2,4,6,...
1 cos nωt n2 − 1 f(x) 1 0
sin ωt T = 2π/ω π/ω
2π/ω t
1.5 COMPLEX ANALYSIS
1.5.1 DEFINITIONS A complex number z has the form z = x + iy where x and y are real numbers, and √ i = −1; the number i is sometimes called the imaginary unit. We write x = Re z and y = Im z. The number x is called the real part of z and y is called the imaginary part of z. This form is also called the Cartesian form of the complex number. Complex numbers can also be written in polar form, z = reiθ , where r, called the modulus, is given by r = |z| = x 2 + y 2 , and θ is called the argument: θ = arg z = tan−1 xy . The geometric relationship between Cartesian and polar forms is shown below z = x + iy = reiθ
r
y
θ x
The complex conjugate of z, denoted√z¯ , is defined as z¯ = x − iy = re−iθ . Note that |z| = |¯z|, arg z¯ = − arg z, and |z| = z¯z. In addition, z¯ = z, z1 + z2 = z¯ 1 + z¯ 2 , and z1 z2 = z¯ 1 z¯ 2 .
1.5.2 OPERATIONS ON COMPLEX NUMBERS Addition and subtraction: z1 ± z2 = (x1 + iy1 ) ± (x2 + iy2 ) = (x1 ± x2 ) + i(y1 ± y2 ). Multiplication: z1 z2 = (x1 + iy1 )(x2 + iy2 ) = (x1 x2 − y1 y2 ) + i(x1 y2 + x2 y1 ) = r1 r2 ei(θ1 +θ2 ) .
©1996 CRC Press LLC
|z1 z2 | = |z1 ||z2 |,
arg(z1 z2 ) = arg z1 + arg z2 = θ1 + θ2 .
Division: z1 z¯ 2 (x1 x2 + y1 y2 ) + i(x2 y1 − x1 y2 ) r1 z1 = = = ei(θ1 −θ2 ) . z2 z2 z¯ 2 r2 x22 + y22 z1 |z1 | z1 = = arg z1 − arg z2 = θ1 − θ2 . , arg z |z | z2 2 2
1.5.3 POWERS AND ROOTS OF COMPLEX NUMBERS Powers: zn = r n einθ = r n (cos nθ + i sin nθ) DeMoivre’s Theorem. Roots:
θ + 2kπ θ + 2kπ z1/n = r 1/n eiθ/n = r 1/n cos + i sin , k = 0, 1, 2, . . . , n − 1. n n
The principal root has −π < θ ≤ π and k = 0.
1.5.4 FUNCTIONS OF A COMPLEX VARIABLE A complex function w = f (z) = u(x, y) + iv(x, y) = |w|eiφ , where z = x + iy, associates one or more values of the complex dependent variable w with each value of the complex independent variable z for those values of z in a given domain.
1.5.5 CAUCHY–RIEMANN EQUATIONS A function w = f (z) is said to be analytic at a point z0 if it is differentiable in a neighborhood (i.e., at each point of a circle centered on z0 with an arbitrarily small (z0 ) radius) of z0 . That is, limh→0 f (z0 +h)−f exists. A function is called analytic in a h connected domain if it is analytic at every point in that domain. A necessary and sufficient condition for f (z) = u(x, y) + iv(x, y) to be analytic is that it satisfy the Cauchy–Riemann equations, ∂u ∂v ∂u ∂v = , and =− . ∂x ∂y ∂y ∂x
(1.5.1)
Examples: 1. f (z) = zn is analytic everywhere when n is a nonnegative integer. If n is a negative integer, then f (z) is analytic except at the origin. 2. f (z) = z¯ is nowhere analytic. 3. f (z) = ez is analytic everywhere.
©1996 CRC Press LLC
1.5.6 CAUCHY INTEGRAL THEOREM If f (z) is analytic at all points within and on a simple closed curve C, then f (z) dz = 0.
(1.5.2)
C
1.5.7 CAUCHY INTEGRAL FORMULA If f (z) is analytic inside and on a simple closed contour C and if z0 is interior to C, then f (z) 1 f (z0 ) = dz. (1.5.3) 2πi C z − z0 Moreover, if the derivatives f (z), f (z), . . . of all orders exist, then f (z) n! (n) f (z0 ) = dz. 2πi C (z − z0 )n+1
(1.5.4)
1.5.8 TAYLOR SERIES EXPANSIONS If f (z) is analytic inside of and on a circle C of radius r centered at the point z0 , then a unique and uniformly convergent series expansion exists in powers of (z − z0 ) of the form f (z) =
∞
an (z − z0 )n ,
|z − z0 | < r,
z0 = ∞,
(1.5.5)
n=0
where an =
1 (n) 1 f (z0 ) = n! 2πi
C
f (z) dz. (z − z0 )n+1
(1.5.6)
If M(r) is an upper bound of |f (z)| on C, then |an | =
1 (n) M(r) |f (z0 )| ≤ n! rn
(Cauchy’s inequality).
(1.5.7)
If the series is truncated with the term an (z − z0 )n , the remainder Rn (z) is given by f (s) (z − z0 )n+1 Rn (z) = ds, (1.5.8) n+1 2πi C (s − z)(s − z0 ) and |Rn (z)| ≤
©1996 CRC Press LLC
|z − z0 | r
n
rM(r) . r − |z − z0 |
(1.5.9)
1.5.9 LAURENT SERIES EXPANSIONS If f (z) is analytic inside the annular domain between the concentric circles C1 and C2 centered at z0 with radii r1 and r2 (r1 < r2 ), respectively, then a unique series expansion exists in terms of positive and negative powers of z − z0 of the following form: f (z)
=
∞
bn (z − z0 )−n +
n=1
=
∞
an (z − z0 )n ,
n=0
··· +
b2 b1 + + a0 + a1 (z − z0 ) + a2 (z − z0 )2 + . . . 2 (z − z0 ) z − z0 (1.5.10)
where 1 an = 2πi
C
f (s) ds, (s − z0 )n+1
n = 0, 1, 2, . . . ,
(1.5.11)
and bn =
1 2πi
C
f (s)(s − z0 )n−1 ds,
n = 1, 2, 3, . . . .
(1.5.12)
1.5.10 ZEROS AND SINGULARITIES The points z for which f (z) = 0 are called zeros of f (z). A function f (z) which is analytic at z0 has a zero of order m there, where m is a positive integer, if and only if the first m coefficients a0 , a1 , . . . , am−1 in the Taylor expansion about z0 vanish. A singular point or singularity of the function f (z) is any point at which f (z) is not analytic. An isolated singularity of f (z) at z0 may be classified in one of three ways: 1. A removable singularity if and only if all coefficients bn in the Laurent series expansion of f (z) about z0 vanish. 2. A pole of order m if and only if (z − z0 )m f (z), but not (z − z0 )m−1 f (z), is analytic at z0 , (i.e., if and only if bm = 0 and 0 = bm+1 = bm+2 = . . . in the Laurent series expansion of f (z) about z0 ). Equivalently, f (z) has a pole of order m if 1/f (z) is analytic at z0 and has a zero of order m there. 3. An isolated essential singularity if and only if the Laurent series expansion of f (z) about z0 has an infinite number of terms involving negative powers of z − z0 .
©1996 CRC Press LLC
1.5.11 RESIDUES Given a point z0 where f (z) is either analytic or has an isolated singularity, the residue of f (z) is the coefficient of (z − z0 )−1 in the Laurent series expansion of f (z) about z0 , or 1 f (z) dz. (1.5.13) Res(z0 ) = b1 = 2πi C If f (z) is either analytic or has a removable singularity at z0 , then b1 = 0 there. If z0 is a pole of order m, then b1 =
1 d m−1 [(z − z0 )m f (z)]z=z0 . m−1 (m − 1)! dz
(1.5.14)
For every simple closed contour C enclosing at most a finite number of singularities z1 , z2 , . . . , zn of an analytic function continuous on C, C
f (z) dz = 2πi
n
Res(zk ),
(1.5.15)
k=1
where Res(zk ) is the residue of f (z) at zk .
1.5.12 THE ARGUMENT PRINCIPLE Let f (z) be analytic on a simple closed curve C with no zeros on C and analytic everywhere inside C except possibly at a finite number of poles. Let AC arg f (z) denote the change in the argument of f (z) (final value − initial value) as z transverses the curve once in the positive sense. Then 1 AC arg f (z) = N − P , 2π
(1.5.16)
where N is number of zeros of f (z) inside C, and P is the number of poles inside C. The zeros and poles are counted according to their multiplicities.
©1996 CRC Press LLC
1.5.13 TRANSFORMATIONS AND MAPPINGS A function w = f (z) = u(z) + iv(z) maps points of the z-plane into corresponding points of the w-plane. At every point z such that f (z) is analytic and f (z) = 0, the mapping is conformal, i.e., the angle between two curves in the z-plane through such a point is reproduced in magnitude and sense by the angle between the corresponding curves in the w-plane. A table giving real and imaginary parts, zeros, and singularities for frequently used functions of a complex variable and a table illustrating a number of special transformations of interest are at the end of this section. A function is said to be simple in a domain D if it is analytic in D and assumes no value more than once in D. Riemann’s mapping theorem states If D is a simply connected domain in the complex z plane, which is neither the z plane nor the extended z plane, then there is a simple function f (z) such that w = f (z) maps D onto the disc |w| < 1.
1.5.14 BILINEAR TRANSFORMATIONS az + b , where a, b, c, and d are cz + d complex numbers and ad = bc. It is also known as the M¨obius or linear fractional transformation. The bilinear transformation is defined for all z = −d/c. The bilinear transformation is conformal and maps circles and lines into circles and lines. −dw + b The inverse transformation is given by z = , which is also a bilinear cw − a transformation. Note that w = a/c. The cross ratio of four distinct complex numbers zk (for k = 1, 2, 3, 4) is given by (z1 − z2 )(z3 − z4 ) (z1 , z2 , z3 , z4 ) = . (z1 − z4 )(z3 − z2 ) The bilinear transformation is defined by w =
If any of the zk is complex infinity, the cross ratio is redefined so that the quotient of the two terms on the right containing zk is equal to 1. Under the bilinear transformation, the cross ratio of four points is invariant: (w1 , w2 , w3 , w4 ) = (z1 , z2 , z3 , z4 ).
©1996 CRC Press LLC
1.5.15 TABLE OF TRANSFORMATIONS f (z) = w(x, y) z z2 1 z 1 z2 1 z − (a + ib) a, b real √
z
u(x, y) = Re w(x, y) x x2 − y2 x 2 x + y2 x2 − y2 (x 2 + y 2 )2 x−a (x − a)2 + (y − b)2 ±
x+
x2 + y2 2
v(x, y) = Im w(x, y) y 2xy −y 2 x + y2 −2xy (x 2 + y 2 )2 −(y − b) (x − a)2 + (y − b)2
1/2
±
−x +
x2 + y2 2
Zeros (and order m) z = 0, m = 1 z = 0, m = 2
Singularities (and order m) Pole (m = 1) at z = ∞ Pole (m = 2) at z = ∞
z = ∞, m = 1
Pole (m = 1) at z = 0
z = ∞, m = 2
Pole (m = 2) at z = 0
z = ∞, m = 1
Pole (m = 1) at z = a + ib
z = 0, m = 1
Branch point (m = 1) at z = 0 Branch point (m = 1) at z = ∞ Essential singularity at z = ∞ Essential singularity at z = ∞
1/2
ez sin z
ex cos y sin x cosh y
ex sin y cos x sinh y
cos z
cos x sinh y
− sin x cosh y
sinh z
sinh x cos y
cosh x sin y
cosh z
cosh x cos y
sinh x sin y
tan z
sin 2x cos 2x + cosh 2y
sinh 2y cos 2x + cosh 2y
z = kπ, m = 1 (k = 0, ±1, ±2, . . . )
Essential singularity at z = ∞ Poles (m = 1) at z = (k + 1/2)π (k = 0, ±1, ±2, . . . )
tanh z
sinh 2x cosh 2x + cos 2y
sin 2y cosh 2x + cos 2y
z = kπi, m = 1 (k = 0, ±1, ±2, . . . )
Essential singularity at z = ∞ Poles (m = 1) at z = (k + 1/2)πi (k = 0, ±1, ±2, . . . )
log z
log(x 2 + y 2 )/2
tan−1
z = 1, m = 1
Branch points at z = 0, z = ∞
©1996 CRC Press LLC
x + 2kπ y (k = 0, ±1, ±2, . . . )
None z = kπ, m = 1 (k = 0, ±1, ±2, . . . ) z = (k + 1/2)π, m = 1 (k = 0, ±1, ±2, . . . ) z = kπi, m = 1 (k = 0, ±1, ±2, . . . ) z = (k + 1/2)πi, m = 1 (k = 0, ±1, ±2, . . . )
Essential singularity at z = ∞ Essential singularity at z = ∞ Essential singularity at z = ∞
1.5.16 TABLE OF CONFORMAL MAPPINGS In the following functions z = x + iy and w = u + iv = ρeiφ .
v
B y C
A' u C'
B' A x
D
y
D'
w = z2 .
v B
B'
A'
C'
D' u
A D
C
x
y
w = z2 .
v
D
A
C
B k
B
A'
x
u
D' B'
C'
y
w = z2 ; A , B on the 2k 2 parabola ρ = . 1 + cos φ
v D'
C
A
u
w = 1/z.
u
w = 1/z.
A' x
C' B'
D y
v
A'
B'
C Bx
A
C' y D
E πi
v F
w = ez . C
B
©1996 CRC Press LLC
A
u
1
x F' E' D' C'
B'
A'
E
D
y
v
πi
C'
w = ez .
C 1 u B'
x B
A
E' A'
D'
y
v E
πi
D
C'
F
w = ez .
F'
C
u
x B
A
E' A'
D'
y
B'
v A
E
−π/2 D
w = sin z.
π/2
D' C'
E'
B
C
u
1
x
y
B'
A'
v A
D
D'
w = sin z. π/2
x B
C
C'
y
π/2 x A
F
w = sin z; BCD: y = k, B C D is on the ellipse
u 2 v 2 + = 1. cosh k sinh k
C'
B
−π/2 E
A'
v
C
D
u
1 B'
D'
y
E'
F'
1 A'
u B'
v B'
A Bι x
C D
A' u 1 E'
C'
E
z−1 . z+1
D' v D'
y
A
w=
B
C
1 D
x E
E' A'
C' u 1
B'
©1996 CRC Press LLC
w=
i−z . i+z
B
y
F' F
E
C
B' A 1 x
G x1
x2
v
G'
A' C'
1
E' R0 u
D' D
1 + x1 x2 +
(1 − x12 )(1 − x22 )
z−a ;a= ; az − 1 x1 + x2 1 − x1 x2 + (1 − x12 )(1 − x22 ) (a > 1 and R0 > 1 when −1 < R0 = x1 − x2 x2 < x1 < 1).
w=
y
B'
AAA AAA B
E
E'
CD 1 x2
A
v
F x1 x
A' F' R0 1 u
C' D'
1 + x1 x2 +
(x12 − 1)(x22 − 1)
z−a ; ;a= az − 1 x1 + x2 x1 x2 − 1 − (x12 − 1)(x22 − 1) R0 = (x2 < a < x1 and 0 < R0 < 1 x1 − x2 when 1 < x2 < x1 ).
w=
y
v
C
B x 1
D E A
E'
w = z + 1/z. D'
A' u
C' B' 2
y
v
AAAAA AAAAA C
1 x B A
E D
w = z + 1/z.
2 E'
D'
u
C' B'
y
v
C
C'
A'
F D
2
B x
A E
1
k
©1996 CRC Press LLC
D' E'
F'
u A' B'
w = z + 1/z; B C D is on the ellipse 2 2 ku kv + = 1. k2 + 1 k2 − 1
y
v C' πi
D'
w = log
1 B
A
u
x
D
C
B'
E
D'
A'
E'
z−1 . z+1
B'
v y B
A x 1
C D
E
F'
E' πi
D'
A'
B' ki
C' u
F
y
πi
v A'
C D 1 A F
B
E
B' x
c1
c2
E' u
v πi
D' kπi x1 1
D
A'
C'
u E'
F'
y
v
πi
C'
w = tan2 (z/2).
C x B
A
y F
k + log 2(1 − k) + 1−k iπ −k log(z+1)−(1−k) log(z−1); x1 = 2k − 1. w = k log
B' G'
x
A B C D
E
w = log
D'
−πi
y
E F G
z+1 ; relationship z−1 between centers and radii: centers of circles at zn = coth cn , radii are csch cn , n = 1, 2.
F'
C'
-1
z−1 ; ABC is on the z+1 2 2 circle x + y − 2y cot k = 1. w = log
D'
E' A'
1 u B'
v πi
A'
E
G x B' C' G' F'
H A
D'
u
w = coth(z/2).
u
w = log coth(z/2).
E'
B D
C y F
πi
H'
E
v πi/2
B'
A'
G x
H A
C' F'
D' E'
B C
D
©1996 CRC Press LLC
G'
H'
y
v 1+πi
E' C'
D' 1 A
B
C D
w = πi + z − log z. u
x E
B'
A'
y
v A' C'
-1 A
B'
u
x
B C D
D'
E
w = 2(z + 1)1/2 + log
E'
(z + 1)1/2 − 1 . (z + 1)1/2 + 1
y
v
E'
-k2 D E F A
1 B
-π/k
x
u
C
B' A'
y
C
x
A'
D
B'
x D
©1996 CRC Press LLC
u
A'
E'
πi
h 2 (z − 1)1/2 + cosh−1 z . π
u D'
2z − k − 1 − w = cosh k−1 1 (k + 1)z − 2k cosh−1 . k (k − 1)z −1
B' C'
w=
D'
v
F' π/k
1 k AB C
ih
C'
y
F
C'
1 + ikt 1+t i + log ; w = log k 1 − ikt 1−t 1/2 z−1 t= . z + k2
v
1
-1 A B
D'
πi
F'
E
πi
1.6 REAL ANALYSIS
1.6.1 RELATIONS For two sets A and B, the product A × B is the set of all ordered pairs (a, b) where a is in A and b is in B. Any subset of the product A × B is called a relation. A relation R on a product A × A is called an equivalence relation if the following three properties hold: 1. Reflexive: (a, a) is in R for every a in A. 2. Symmetric: If (a, b) is in R, then (b, a) is in R. 3. Transitive: If (a, b) and (b, c) are in R, then (a, c) is in R. When R is an equivalence relation then the equivalence class of an element a in A is the set of all b in A such that (a, b) is in R. Example: The set of rational numbers has an equivalence relation “=” defined by the requirement that an ordered pair ( ab , dc ) belongs in the relation if and only if −1 −2 , −4 , . . . }. ad = bc. The equivalence class of 21 is the set { 21 , 24 , 36 , . . . −2
1.6.2 FUNCTIONS (MAPPINGS) A relation f on a set X × Y is a function (or mapping) from X into Y if (x, y) and (x, z) in the relation implies that y = z, and each x ∈ X has a y ∈ Y such that (x, y) is in the relation. The last condition means that there is a unique pair in f whose first element is x. We write f (x) = y to mean that (x, y) is in the relation f , and emphasize the idea of mapping by the notation f : X → Y . The domain of a function f is the set of all x for which there is a pair (x, y) in the relation. The range of a function f is a set containing all the y for which there is a pair (x, y) in the relation. The image of a set A in the domain of a function f is the set of y in Y such that y = f (x) for some x in A. The notation for the image of A under f is f [A]. The inverse image of a set B in the range of a function f is the set of all x in X such that f (x) = y for some y in B. The notation is f −1 [B]. A function f is one-to-one (or univalent, or injective) if f (x1 ) = f (x2 ) implies x1 = x2 . A function f : X → Y is onto (or surjective) if for every y in Y there is some x in X such that f (x) = y. A function is bijective if it is both one-to-one and onto. Examples: • f (x) = ex , as a mapping from R to R, is one-to-one because ex1 = ex2 implies x1 = x2 (by taking the natural logarithm). It is not onto because −1 is not the value of ex for any x in R.
©1996 CRC Press LLC
• g(x) = x 3 − x, as a mapping from R to R, is onto because every real number is attained as a value of g(x), for some x. It is not one-to-one because g(−1) = g(0) = g(1). • h(x) = x 3 , as a mapping from R to R, is bijective. For an injective function f mapping X into Y , there is an inverse function f −1 mapping the range of f into X which is defined by: f −1 (y) = x if and only if f (x) = y. Example: The function f (x) = ex mapping R into R+ (the set of positive reals) is bijective. Its inverse is f −1 (x) = ln(x) which maps R+ into R. For functions f : X → Y and g : Y → Z, with the range of f contained in the domain of g, the composition (g ◦ f ) : X → Z is a function defined by (g ◦ f )(x) = g(f (x)) for all x in domain of f . 1. Note that g ◦ f may not be the same as f ◦ g. For example, for f (x) = x + 1, and g(x) = 2x, we have (g ◦ f )(x) = g(f (x)) = 2f (x) = 2(x + 1) = 2x + 2. However (f ◦ g)(x) = f (g(x)) = g(x) + 1 = 2x + 1. 2. For every function f and its inverse f −1 , we have (f ◦ f −1 )(x) = x, for all x in the domain of f −1 , and (f −1 ◦ f )(x) = x for all x in the domain of f . Note that the inverse function f −1 is not the same as f1 (unless f (x) = x).
1.6.3 SETS OF REAL NUMBERS A sequence is the range of a function having the natural numbers as its domain. It can be denoted by {xn | n is a natural number} or simply {xn }. For a chosen natural number N, a finite sequence is the range of a function having natural numbers less than N as its domain. Sets A and B are in a one-to-one correspondence if there is a bijective function from A into B. Two sets A and B have the same cardinality if there is a one-to-one correspondence between them. A set which is equivalent to the set of natural numbers is denumerable (or countably infinite). A set which is empty or is equivalent to a finite sequence is finite (or finite countable). Examples: The set of letters in the English alphabet is finite. The set of rational numbers is denumerable. The set of real numbers is uncountable.
Axioms of order 1. There is a subset P (positive numbers) of R for which x + y and xy are in P for every x and y in P . 2. Exactly one of the following conditions can be satisfied by a number x in R (trichotomy): x ∈ P , −x ∈ P , or x = 0.
Definitions A number b is an upper (lower) bound of a subset S in R if x ≤ b (x ≥ b) for every x in S. A number c is a least upper bound (lub, supremum, or sup) of a subset S in R if c is an upper bound of S and b ≥ c for every upper bound b of S. A number c is
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a greatest lower bound (glb, infimum, or inf) if c is a lower bound of S and c ≥ b for every lower bound b of S.
Completeness (or least upper bound) axiom If a nonempty set of real numbers has an upper bound, then it has a least upper bound.
Characterization of the real numbers The real numbers are the smallest complete ordered field that contain the rationals. Alternatively, the properties of a field, the order properties, and the least upper bound axiom characterize the set of real numbers. The least upper bound axiom distinguishes the set of real numbers from other ordered fields. Archimedean property of R: For every real number x, there is an integer N such that x < N. For every pair of real numbers x and y with x < y, there is a rational number r such that x < r < y. This is sometimes stated: The set of rational numbers is dense in R.
Definitions The extension of R by +∞ is accomplished by including the symbols ∞ and −∞ with the following definitions (for all x ∈ R) • −∞ < x < ∞, for all x in R • x + ∞ = ∞, for all x in R • x − ∞ = −∞, for all x in R •
x ∞
=
x −∞
= 0, for all x in R
• x · ∞ = ∞, if x > 0 • x · (−∞) = −∞, if x > 0 • ∞+∞=∞ • −∞ − ∞ = −∞ • ∞·∞=∞ • −∞ · (−∞) = ∞
1.6.4 TOPOLOGY A topology on a set X is a collection T of subsets of X (called open sets) having the following properties: 1. The empty set and X are in T . 2. The union of elements in an arbitrary subcollection of T is in T . 3. The intersection of elements in a finite subcollection of T is in T .
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The complement of an open set is a closed set. A set is compact if every open cover has a finite subcover.
Notes 1. 2. 3. 4.
A subset E of X is closed if and only if E contains all its limit points. The union of finitely many closed sets is closed. The intersection of an arbitrary collection of closed sets is closed. The image of a compact set under a continuous function is compact.
1.6.5 METRIC SPACE A norm on a vector space E with scalar field R is a function || · || from E into R that satisfies the following conditions: 1. Positive definiteness: ||x|| ≥ 0 for all x in E, and ||x|| = 0 if and only if x = 0. 2. Scalar homogeneity: For every x in E and a in R,||ax|| = |a| ||x||. 3. Triangle inequality: ||x + y|| ≤ ||x|| + ||y|| for all x, y in E. A metric (or distance function) on a set E is a function ρ : E × E → R that satisfies the following conditions: 1. Positive definiteness: ρ(x, y) > 0 for all x, y in E, and ρ(x, y) = 0 if and only if x = y. 2. Symmetry: ρ(x, y) = ρ(y, x) for all x, y in E. 3. Triangle inequality: ρ(x, y) ≤ ρ(x, z) + ρ(z, y) for all x, y, z in E. Every norm || · || gives rise to a metric ρ by defining: ρ(x, y) = ||x − y||. Examples: 1. R with absolute value as norm has metric ρ(x, y) = |x − y|. 2 2 2. R × R (denoted R2 ) with Euclidean norm ||(x, y)|| = x + y , has metric 2 2 ρ((x1 , y1 ), (x2 , y2 )) = (x1 − x2 ) + (y1 − y2 ) A δ neighborhood of a point x in a metric space E is the set of all y in E such that ρ(x, y) < δ. For example, a δ neighborhood of x in R is the interval centered at x with radius δ, (x − δ, x + δ). In a metric space the topology is generated by the δ neighborhoods. 1. A subset G of R is open if, for every x in G, there is a δ > 0 such that every y in R with |x − y| < δ is in G also. For example, intervals (a,b), (a,∞), (−∞,b) are open in R. 2. A number x is a limit (or a point of closure, or an accumulation point) of a set F if, for every δ > 0, there is a point y in F such that |x − y| < δ.
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3. A subset F of R is closed if it contains all of its limits. For example, intervals [a,b], (−∞, b], and [a, ∞) are closed in R. 4. A subset F is dense in R if every element of R is a limit point of F . 5. A metric space is separable if it contains a denumerable dense set. For example, R is separable because the subset of rationals is a denumerable dense set. • (Bolzano–Weierstrass theorem) Any bounded infinite set of real numbers has a limit point in R. • (Heine–Borel theorem) A closed, bounded subset of R is compact.
1.6.6 CONVERGENCE IN R A number L is a limit point of a sequence {xn } if, for every F > 0, there is a natural number N such that |xn −L| < F for all n > N. If it exists, a limit point of a sequence is unique. A sequence is said to converge if it has a limit. A number L is a cluster point of a sequence {xn } if, for every F > 0, there is an index N such that |xn − L| < F for some n > N. Example: The limit of a sequence is a cluster point, as in { n1 }, which converges to 0. However, cluster points are not necessarily limits, as in {(−1)n }, which has cluster points +1 and −1 but no limit.
Limits Let {xn } be a sequence. A number L is the limit superior (limsup) if, for every F > 0, there is a natural number N such that xn > L − F for infinitely many n ≥ N, and xn > L+F for only finitely many terms. An equivalent definition of the limit superior is given by lim xn = inf sup xk . N k≥N
The limit inferior (liminf) is defined in a similar way by lim xn = sup inf xk . N k≥N
For example, the sequence {xn } with xn = 1 + (−1)n + 21n has lim xn = 2, and lim xn = 0. (Theorem) Every bounded sequence {xn } in R has a lim and a lim. In addition, if lim xn = lim xn , then the sequence converges to their common value. A sequence {xn } is a Cauchy sequence if, for any F > 0, there exists a positive integer N such that |xn − xm | < F for every n > N and m > N. (Theorem) A sequence {xn } in R converges if and only if it is a Cauchy sequence. A metric space, in which every Cauchy sequence converges to a point in the space, is called complete. For example, R with the metric ρ(x, y) = |x − y| is complete. A number L is a limit of a function f as x approaches a number a if, for every F > 0, there is a δ > 0 such that |f (x) − L| < F for all x with |x − a| < δ. This
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is represented by the notation limx→a f (x) = L. The symbol ∞ is the limit of a function f as x approaches a number a if, for every positive number M, there is a δ > 0 such that f (x) > M for all x with |x − a| < δ. Notation is limx→a f (x) = ∞. A number L is a limit of a function f as x approaches ∞ if, for every F > 0, there is a positive number M such that |f (x) − L| < F for all x > M; this is written limx→∞ f (x) = L. The number L is said to be the limit at infinity. 1 1 For example, lim 3x − 1 = 5, lim 2 = ∞, and lim = 0. x→∞ x x→2 x→0 x
Pointwise and uniform convergence A sequence of functions {fn (x)} is said to converge pointwise to the function f (x) on a set E if for every F > 0 and x ∈ E there is a positive integer N such that |f (x) − fn (x)| < F for every n ≥ N. A sequence of functions {fn (x)} is said to converge uniformly to the function f on a set E if, for every F > 0, there exists a positive integer N such that |f (x) − fn (x)| < F for all x in E and n ≥ N. Note that these formulations of convergence are not equivalent. For example, the functions fn (x) = x n on the interval [0, 1] converge pointwise to the function f (x) = 0 for 0 ≤ x < 1, f (1) = 1. They do not converge uniformly because, for F = 1/2, there is no N such that |fn (x) − f (x)| < 1/2 for all x in [0, 1] and every n $ N. A function f is Lipschitz if there exists k > 0 in R such that |f (x) − f (y)| ≤ k|x − y| for all x and y in its domain. The function is a contraction if 0 < k < 1. (Fixed point or contraction mapping theorem) If the function f : [a, b] → [a, b] is a contraction, then there is a unique point x in [a, b] such that f (x) = x. The point x is called a fixed point of f . Example: Newton’s method for finding a zero of f (x) = (x + 1)2 − 2 on the 1 interval [0, 1] produces xn+1 = g(xn ) with the contraction g(x) = x2 − 21 + x+1 . This √ has the unique fixed point 2 − 1 in [0, 1].
1.6.7 CONTINUITY IN R A function f : R → R is continuous at a point a if f is defined at a and limx→a f (x) = f (a). The function f is continuous on a set E if it is continuous at every point of E. A function f is uniformly continuous on a set E if, for every F > 0, there exists a δ > 0 such that |f (x) − f (y)| < F for every x and y in its domain with |x − y| < δ. A sequence {fn (x)} of continuous functions on the interval [a, b] is equicontinuous if, for every F > 0, there exists a δ > 0 such that |fn (x) − fn (y)| < F for every n and for all x and y in [a, b] with |x − y| < δ. Examples: 1. A function can be continuous without being uniformly continuous. The function g(x) = x1 is continuous but not uniformly continuous on the open interval (0, 1). 2. A collection of continuous functions can be bounded on a closed interval without having a uniformly convergent sub-sequence. The continuous functions fn (x) = x2 are each bounded by 1 in the closed interval [0, 1] and for every x x 2 +(1−nx)2 there is the limit: limn→∞ fn (x) = 0. However, fn ( n1 ) = 1 for every n, so
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that no sub-sequence can converge uniformly to 0 everywhere on [0, 1]. This sequence is not equicontinuous. (Theorem) Let {fn (x)} be a sequence of functions mapping R into R which converges uniformly to a function f . If each fn (x) is continuous at a point a, then f (x) is also continuous at a. (Theorem) If a function f is continuous on a closed bounded set E, then it is uniformly continuous on E. (Ascoli–Arzela theorem) Let K be a compact set in R. If {fn (x)} is uniformly bounded and equicontinuous on K, then {fn (x)} contains a uniformly convergent sub-sequence on K.
1.6.8 CONVERGENCE IN Lp In the context of elementary measure theory, two measurable functions f and g are equivalent if they are equal except on a set of measure zero. They are said to be equal almost everywhere. This is denoted by f = g a.e. The (vector) space of measurable functions f on [a, b], for which b |f (x)|p dx < ∞ with 0 < p < ∞, is denoted by Lp [a, b] or simply Lp . The a space of bounded measurable functions on [a, b] is denoted by L∞ .
1/p b . The The Lp norm for 0 < p < ∞ is defined by f p = a |f (x)|p dx L∞ norm is defined by f ∞ = ess sup |f (x)| ,
(1.6.1)
ess sup |f (x)| = inf{M|m{t : f (t) > M} = 0}.
(1.6.2)
a≤x≤b
where a≤x≤b
Let {fn (x)} be a sequence of functions in Lp (1 ≤ p < ∞) and f be some function in Lp . We say that {fn }converges in the mean of order p to f if limn→∞ fn − f p = 0. (Riesz–Fischer theorem) The Lp spaces are complete.
Inequalities 1. Minkowski inequality: If f and g are in Lp with 1 ≤ p ≤ ∞, then f + gp ≤ f p + gp . That is,
b a
|f + g|
p
1/p
≤
b a
p
|f |
1/p
+
a
b
|g|
p
1/p for (1 ≤ p < ∞), (1.6.3)
ess sup |f + g| ≤ ess sup |f | + ess sup |g|.
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2. H¨older inequality: If p and q are nonnegative extended real numbers such that 1/p + 1/q = 1 and f ∈ Lp and g ∈ Lq , then fg1 ≤ f p gq . That is 1/p b 1/q b b p q |fg| ≤ |f | |g| for (1 ≤ p < ∞), a a a (1.6.4)
b a
|fg| ≤ (ess sup |f |)
a
b
|g|.
(1.6.5)
3. Schwartz (or Cauchy–Schwartz) inequality: If f and g are in L2 , then fg1 ≤ f 2 g2 . This is the special case of H¨older’s inequality with p = q = 2.
1.6.9 CONVERGENCE IN L2
b Two functions f and g in L2 [a, b] are orthogonal if a fg = 0. A set of L2 functions b {φn } are orthogonal if a φm φn = 0 for m = n . The set is orthonormal if, in addition, each member has norm 1. That is, φn 2 = 1. For example, the functions {sin nx} are mutually orthogonal on (−π, π). The functions { sin√πnx } form an orthonormal set on (−π, π). Let φn be an orthonormal set in L2 and f be in L2 . The numbers cn =
b
a
f φn dx
are the generalized Fourier coefficients of f with respect to {φn }, and the series ∞ n=1 cn φn (x) is called the generalized Fourier series of f with respect to {φn }. For a function f in L2 , the mean square error of approximating f by the sum b N N 1 2 n=1 an φn is b−a a |f (x) − n=1 a n φn (x)| dx. An orthonormal set {φn } is complete if the only measurable function f that is orthogonal to every φn is zero. That is, f = 0 a.e. (Theorem) The generalized Fourier series of f in L2 converges in the mean (of order 2) to f. (Theorem) Parseval’s identity holds: b ∞ |f (x)|2 dx = cn 2 . a
n=1
Bessel’s inequality: For a function f in L2 having generalized Fourier coefficients b 2 2 {cn }, ∞ n=1 cn ≤ a |f (x)| dx. (Theorem) The mean square error of approximating f by the series ∞ n=1 an φn is minimum when all coefficients an are the Fourier coefficients of f with respect to {φn }. (Riesz–Fischer theorem) Let {φn } be an orthonormal set in L2 and let {cn } be 2 constants such that ∞ c L2 exists n=1 n converges. Then a unique function f in such that the cn are the Fourier coefficients of f with respect to {φn } and ∞ n=1 cn φn converges in the mean (or order 2) to f . 2 2 Example: suppose that the series a2o + ∞ n=1 (an + bn ) converges. Then the ∞ ao trigonometric series 2 + n=1 (an cos nx + bn sin nx) is the Fourier series of some function in L2 .
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1.6.10 ASYMPTOTIC RELATIONSHIPS Asymptotic relationships are indicated by the symbols O, J, K, o, and ∼. 1. The symbol O (pronounced “big-oh”): f (x) ∈ O(g(x)) as x → x0 if a positive constant C exists such that |f (x)| ≤ C |g(x)| for all x sufficiently close to x0 . Note that O(g(x)) is a class of functions. Sometimes the statement f (x) ∈ O(g(x)) is written (imprecisely) as f = O(g). 2. The symbol J : f (x) ∈ J(g(x)) as x → x0 if a positive constant C exists such that g(x) ≤ Cf (x) for all x sufficiently close to x0 . 3. The symbol K : f (x) ∈ K(g(x)) as x → x0 if positive constants c1 and c2 exist such that c1 g(x) ≤ f (x) ≤ c2 g(x) for all x sufficiently close to x0 . This is equivalent to: f (x) = O(g(x)) and g(x) = O(f (x)). The symbol ≈ is often use for K. 4. The symbol o (pronounced “little-oh”): f (x) ∈ o(g(x)) as x → x0 if, given any µ > 0, we have |f (x)| < µ|g(x)| for all x sufficiently close to x0 . 5. The symbol ∼ (pronounced “asymptotic to”): f (x) ∼ (g(x)) as x → x0 if f (x) = g(x) [1 + o(1)] as x → x0 . 6. Two functions, f (x) and g(x), are asymptotically equivalent as x → x0 if f (x)/g(x) ∼ 1 as x → x0 . 7. A sequence of functions, {gk (x)}, forms an asymptotic series at x0 if gk+1 (x) = o(gk (x)) as x → x0 . the formal series 8. Given ` a function f (x) and an asymptotic series {gk (x)} at x0 , n k=0 ak gk (x) = k=0 ak gk (x) is an asymptotic expansion of f (x) if f (x) − ` o(gn (x)) as x → x0 for every n; this is expressed as f (x) ∼ k=0 ak gk (x). Partial sums of this formal series are called asymptotic approximations to f (x). This formal series need not converge. Think of O being an upper bound on a function, J being a lower bound, and K being both an upper and lower bound. For example: sin x ∈ O(x) as x → 0, log n ∈ o(n) as n → ∞, and n9 ∈ J(n9 + n2 ) as n → ∞.
1.7 GENERALIZED FUNCTIONS 0 x = 0 Dirac’s delta function is defined by δ(x) = , and is normalized so that ∞ x=0 ∞ −∞ δ(x) dx = 1. Properties include (assuming that f (x) is continuous): 1. 2.
∞
−∞
f (x)δ(x − a) dx = f (a).
−∞
m d f (0) f (x) d dxδ(x) . m dx = (−1) dx m
∞
m
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m
3. xδ(x), as a distribution, equals zero 1 δ(x) when a = 0. 4. δ(ax) = |a| 5. δ(x 2 − a 2 ) = 6. 7. 8. 9.
1 [δ(x + a) + δ(x − a)]. 2a 1 1 ∞ δ(x) = 2L + L n=1 cos nπx (Fourier series). L nπξ 2 ∞ nπx δ(x) = L n=1 sin L sin L for 0 < ξ < L ∞ ikx 1 dk (Fourier transform). δ(x) = 2π −∞ e ∞ δ(ρ − ρ ) = ρ 0 kJm (kρ)Jm (kρ ) dk.
(Fourier sine series).
Sequences of functions {φn } that approximate the delta function as n → ∞ are known as delta sequences. For example: • φn (x) =
1 n π a+n2 x 2
• φn (x) =
2 2 √n e−n x π
• φn (x) =
1 sin2 nx nπ x 2
0 n/2
• φn (x) =
|x| ≥ 1/n |x| < 1/n
The Heaviside function, or step function, is defined as x 0 x<0 . Sometimes H (0) is stated to be 1/2. This H (x) = −∞ δ(t) dt = 1 x>1 function has the representations: 1 + π2 ∞ n=odd n sin ikx ∞ e 1 2. H (x) = 2πi −∞ k dk
1. H (x) =
1 2
nπx L
The related signum function gives the sign of its argument: sgn(x) = 2H (x) − 1 =
−1 1
if x < 0 if x > 0
(1.7.1)
The delta function δ(x − x ) = δ(x1 − x1 )δ(x2 − x2 )δ(x3 − x3 ) in terms of the coordinates (ξ1 , ξ2 , ξ3 ), related to (x1 , x2 , x3 ), via the Jacobian J (xi , ξj ), is written 1 δ(ξ − ξ1 )δ(ξ2 − ξ2 )δ(ξ3 − ξ3 ). δ(x − x ) = J (xi , ξj ) 1
(1.7.2)
For example, in spherical polar coordinates 1 δ(r − r )δ(φ − φ )δ(cos θ − cos θ ). (1.7.3) r2 The solutions to differential equations involving delta functions are called Green’s functions (see Sections 5.6.5 and 5.7.4). δ(x − x ) =
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References 1. L. B. W. Jolley, Summation of Series, Dover Publications, New York, 1961. 2. S. G. Krantz, Real Analysis and Foundations, CRC Press, Boca Raton, FL, 1991. 3. S. G. Krantz, The Elements of Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. 4. N. J. A. Sloane and S. Plouffe, Encyclopedia of Integer Sequences, Academic Press, New York, 1995.
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Chapter
2
Algebra 2.1
ELEMENTARY ALGEBRA 2.1.1 2.1.2 2.1.3 2.1.4
2.2
POLYNOMIALS 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.2.8
2.3
Quadratic polynomials Cubic polynomials Quartic polynomials Quintic polynomials Tschirnhaus’ transformation Polynomial norms Galois group of a polynomial Other polynomial properties
NUMBER THEORY 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6 2.3.7 2.3.8 2.3.9 2.3.10 2.3.11 2.3.12 2.3.13 2.3.14 2.3.15 2.3.16
2.4
Basic algebra Progressions DeMoivre’s theorem Partial fractions
Congruences Chinese remainder theorem Continued fractions Diophantine equations Greatest common divisor Least common multiple Farey sequences M¨obius function Prime numbers Prime numbers less than 10,000 Prime numbers of special forms Prime period lengths Factorization table Factorization of 2m − 1 Magic squares Totient function
VECTOR ALGEBRA 2.4.1 2.4.2 2.4.3 2.4.4 2.4.5
Notation for vectors and scalars Physical vectors Fundamental definitions Laws of vector algebra Vector norms
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2.4.6 2.4.7 2.4.8
2.5
LINEAR AND MATRIX ALGEBRA 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5 2.5.6 2.5.7 2.5.8 2.5.9 2.5.10 2.5.11 2.5.12 2.5.13 2.5.14 2.5.15 2.5.16 2.5.17 2.5.18 2.5.19 2.5.20
2.6
Dot, scalar, or inner product Vector or cross product Scalar and vector triple products
Definitions Types of matrices Conformability for addition and multiplication Determinants and permanents Matrix norms Singularity, rank, and inverses Systems of linear equations Other matrix transformations Linear spaces and linear mappings Traces Generalized inverses Eigenstructure Eigenvalue diagonalization Matrix exponentials Quadratic forms Matrix factorizations Theorems Kronecker products or tensor products Kronecker sums The vector operation
ABSTRACT ALGEBRA 2.6.1 2.6.2 2.6.3 2.6.4 2.6.5 2.6.6 2.6.7 2.6.8 2.6.9
Basic concepts Groups Rings Fields Finite fields Homomorphisms and isomorphisms Matrix classes that are groups Permutation groups Tables
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2.1 ELEMENTARY ALGEBRA 2.1.1 BASIC ALGEBRA Algebraic equations A polynomial equation in one variable is an equation of the form f (x) = an x n + an−1 x n−1 + · · · + a1 x + a0 . The degree of the equation is n (where an = 0). A complex number z is a root of the polynomial f (x) if f (z) = 0. A complex number z is a root of multiplicity k if f (z) = f (z) = f (z) = · · · = f (k−1) (z) = 0, but f (k) (z) = 0. A root of multiplicity 1 is called a simple root. A root of multiplicity 2 is called a double root, and a root of multiplicity 3 is called a triple root.
Fundamental theorem of algebra A polynomial equation of degree n has exactly n complex roots, where a double root is counted twice, a triple root three times, and so on. If the n roots of the polynomial f (x) are z1 , z2 , . . . , zn (where a double root is listed twice, a triple root three times, and so on), then f (x) = an (x − z1 )(x − z2 ) · · · (x − zn ).
(2.1.1)
If the coefficients a0 , a1 , . . . an are real numbers, then the polynomial will always have an even number of complex roots occurring in pairs. That is, if z is a complex root, then so is z. If the polynomial has an odd degree and the coefficients are real, then it must have at least one real root. The coefficients of the polynomial may be expressed as symmetric functions of the roots. For example, the elementary symmetric functions are an−1 , s1 = z1 + z2 + · · · + zn = − an an−2 s2 = z1 z2 + z1 z3 + z2 z3 + · · · = zi zj = , an i>j .. . sn = z1 z2 z3 · · · zn = (−1)n
(2.1.2)
a0 . an
where sk is the sum of nk products, each product combining k factors without repetition. The discriminant of the polynomial is defined by i>j (zi − zj )2 , where the ordering of the roots is irrelevant. The discriminant can always be written as a polynomial combination of a0 , a1 , . . . , an , divided by an .
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Resultants Let f (x) = an x n + an−1 x n−1 + · · · + a1 x + a0 and g(x) = bm x m + bm−1 x m−1 + · · · + b1 x + b0 , where an = 0 and bm = 0. The resultant of f and g is the determinant of the (m + n) × (m + n) matrix
an an−1 . . a0 0 ... 0 0 an−1 . a1 a0 0 an . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 an an−1 . a1 a0 det . b0 0 ... . 0 bm bm−1 0 b0 0 ... 0 bm bm−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0 bm . b1 b0
(2.1.3)
The resultant of f (x) and g(x) is 0 if and only if f (x) and g(x) have a common root. The resultant of f (x) and f (x) is zero if and only if f (x) has a multiple root. For example: 1. If f (x) = x 2 + 2x + 3 and g(x; α) = 4x 3 + 5x 2 + 6x + (7 + α), then the resultant of f (x) and g(x; α) is
1 0 det 0 4 0
2 1 0 5 4
3 2 1 6 5
0 3 2 7+α 6
0 0 2 3 = (16 + α) 0 7+α
Note that g(x, −16) = (4x − 3)(x 2 + 2x + 3) = (4x − 3)f (x). 2. The resultant of ax + b and cx + d is da − bc. 3. The resultant of (x + a)5 and (x + b)5 is (b − a)25 .
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Algebraic identities (a ± b)2 = a 2 ± 2ab + b2 . (a ± b)3 = a 3 ± 3a 2 b + 3ab2 ± b3 . (a ± b)4 = a 4 ± 4a 3 b + 6a 2 b2 ± 4ab3 + b4 . n n k n−k n n! n (a + b) = a b where . = k k!(n − k)! k k=0 a 2 + b2 = (a + bi)(a − bi). √ √ a 4 + b4 = (a 2 + 2ab + b2 )(a 2 − 2ab + b2 ). a 2 − b2 = (a − b)(a + b). a 3 − b3 = (a − b)(a 2 + ab + b2 ). a n − bn = (a − b)(a n−1 + a n−2 b + · · · + abn−2 + bn−1 ). (a + b + c)2 = a 2 + b2 + c2 + 2ab + 2ac + 2bc. (a + b + c)3 = a 3 + b3 + c3 + 3(a 2 b + ab2 + a 2 c + ac2 + b2 c + bc2 ) + 6abc.
Laws of exponents Assuming all quantites are real, then
a x a y = a x+y , a 0 = 1 if a = 0, (a x )y = a xy , √ x √ y a = xy a,
ax = a x−y , ay 1 a −x = x , a √ 1 a x = x a, √ √ x y a y = a x = ( y a)x ,
(ab)x = a x bx ,
a x ax = x, b b √ √ √ x x ab = x a b, √ x a a x = √ . x b b
Proportion = dc , then ac = db , ad = bc, a+b = c+d , a−b = c−d , and a−b = c−d . b d b d a+b c+d a c If b = d , where a, b, c, and d are all positive numbers and a is the largest of the four numbers, then a + d > b + c.
If
a b
2.1.2 PROGRESSIONS Arithmetic progression An arithmetic progression is a sequence of numbers such that the difference of any two consecutive numbers is constant. If the sequence is a1 , a2 , . . . an , where ai+1 −ai = d, then ak = a1 + (k − 1)d and a1 + a2 + · · · an =
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n (2a1 + (n − 1)d). 2
In particular, the sequence 1, 2, . . . n is an arithmetic progression with sum n(n+1)/2.
Geometric progression A geometric progression is a sequence of numbers such that the ratio of any two consecutive numbers is constant. If the sequence is a1 , a2 , . . . an , where ai+1 /ai = r, then ak = a1 r k−1 .
n
a1 1−r 1−r a1 + a2 + · · · an = na1
r= 1 r = 1.
(2.1.4)
If |r| < 1, then the infinite geometric series a1 (1 + r + r 2 + r 3 + · · · ) converges to a1 . For example, 1 + 21 + 41 + 18 + · · · = 2. 1−r
Means The arithmetic mean of a and b is given by a+b . More generally, the arithmetic mean 2 of a1 , a2 , . . . , an is given by (a1 + a2 + · · · + an )/n. √ The geometric mean of a and b is given by ab. More generally, the geometric √ mean of a1 , a2 , . . . , an is given by n a1 a2 · · · an . The geometric mean of n numbers is less than the arithmetic mean, unless all of the numbers are equal. 2ab The harmonic mean of a and b is given by a+b . If A, G, and H represent the arithmetic, geometric, and harmonic means of a and b, then AH = G2 .
2.1.3 DEMOIVRE’S THEOREM A complex number a + bi can be written in the form reiθ , where r 2 = a 2 + b2 and tan θ = b/a. Because eiθ = cos θ + i sin θ, (a + bi)n = r n (cos nθ + i sin nθ), √ 2kπ 2kπ n and 1 = cos + i sin , n n
k = 0, 1, . . . , n − 1.
2.1.4 PARTIAL FRACTIONS The technique of partial fractions allows a quotient of two polynomials to be written as a sum of simpler terms. (x) Given the fraction fg(x) , where both f (x) and g(x) are polynomials, begin by dividing f (x) by g(x) to produce a quotient q(x) and a remainder r(x), where the (x) r(x) degree of r(x) is less than the degree of g(x), so that fg(x) = q(x) + g(x) . Therefore, r(x) , where the degree of the numerator assume that the rational function has the form g(x) is less than the degree of the denominator. The techniques used depend on the factorization of g(x).
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Single linear factor Suppose that g(x) = (x − a)h(x), where h(a) = 0. Then s(x) A r(x) + , = x−a h(x) g(x) where the number A is given by r(a)/ h(a). For example, 2x 1 1 = + . x2 − 1 x−1 x+1
Repeated linear factor Suppose that g(x) = (x − a)k h(x), where h(a) = 0. Then A2 Ak s(x) A1 r(x) + , + ··· + + = 2 k x−a (x − a) (x − a) h(x) g(x) where r(a) , h(a) d r(x) = , dx g(x) x=a 1 d 2 r(x) = , 2! dx 2 g(x) x=a r(x) 1 dj = . j ! dx j g(x) x=a
Ak = Ak−1 Ak−2 Ak−j
Single quadratic factor Suppose that g(x) = (x 2 + bx + c)h(x), where b2 − 4c < 0 (so that x 2 + bx + c does not factor into real linear factors) and h(x) is relatively prime to x 2 + bx + c. Then Ax + B s(x) r(x) = 2 + . g(x) x + bx + c h(x) In order to determine A and B, multiply the equation by g(x) so that there are no denominators remaining, and substitute any two values for x, yielding two equations for A and B.
Repeated quadratic factor Suppose that g(x) = (x 2 + bx + c)k h(x), where b2 − 4c < 0 (so that x 2 + bx + c does not factor into real linear factors) and h(x) is relatively prime to x 2 + bx + c. Then r(x) A1 x + B1 A2 x + B2 A3 x + B3 = 2 + 2 + 2 2 g(x) x + bx + c (x + bx + c) (x + bx + c)3 Ak x + Bk s(x) . + ··· + 2 + (x + bx + c)k h(x)
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In order to determine Ai and Bi , multiply the equation by g(x) so that there are no denominators remaining, and substitute any 2k values for x, yielding 2k equations for Ai and Bi .
2.2 POLYNOMIALS 2.2.1 QUADRATIC POLYNOMIALS The solution of the equation ax 2 + bx + c = 0, where a = 0, is given by x=
−b ±
√
b2 − 4ac . 2a
(2.2.1)
The discriminant of this equation is b2 − 4ac. Suppose that a, b, and c are all real. If the discriminant is negative, then the two roots are complex numbers which are conjugate. If the discriminant is positive, then the two roots are unequal real numbers. If the discriminant is 0, then the two roots are equal.
2.2.2 CUBIC POLYNOMIALS To solve the equation ax 3 + bx 2 + cx + d = 0, where a = 0, begin by making the 2 b substitution y = x + 3a . That gives the equation y 3 +3py +q = 0, where p = 3ac−b 9a 2 3 2 d and q = 2b −9abc+27a . The discriminant of this polynomial is 4p3 + q 2 . 27a 3 √ √ 2πi √ 4πi √ 4πi √ 2πi √ The solutions are given by 3 α − 3 β, e 3 3 α − e 3 3 β, and e 3 3 α − e 3 3 β, where −q + q 2 + 4p3 −q − q 2 + 4p3 α= and β = . 2 2 Suppose that p and q are real numbers. If the discriminant is positive, then one root is real, and two are complex conjugates. If the discriminant is 0, then there are three real roots, of which at least two are equal. If the discriminant is negative, then there are three unequal real roots.
Trigonometric solution of cubic polynomials In the event that the roots of the polynomial y 3 + 3py + q = 0 are all real, meaning that q 2 + 4p3 ≤ 0, then the expressions above involve complex numbers. In that case one can also express the solution in terms of trigonometric functions. Define r and θ by r=
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−p3 and θ = cos−1
−q . 2r
Then the three roots are given by √ √ θ θ + 2π 2 3 r cos , 2 3 r cos , 3 3
and
√ θ + 4π 2 3 r cos 3
2.2.3 QUARTIC POLYNOMIALS To solve the equation ax 4 + bx 3 + cx 2 + dx + e = 0, where a = 0, start with the 2 b . That gives y 4 + py 2 + qy + r = 0, where p = 8ac−3b , substitution y = x + 4a 8a 2 3 2 2 3 4 2 b −4abc+8a d 16ab c+256a e−3b −64a bd q= , and r = . 8a 3 256a 4 The cubic resolvent of this polynomial is defined as t 3 −pt 2 −4rt +(4pr −q 2 ) = 0. If u is a root of the cubic resolvent, then the solutions of the original quartic are given by the solutions of √ u q 2 y ± u−p y− + = 0. 2(u − p) 2
2.2.4 QUINTIC POLYNOMIALS Some quintic equations are solvable by radicals. If the function f (x) = x 5 + ax + b (with a and b rational) is irreducible, then f (x) = 0 is solvable by radicals if, and only if, numbers &, c, and e exist (with & = ±1, c ≥ 0, and e = 0) such that −4e5 (11& + 2c) 5e4 (3 − 4&c) and b = . c2 + 1 c2 + 1 In this case, the roots are given by x = e ωj u1 + ω2j u2 + ω3j u3 + ω4j u4 for j = 0, 1, 2, 3, 4, where ω is a fifth root of unity (ω = exp (2πi/5)) and 2 2 2 2 v3 v4 v1 v3 v2 v1 v4 v2 , u2 = , u3 = , u4 = , u1 = D2 D2 D2 D2 √ √ √ √ v1 = D + D − & D, v2 = − D − D + & D, √ √ √ √ v3 = − D + D + & D, v4 = D − D − & D, and a=
D = c2 + 1.
EXAMPLE 2.2.1 The quintic f (x) = x 5 + 15x + 12 has the values & = −1, c = 4/3, and e = 1. Hence the unique real root is given by √ 1/5 √ 1/5 −75 − 21 10 −75 + 21 10 + x= 125 125 √ 1/5 √ 1/5 225 + 72 10 225 − 72 10 + + . 125 125
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2.2.5 TSCHIRNHAUS’ TRANSFORMATION The nth degree polynomial equation an x n + an−1 x n−1 + · · · + a1 x + a0 = 0 can be transformed to one with up to three fewer terms, zn + bn−4 zn−4 + · · · + b1 z + b0 = 0 by making a transformation of the form zj = γ4 xj4 + γ3 xj3 + γ2 xj2 + γ1 xj + γ0 for j = 1, . . . , n where the {γi } can be computed, in terms of radicals, from the {ai }. Hence, the general quintic polynomial can be transformed to the form z5 +az+b = 0.
2.2.6 POLYNOMIAL NORMS The polynomial P (x) =
n
aj x j has the norms:
j =0
||P ||1 =
2π 0
||P ||2 =
iθ dθ P e 2π
2π 0
iθ 2 dθ P e 2π
||P ||∞ = max |P (z)| |z|=1
|P |1 =
n aj . j =0
1/2
n aj 2 |P |2 =
(2.2.2) 1/2 . (2.2.3)
j =1
|P |∞ = max aj . j
(2.2.4)
For the double bar norms, P is considered as a function on the unit circle; for the single bar norms, P is identified with its coefficients. These norms are comparable: |P |∞ ≤ ||P ||1 ≤ |P |2 = ||P ||2 ≤ ||P ||∞ ≤ |P |1 ≤ n|P |∞ .
(2.2.5)
2.2.7 GALOIS GROUP OF A POLYNOMIAL Consider the polynomial P (x) = an x n + an−1 x n−1 + · · · + a0 = an (x − z1 )(x − z2 ) . . . (x − zn ). There are certain relations among the roots that do not depend on how the roots are numbered. For example, this is true of the elementary symmetric functions; see Equation (2.1.2). The set of permutations that leave all polynomials with rational coefficients, H (z1 , . . . , zn ), invariant is the Galois group of the equation.
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2 2 For √ example, √ consider the√polynomial √ f (x) = (x − 2)(x − 3), with roots {z1 = − 2, z2 = 2, z3 = − 3, z4 = 3}. There are four permutations that leave all allowable relationships between the roots invariant:
1 π= 1 1 p2 = 1
2 2
3 3
4 4
2 2
3 4
4 3
1 p1 = 2 1 p3 = 2
2 1
3 4 3 4
2 1
3 4
4 3
That is, all permutations that switch z1 and z2 , or switch z3 and z4 , or both, will leave allowable relations invariant. To demonstrate that neither z1 nor z2 can be switched with z3 or z4 , consider the expressions H1 (zi ) = z1 z2 + 2 and H2 (zi ) = z3 z4 + 3.
2.2.8 OTHER POLYNOMIAL PROPERTIES 1. For polynomial P (x) =
n
aj x j , with a0 = 0, Jensen’s inequality is
j =0
dθ log P eiθ ≥ log |a0 | 2π 0 2. The polynomial P (x1 , . . . , xn ) = aα x1α1 x2α2 . . . xNαN , where α = (α1 , . . . , αN ) 2π
|α|=m
can be written in the symmetric form P (x1 , . . . , xn ) =
N
ci1 ,...,im xi1 xi2 . . . xim
i1 ,...,im =1 ∂mP . This means that the x1 x2 term is written as 21 (x1 x2 + ∂xi1 ...∂xim x2 x1 ), the term x1 x22 becomes 13 (x1 x2 x2 + x2 x1 x2 + x2 x2 x1 ). n−1 A valuation of the polynomial P (x) = an x n + + · · · + a0 = an (x − ann−1 x z1 )(x−z2 ) . . . (x−zn ) is given by M(P ) = an i=1 max(1, |zi |). This valuation
with ci1 ,...,im = 3.
satisfies the properties:
• M(P ) M(Q) = M(P Q) • M(P (x)) = M(P (x k )) for k ≥ 1 • M(x n P (x −1 )) = M(P (x))
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2.3 NUMBER THEORY
2.3.1 CONGRUENCES Definitions 1. If the integers a and b leave the same remainder when divided by the number n, then a and b are congruent modulo n. This is written a ≡ b (mod n). 2. If the congruence x 2 ≡ a (mod p) has a solution, then a is a quadratic residue of p. Otherwise, a is a quadratic nonresidue of p.
• Let p be a prime. Legendre’s symbol pa has the value +1 if a is a quadratic residue of p, and the value −1 if a is a quadratic nonresidue of p. • The Jacobi symbol generalizes the Legendre symbol to non-prime moduli. If n = ki=1 pibi then the Jacobi symbol can be written in terms of the Legendre symbol
a n
=
k a bi pi i=1
(2.3.1)
3. Carmichael numbers are composite numbers {n} that satisfy a n−1 ≡ 1 (mod n) for every a (1 < a < n) relatively prime to n.
Properties 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
If a ≡ b (mod n), then b ≡ a (mod n). If a ≡ b (mod n), and b ≡ c (mod n), then a ≡ c (mod n). If a ≡ a (mod n), and b ≡ b (mod n), then a ± b ≡ a ± b (mod n). If a ≡ a (mod n), then a 2 ≡ (a )2 (mod n), a 3 ≡ (a )3 (mod n), etc. If (k, m) = d, then the congruence kx ≡ n (mod m) is solvable if and only if d divides n. It then has d solutions. If p is a prime, then a p ≡ a (mod p). If p is a prime, and p does not divide a, then a p−1 ≡ 1 (mod p). If (a, m) = 1 then a φ(m) ≡ 1 (mod m). (See Section 2.3.16 for φ(m).)
If p is an odd prime and a is not a multiple of p, then (p − 1)! ≡ − pa a (p−1)/2 (mod p). If p and q are odd primes, then the law of quadratic reciprocity states that q p = (−1)(p−1)(q−1)/4 . q p
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11. The number −1 is a quadratic residue of primes of the form 4k + 1 and a nonresidue of primes of the form 4k + 3. 12. The number 2 is a quadratic residue of primes of the form 8k±1 and a nonresidue of primes of the form 8k ± 3. 13. The number −3 is a quadratic residue of primes of the form 6k + 1 and a nonresidue of primes of the form 6k + 5. 14. The number 3 is a quadratic residue of primes of the form 12k ± 1 and a nonresidue of primes of the form 12k ± 5.
Values There are infinitely many Carmichael numbers. There are 32 Carmichael numbers less than one million; they are 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, and 488881.
2.3.2 CHINESE REMAINDER THEOREM Let m1 , m2 , . . . , mr be pairwise relatively prime integers. congruences x ≡ a1
(mod m1 )
x ≡ a2 .. .
(mod m2 )
x ≡ ar
(mod mr )
Then the system of
has a unique solution modulo M = m1 m2 · · · mr . This unique solution can be written as x = a1 M1 y1 + a2 M2 y2 + · · · + ar Mr yr
(2.3.2)
where Mk = M/mk , and yk is the inverse of Mk (modulo mk ). Example: For the system of congruences x ≡ 1 (mod 3) x ≡ 2 (mod 5) x ≡ 3 (mod 7) we have M = 3·5·7 = 105. Hence M1 = 35, M2 = 21, and M3 = 15. The equation for y1 is M1 y1 = 35y1 ≡ 1 (mod 3) with solution y1 ≡ 2 (mod 3). Likewise, y2 ≡ 1 (mod 5) and y3 ≡ 1 (mod 7). This results in x = 1 · 35 · 2 + 2 · 21 · 1 + 3 · 15 · 1 ≡ 52 (mod 105).
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2.3.3 CONTINUED FRACTIONS The symbol [a0 , a1 , . . . , aN ] (with N finite or infinite and the {ai } being positive integers) represents the simple continued fraction, 1
[a0 , a1 , . . . , aN ] = a0 +
(2.3.3)
1
a1 +
1
a2 +
1
a3 + a4 +
... ··· +
1 . aN
The nth convergent (with 0 < n < N ) of [a0 , a1 , . . . , aN ] is defined to be [a0 , a1 , . . . , an ]. If {pn } and {qn } are defined by p0 = a0 , q0 = 1,
p1 = a1 a0 + 1, q1 = a1 ,
pn = an pn−1 + pn−2 qn = an qn−1 + qn−2
(2 ≤ n ≤ N) (2 ≤ n ≤ N)
then [a0 , a1 , . . . , a n ] = pn /qn . The continued fraction is convergent if and only if the infinite series ∞ ai is divergent. If the positive rational number x can be represented by a simple continued fraction with an odd (even) number of terms, then it is also representable by one with an even (odd) number of terms. (Specifically, if a = 1 then a , a , . . . , an−1 , 1 = n 0 1 a0 , a1 , . . . , an−1 + 1 , and if an ≥ 2, then [a0 , a1 , . . . , an ] = [a0 , a1 , . . . , an − 1, 1].) Aside from this indeterminacy, the simple continued fraction of x is unique. The error in approximating by a convergent is bounded by 1 1 x − pn ≤ < 2. (2.3.4) qn qn qn+1 qn The algorithm for finding a continued fraction expansion of a number is to remove the integer part of the number (this becomes ai ), take the reciprocal, and repeat. For the number π: β0 = π ≈ 3.14159 β1 = 1/(β0 − a0 ) ≈ 7.062 β2 = 1/(β1 − a1 ) ≈ 15.997 β3 = 1/(β2 − a2 ) ≈ 1.0034
a0 = β0 = 3 a1 = β1 = 7 a2 = β2 = 15 a3 = β3 = 1
β4 = 1/(β3 − a3 ) ≈ 292.6
a4 = β4 = 292
Approximations to π and e may be found from π = [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, . . . ] and e = [2, 1, 2, 1, 1, 4, 1, 1, 6, . . . , 1, 1, 2n, . . . ]. The convergents for , 333 , 355 , 103993 , . . .. The convergents for e are 83 , 11 , 19 , 87 , . . .. π are 22 7 106 113 33102 4 7 32
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A periodic continued fraction is an infinite continued fraction in which al = al+k for all l ≥ L. The set of partial quotients aL , aL+1 , . . . , aL+k−1 is the period. A periodic continued fraction may be written as a0 , a1 , . . . , aL−1 , a˙ L , aL+1 , . . . , a˙ L+k−1 . (2.3.5) √ √ √ √ ˙ ˙ ˙ ˙ 1, 1, 4]. ˙ If ˙ 7 = [2, 1, For example, 2 = [1, 2], 3 = [1, 1, 2], 5 = [2, 4], and √ √ 4b 1 ˙ = (1+ 5)/2, [2] ˙ = 1+ 2, ˙ a] ˙ then x = 2 (b+ b2 + a ). For example, [1] x = [b, √ ˙ 1] ˙ = 1 + 3. and [2, Functions can be represented as continued fractions. Using the notation a1
b0 +
= b0 +
a2
b1 +
a1 a2 a3 a4 ... b1 + b2 + b3 + b4 +
a3
b2 + b3 +
(2.3.6)
a4 b4 + . . .
then (allowable values of z may be restricted in the following) z z z 4z 4z 9z 1+ 2+ 3+ 4+ 5+ 6+
• ln(1 + z) = • ez =
1 z z z z z z 1− 1+ 2− 3+ 2− 5+ 2−
• tan z =
2
2
2
z z z z 1− 3− 5− 7−
• tanh z =
2
2
...
··· = 1 +
z z z z z z z 1− 2+ 3− 2+ 5− 2+ 7−
...
... 2
z z z z 1+ 3+ 5+ 7+
...
2.3.4 DIOPHANTINE EQUATIONS A diophantine equation is one which requires the solutions to come from the set of integers. Apart from the trivial solutions (with x = y = 0 or x = u), the general solution to the equation x 3 + y 3 = u3 + v 3 is given by x = λ 1 − (a − 3b)(a 2 + 3b2 ) y = λ (a + 3b)(a 2 + 3b2 ) − 1 u = λ (a + 3b) − (a 2 + 3b2 )2 v = λ (a 2 + 3b2 )2 − (a − 3b) (2.3.7) where {λ, a, b} are any rational numbers except that λ = 0. A parametric solution to x 4 + y 4 = u4 + v 4 is given by x = a 7 + a 5 b2 − 2a 3 b4 + 3a 2 b5 + ab6 y = a 6 b − 3a 5 b2 − 2a 4 b3 + a 2 b5 + b7 u = a 7 + a 5 b2 − 2a 3 b4 − 3a 2 b5 + ab6
(2.3.8)
v = a 6 b + 3a 5 b2 − 2a 4 b3 + a 2 b5 + b7 Fermat’s last theorem states that there are no integer solutions to x n + y n = zn , when n > 2. This was proven by Andrew Wiles in 1995.
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Pell’s equation Pell’s equation is x 2 − My 2 = 1. The √ solutions, integral values of (x, y), arise from continued fraction√convergents of M (see page 88). The number 2 has the continued fraction expansion [1, 2, 2, 2, 2, . . . ]. Hence, 17 41 99 the first few convergents are 23 , 75 , 12 , 29 , 70 , . . . . In this case, every second convergent represents a solution: 32 − 2 · 22 = 1, 172 − 2 · 122 = 1, and 992 − 2 · 702 = 1.
Pythagorean triples If the positive integers A, B, and C satisfy the relationship A2 + B 2 = C 2 , then the triplet (A, B, C) is a Pythagorean triple. It is possible to construct a right triangle with sides of length A and B and a hypotenuse of C. There are infinitely many Pythagorean triples. The most general solution to A2 + B 2 = C 2 , with (A, B) = 1 and A even, is given by B = x2 − y2
A = 2xy
C = x2 + y2,
(2.3.9)
where x and y are relatively prime integers of opposite parity with x > y > 0. The following shows some Pythagorean triples with the associated (x, y) values. x
y
A
B
C
2 4 6 8 10 3 5 7 4
1 1 1 1 1 2 2 2 3
4 8 12 16 20 12 20 28 24
3 5 15 17 35 37 63 65 99 101 5 13 21 29 45 51 7 25
Waring’s problem If each positive integer can be expressed as a sum of n k th powers, then there is a least value of n for which this is true: this is the number g(k). For all sufficiently large numbers, however, a smaller value of n may suffice: this is the number G(k). Lagrange’s theorem states: “Every positive integer is the sum of four squares;” this is equivalent to the statement g(2) = 4. The following identity shows how a product can be written as the sum of four squares: (x12 + x22 + x32 + x42 )(y12 + y22 + y32 + y42 ) = (x1 y1 + x2 y2 + x3 y3 + x4 y4 )2 + (x1 y2 − x2 y1 + x3 y4 − x4 y3 )2 + (x1 y3 − x3 y1 + x4 y2 − x2 y4 )2 + (x1 y4 − x4 y1 + x2 y3 − x3 y2 )2
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(2.3.10)
Consider k = 3; all numbers can be written as the sum of not more than 9 cubes, so that g(3) = 9. However, only the two numbers 23 = 23 + 23 + 13 + 13 + 13 + 13 + 13 + 13 + 13 , 239 = 43 + 43 + 33 + 33 + 33 + 33 + 13 + 13 + 13 , require the use of 9 cubes; so G(3) ≤ 8. The current known values of g(k) include: g(3) = 9, g(4) ≥ 19, g(5) = 37 and 3 k + 2k − 2 for 6 ≤ k ≤ 471600000. g(k) = 2 The value of G(k) is only known for two values of k, G(2) = 4 and G(4) = 16. It is known that: 4 ≤ G(3) ≤ 7, G(5) ≤ 23, G(6) ≤ 36, G(7) ≤ 137, and G(8) ≤ 163. It is also known that 2 G(k) ≤ 6k log k + 4 + 3 log 3 + k + 3. (2.3.11) k
2.3.5 GREATEST COMMON DIVISOR The greatest common divisor (GCD) of the integers n and m is the largest integer that evenly divides both n and m. The Euclidean algorithm is frequently used for computing the GCD of two numbers; it utilizes the fact that a = ab b + c where 0 ≤ c < b. For example, consider 78 and 21. Since 78 = 3 · 21 + 15, the largest integer that evenly divides both 78 and 21 is also the largest integer that evenly divides both 21 and 15. Iterating results in 78 = 3 · 21 + 15 21 = 1 · 15 + 6 15 = 2 · 6 + 3 6=2·3+0 Hence GCD(78,21) = GCD(6,3) = 3. A common notation for GCD(n, m) is (n, m). Two numbers, a and b, are said to be relatively prime if they have no divisors in common; i.e., if GCD(a, b) = 1. The probability that two integers chosen randomly are relatively prime is π/6.
2.3.6 LEAST COMMON MULTIPLE The least common multiple of the integers a and b (denoted LCM(a, b)) is the smallest integer r that is divisible by both a and b. The simplest way to find the LCM of a and b is via the formula LCM(a, b) = ab/GCD(a, b). For example, LCM(10, 4) = 10·4 = 10·4 = 20. GCD(10,4) 2
©1996 CRC Press LLC
2.3.7 FAREY SEQUENCES The Farey series of order n, Fn , is the ascending series of irreducible fractions between zero and one whose denominators do not exceed n. The fraction k/m belongs to Fn if 0 ≤ k ≤ m ≤ n and k and m are relatively prime. If k/m and k /m are two successive terms of Fn then mk − km = 1. Fn−1 can be obtained from Fn by deleting terms with denominators of n. F3
=
F5
=
0 , 1 0 , 1
1 , 3 1 , 5
1 , 2 1 , 4
2 , 3 1 , 3
1 , 1 2 1 3 2 3 4 1 , , , , , , . 5 2 5 3 4 5 1
¨ 2.3.8 MOBIUS FUNCTION The M¨obius function is defined by • µ(1) = 1 • µ(n) = 0 if n has a squared factor • µ(p1 p2 . . . pk ) = (−1)k if all the primes {p1 , . . . , pk } are distinct The M¨obius inversion formula states that, if g(n) = f (d), then d|n
f (n) =
d|n
µ
n d
g(d) =
d|n
µ(d)g
n d
.
. For example, φ(n) = n d|n µ(d) d α The generalized in one dimension is that: if g(x) = ∞ n=1 f (n x) , ∞ M¨obius formula then f (x) = n=1 µ(n)g (nα x), for any real α except 0. The table below can be derived from the table in Section 2.3.13 (For example, µ(2) = −1, µ(4) = 0, and µ(6) = 1).
©1996 CRC Press LLC
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1 2 1 −1 1 −1 0 0 1 1 −1 −1 0 0 −1 −1 0 1 0 0 −1 1 −1 −1 0 0 0 1 0 1 0 0 −1 −1 −1 1 0 0 0 1 −1 −1 0 0 1 1 0 −1 0 0 1 0 −1 0 0 0 −1 −1 −1 −1 0
3 4 −1 0 −1 1 −1 0 1 1 −1 0 −1 0 0 0 −1 1 −1 0 1 1 −1 0 −1 −1 1 0 1 1 1 0 0 −1 −1 0 −1 −1 1 0 −1 1
5 6 −1 1 1 0 0 1 1 0 0 1 1 0 1 −1 0 0 1 1 1 0 −1 1 1 0 0 0 0 0 1 1 1 0 −1 1 0 0 1 −1 −1 0
7 8 −1 0 −1 0 0 0 −1 1 −1 0 1 1 −1 0 1 −1 1 0 −1 0 −1 0 0 1 −1 0 −1 −1 0 0 −1 1 −1 0 1 1 1 0 −1 0
9 0 −1 −1 1 0 −1 1 −1 −1 0 −1 1 1 −1 −1 1 0 −1 0 −1
2.3.9 PRIME NUMBERS • A prime number is a positive integer greater than 1 with no positive, integral divisors other than 1 and itself. There are infinitely many prime numbers, 2, 3, 5, 7, . . . . • Twin primes are prime numbers that differ by two: (3, 5), (5, 7), (11, 13), (17, 19), . . . . It is not known whether there are infinitely many twin primes. • For every integer n ≥ 2, the numbers {n! + 2, n! + 3, . . . , n! + n} are a sequence of n − 1 consecutive composite numbers.
The function π(x) represents the number of primes less than x. The prime number theorem states that π(x) ∼ x/ log x as x → ∞.
x π(x)
100 25
1000 168
10000 1229
105 9592
106 78498
107 664579
Prime formula The set of prime numbers is identical with the set of positive values taken on by the polynomial of degree 25 in the 26 variables {a, b, . . . , z}:
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(k + 2){1 − [wz + h + j − q]2 − [(gk + 2g + k + 1)(h + j ) + h − z]2 − [2n + p + q + z − e]2 − [16(k + 1)3 (k + 2)(n + 1)2 + 1 − f 2 ]2 − [e3 (e + 2)(a + 1)2 + 1 − o2 ]2 − [(a 2 − 1)y 2 + 1 − x 2 ]2 −[16r 2 y 4 (a 2 −1)+1−u2 ]2 −[((a+u2 (u2 −a))2 −1)(n+4dy)2 +1−(x+cu)2 ]2 −[n+l+v−y]2 −[(a 2 −1)l 2 +1−m2 ]2 −[ai +k+1−l−i]2 −[p+l(a−n−1)+b(2an+2a−n2 −2n−2)−m]2 −[q+y(a−p−1)+s(2ap+2a−p2 −2p−2)−x]2 −[z+pl(a−p)+t (2ap−p2 −1)−pm]2 }. (2.3.12)
Although this polynomial appears to factor, the factors are improper, P = P · 1. Note that this formula will also take on negative values, such as −76. There also exists a prime representing polynomial which contains 12 variables. Dirichlet’s theorem on primes in arithmetic progressions: Let a and b be relatively prime positive integers. Then the arithmetic progression an + b (for n = 1, 2, . . . ) contains infinitely many primes.
Proofs of primality Lucas–Lehmer primality test: Define the sequence, r1 = 3, rm+1 = rm2 − 2. If p is a prime of the form 4n + 3 and Mp = 2p − 1, then Mp will be prime (called a Mersenne prime) if, and only if, Mp divides rp−1 . This simple test is the reason that the largest known prime numbers are Mersenne primes. For example, consider p = 7 and M7 = 127. The {rn } sequence is {3, 7, 47, 2207 ≡ 48, 2302 ≡ 16, 254 ≡ 0}; hence M7 is prime. It is also possible to give, for a general number p, a “certificate” that p is prime. It is easy to use the certificate to verify that a given number is prime (easier than it was to determine that it was prime in the first place). There are several types of certificates that can be given. Pratt’s certificate consists of a number a and the factorization of the number p − 1. This method is feasible if p − 1 is easy to factor. The number p will be prime if there exists a primitive root a in the field GF[p]. This primitive root must satisfy the conditions a p−1 = 1 (mod p) and a (p−1)/q = 1 (mod p) for any prime q that divides p. For example, the number p = 31 has p − 1 = 30 = 2 · 3 · 5, and a primitive root is given by a = 3. Hence, to verify that p = 31 is prime, we compute 3(31−1)/2 = 315 ≡ 14348907 ≡ −1 = 1 (31−1)/3
3 3
(31−1)/5
3(31−1)
10
=3
≡ 59049 ≡ 25 = 1
6
= 3 ≡ 719 ≡ 16 = 1 2 = 3(31−1)/2 ≡ (−1)2 = 1
(mod 31), (mod 31), (mod 31), (mod 31).
Probabilistic primality test Let n be a number whose primality is to be determined. Probabilistic primality tests can return one of two results: either a proof that the number n is composite or a
©1996 CRC Press LLC
statement of the form, “The probability that the number n is not prime is less than &”, where & can be specified by the user. Typically, we take & = 2−200 < 10−60 . From Fermat’s theorem, if b = 0, then bn−1 = 1 (mod n) whenever n is prime. If this holds, then n is a probable prime to the base b. Given a value of n, if a value of b can be found such that this does not hold, then n cannot be prime. It can happen, however, that a probable prime is not prime. Let P (x) be the probability that n is composite under the hypotheses: 1. n is an odd integer chosen randomly from the range [2, x]; 2. b is an integer chosen randomly from the range [2, n − 2]; 3. n is a probable prime to the base b. Then P (x) ≤ (log x)−197 for x ≥ 1010000 . A different test can be obtained from the following theorem. Given the number n, find s and t with n − 1 = 2s t, with t odd. Then choose a random integer b from the range [2, n − 2]. If either bt = 1 (mod n)
i
b2 t = −1 (mod n),
or
for some i < s,
then n is a strong probable prime to the base b. Every odd prime must pass this test. If n > 1 is an odd composite, then the probability that it is a strong probable prime to the base b, when b is chosen randomly, is less than 1/4. A stronger test can be obtained by choosing k independent values for b in the range [2, n − 2] and checking the above relation for each value of b. Let Pk (x) be the probability that n is found to be a strong probable prime to each base b. Then Pk (x) ≤ 4−(k−1) P (x)/(1 − P (x)).
2.3.10 PRIME NUMBERS LESS THAN 10,000 The prime number p10n+k is found by looking at the row begining with n and at the column beginning with k. 0
1
2
3
4
5
6
7
8
9
1 2 3 4 5
29 71 113 173 229
2 31 73 127 179 233
3 37 79 131 181 239
5 41 83 137 191 241
7 43 89 139 193 251
11 47 97 149 197 257
13 53 101 151 199 263
17 59 103 157 211 269
19 61 107 163 223 271
23 67 109 167 227 277
6 7 8 9 10
281 349 409 463 541
283 353 419 467 547
293 359 421 479 557
307 367 431 487 563
311 373 433 491 569
313 379 439 499 571
317 383 443 503 577
331 389 449 509 587
337 397 457 521 593
347 401 461 523 599
11 12 13 14 15
601 659 733 809 863
607 661 739 811 877
613 673 743 821 881
617 677 751 823 883
619 683 757 827 887
631 691 761 829 907
641 701 769 839 911
643 709 773 853 919
647 719 787 857 929
653 727 797 859 937
©1996 CRC Press LLC
0
1
2
3
4
5
6
7
8
9
16 17 18 19 20
941 1013 1069 1151 1223
947 1019 1087 1153 1229
953 1021 1091 1163 1231
967 1031 1093 1171 1237
971 1033 1097 1181 1249
977 1039 1103 1187 1259
983 1049 1109 1193 1277
991 1051 1117 1201 1279
997 1061 1123 1213 1283
1009 1063 1129 1217 1289
21 22 23 24 25
1291 1373 1451 1511 1583
1297 1381 1453 1523 1597
1301 1399 1459 1531 1601
1303 1409 1471 1543 1607
1307 1423 1481 1549 1609
1319 1427 1483 1553 1613
1321 1429 1487 1559 1619
1327 1433 1489 1567 1621
1361 1439 1493 1571 1627
1367 1447 1499 1579 1637
26 27 28 29 30
1657 1733 1811 1889 1987
1663 1741 1823 1901 1993
1667 1747 1831 1907 1997
1669 1753 1847 1913 1999
1693 1759 1861 1931 2003
1697 1777 1867 1933 2011
1699 1783 1871 1949 2017
1709 1787 1873 1951 2027
1721 1789 1877 1973 2029
1723 1801 1879 1979 2039
31 32 33 34 35
2053 2129 2213 2287 2357
2063 2131 2221 2293 2371
2069 2137 2237 2297 2377
2081 2141 2239 2309 2381
2083 2143 2243 2311 2383
2087 2153 2251 2333 2389
2089 2161 2267 2339 2393
2099 2179 2269 2341 2399
2111 2203 2273 2347 2411
2113 2207 2281 2351 2417
36 37 38 39 40
2423 2531 2617 2687 2741
2437 2539 2621 2689 2749
2441 2543 2633 2693 2753
2447 2549 2647 2699 2767
2459 2551 2657 2707 2777
2467 2557 2659 2711 2789
2473 2579 2663 2713 2791
2477 2591 2671 2719 2797
2503 2593 2677 2729 2801
2521 2609 2683 2731 2803
41 42 43 44 45
2819 2903 2999 3079 3181
2833 2909 3001 3083 3187
2837 2917 3011 3089 3191
2843 2927 3019 3109 3203
2851 2939 3023 3119 3209
2857 2953 3037 3121 3217
2861 2957 3041 3137 3221
2879 2963 3049 3163 3229
2887 2969 3061 3167 3251
2897 2971 3067 3169 3253
46 47 48 49 50
3257 3331 3413 3511 3571
3259 3343 3433 3517 3581
3271 3347 3449 3527 3583
3299 3359 3457 3529 3593
3301 3361 3461 3533 3607
3307 3371 3463 3539 3613
3313 3373 3467 3541 3617
3319 3389 3469 3547 3623
3323 3391 3491 3557 3631
3329 3407 3499 3559 3637
51 52 53 54 55
3643 3727 3821 3907 3989
3659 3733 3823 3911 4001
3671 3739 3833 3917 4003
3673 3761 3847 3919 4007
3677 3767 3851 3923 4013
3691 3769 3853 3929 4019
3697 3779 3863 3931 4021
3701 3793 3877 3943 4027
3709 3797 3881 3947 4049
3719 3803 3889 3967 4051
56 57 58 59 60
4057 4139 4231 4297 4409
4073 4153 4241 4327 4421
4079 4157 4243 4337 4423
4091 4159 4253 4339 4441
4093 4177 4259 4349 4447
4099 4201 4261 4357 4451
4111 4211 4271 4363 4457
4127 4217 4273 4373 4463
4129 4219 4283 4391 4481
4133 4229 4289 4397 4483
61 62 63 64 65
4493 4583 4657 4751 4831
4507 4591 4663 4759 4861
4513 4597 4673 4783 4871
4517 4603 4679 4787 4877
4519 4621 4691 4789 4889
4523 4637 4703 4793 4903
4547 4639 4721 4799 4909
4549 4643 4723 4801 4919
4561 4649 4729 4813 4931
4567 4651 4733 4817 4933
66
4937
4943
4951
4957
4967
4969
4973
4987
4993
4999
©1996 CRC Press LLC
0
1
2
3
4
5
6
7
8
9
67 68 69 70
5003 5087 5179 5279
5009 5099 5189 5281
5011 5101 5197 5297
5021 5107 5209 5303
5023 5113 5227 5309
5039 5119 5231 5323
5051 5147 5233 5333
5059 5153 5237 5347
5077 5167 5261 5351
5081 5171 5273 5381
71 72 73 74 75
5387 5443 5521 5639 5693
5393 5449 5527 5641 5701
5399 5471 5531 5647 5711
5407 5477 5557 5651 5717
5413 5479 5563 5653 5737
5417 5483 5569 5657 5741
5419 5501 5573 5659 5743
5431 5503 5581 5669 5749
5437 5507 5591 5683 5779
5441 5519 5623 5689 5783
76 77 78 79 80
5791 5857 5939 6053 6133
5801 5861 5953 6067 6143
5807 5867 5981 6073 6151
5813 5869 5987 6079 6163
5821 5879 6007 6089 6173
5827 5881 6011 6091 6197
5839 5897 6029 6101 6199
5843 5903 6037 6113 6203
5849 5923 6043 6121 6211
5851 5927 6047 6131 6217
81 82 83 84 85
6221 6301 6367 6473 6571
6229 6311 6373 6481 6577
6247 6317 6379 6491 6581
6257 6323 6389 6521 6599
6263 6329 6397 6529 6607
6269 6337 6421 6547 6619
6271 6343 6427 6551 6637
6277 6353 6449 6553 6653
6287 6359 6451 6563 6659
6299 6361 6469 6569 6661
86 87 88 89 90
6673 6761 6833 6917 6997
6679 6763 6841 6947 7001
6689 6779 6857 6949 7013
6691 6781 6863 6959 7019
6701 6791 6869 6961 7027
6703 6793 6871 6967 7039
6709 6803 6883 6971 7043
6719 6823 6899 6977 7057
6733 6827 6907 6983 7069
6737 6829 6911 6991 7079
91 92 93 94 95
7103 7207 7297 7411 7499
7109 7211 7307 7417 7507
7121 7213 7309 7433 7517
7127 7219 7321 7451 7523
7129 7229 7331 7457 7529
7151 7237 7333 7459 7537
7159 7243 7349 7477 7541
7177 7247 7351 7481 7547
7187 7253 7369 7487 7549
7193 7283 7393 7489 7559
96 97 98 99 100
7561 7643 7723 7829 7919
7573 7649 7727 7841 7927
7577 7669 7741 7853 7933
7583 7673 7753 7867 7937
7589 7681 7757 7873 7949
7591 7687 7759 7877 7951
7603 7691 7789 7879 7963
7607 7699 7793 7883 7993
7621 7703 7817 7901 8009
7639 7717 7823 7907 8011
101 102 103 104 105
8017 8111 8219 8291 8387
8039 8117 8221 8293 8389
8053 8123 8231 8297 8419
8059 8147 8233 8311 8423
8069 8161 8237 8317 8429
8081 8167 8243 8329 8431
8087 8171 8263 8353 8443
8089 8179 8269 8363 8447
8093 8191 8273 8369 8461
8101 8209 8287 8377 8467
106 107 108 109 110
8501 8597 8677 8741 8831
8513 8599 8681 8747 8837
8521 8609 8689 8753 8839
8527 8623 8693 8761 8849
8537 8627 8699 8779 8861
8539 8629 8707 8783 8863
8543 8641 8713 8803 8867
8563 8647 8719 8807 8887
8573 8663 8731 8819 8893
8581 8669 8737 8821 8923
111 112 113 114 115
8929 9011 9109 9199 9283
8933 9013 9127 9203 9293
8941 9029 9133 9209 9311
8951 9041 9137 9221 9319
8963 9043 9151 9227 9323
8969 9049 9157 9239 9337
8971 9059 9161 9241 9341
8999 9067 9173 9257 9343
9001 9091 9181 9277 9349
9007 9103 9187 9281 9371
116 117
9377 9439
9391 9461
9397 9463
9403 9467
9413 9473
9419 9479
9421 9491
9431 9497
9433 9511
9437 9521
©1996 CRC Press LLC
0
1
2
3
4
5
6
7
8
9
118 119 120
9533 9631 9733
9539 9643 9739
9547 9649 9743
9551 9661 9749
9587 9677 9767
9601 9679 9769
9613 9689 9781
9619 9697 9787
9623 9719 9791
9629 9721 9803
121 122
9811 9887
9817 9901
9829 9907
9833 9923
9839 9929
9851 9931
9857 9941
9859 9949
9871 9967
9883 9973
2.3.11 PRIME NUMBERS OF SPECIAL FORMS 1. As of September 1996, the largest known prime numbers, in descending order, are Number 2
1257787 859433
−1
2 −1 2756839 − 1 391581 · 2216193 − 1 2216091 2 1 3 · 2157169 + 1 9 · 2149143 + 1 9 · 2147073 + 1 9 · 2145247 1 1 2132049 2 1 9 · 2127003 + 1 5 · 2125413 + 1 9 · 2114854 + 1 13 · 2114296 1 1 2110503 2 1
a Largest b Largest
Number of digits a
378,632 258,716 227,832 b 65,087 65,050 47,314 44,898 44,275 43,725 39,751 38,233 37,754 34,576 34,408 33,265
prime of the form 2n − 1, a Mersenne prime. non-Mersenne prime.
Other large primes include: • 134088 · 1015030 + 1 (largest prime of the form k · 10n + 1, has 15,036 digits) • 1011810 +1465641·105902 +1 (largest palindromic prime, has 11,811 digits) • 3610! − 1 (largest prime factorial minus 1, has 11,277 digits) • 2 · 313782 + 1 (largest prime of the form k · 3n + 1, has 6,576 digits) • 3476!! 2 1 (largest double factorial minus one, has 5,402 digits)
©1996 CRC Press LLC
2. All of the known Mersenne primes of the form 2n − 1 are given by the values (note that n must be prime itself), n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, and 859433. 3. The largest known twin primes are: 242206083· 238880 ± 1 (with 11,713 digits), 570918348 · 10 5120 ± 1 (with 5,129 digits), and 697053813 · 216352 ± 1 (with 4,932 digits). 4. The nth repunit is Rn = (10n −1)/9 = 11 . . . 11. The only known prime repunits correspond to n = 2, 19, 23, 317, 1031. 5. Prime numbers of the forms 2n ± a and 10n ± b : Large prime numbers of a specified size are sometimes needed. In the following table, for a given n, a± and b± are the least values such that 2n + a± and 10n + b± are probably primes (a probabilistic primality test was used). For example, for n = 3, the numbers 23 − 1 = 7, 23 + 3 = 11, 103 − 3 = 997, and 103 + 9 = 1009 are all prime.
©1996 CRC Press LLC
2n + a n
©1996 CRC Press LLC
10n + b
2 3 4 5 6
a− −1 −1 −3 −1 −3
a+ 1 3 1 5 3
b− −3 −3 −27 −9 −17
b+
7 8 9 10 11
−1 −5 −3 −3 −9
3 1 9 7 5
−9 −11 −63 −33 −23
19 7 7 19 3
12 13 14 15 16
−3 −1 −3 −19 −15
3 17 27 3 1
−11 −29 −27 −11 −63
39 37 31 37 61
17 18 19 20 50
−1 −5 −1 −3 −27
29 3 21 7 55
−3 −11 −39 −11 −57
3 3 51 39 151
100 150 200 300 400
−15 −3 −75 −153 −593
277 147 235 157 181
−797 −273 −189 −69 −513
267 67 357 331 69
500 600 700 800 900 1000
−863 −95 −1113 −105 −207 −1245
55 187 535 25 693 297
−1037 −1791 −2313 −1007 −773 −1769
961 543 7 1537 1873 453
1 9 7 3 3
6. Define p# to be the product of the prime numbers less than or equal to p. Form n! + 1 n! − 1 p# + 1 p# − 1
Values of n for which the form is prime 1, 2, 3, 11, 27, 37, 41, 73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, . . . 3, 4, 6, 7, 12, 14, 30, 32, 33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, . . . 2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, . . . 3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, . . .
2.3.12 PRIME PERIOD LENGTHS When an integer is divided by a prime p, and the result is represented as a decimal, the number of digits in the period is called λ(p). For example, 1/7 = 0. 142856 142856 . . . so that λ(7) = 6, and 1/37 = 0. 027 027 . . . so that λ(37) = 3. The period λ(p) always divides p − 1. Prime Period length
3 5 7 1 1 6
11 2
13 17 19 23 29 31 6 16 18 22 28 15
37 3
2.3.13 FACTORIZATION TABLE The following is a list of the factors of numbers up to 1,000. When a number is prime, it is shown in a boldface.
©1996 CRC Press LLC
©1996 CRC Press LLC
0
1
2
3
4 22
5
6
7
8 23
9 32
0 1 2 3 4
2·5 22 · 5 2·3·5 23 · 5
1 11 3·7 31 41
2 22 · 3 2 · 11 25 2·3·7
3 13 23 3 · 11 43
2·7 23 · 3 2 · 17 22 · 11
5 3·5 52 5·7 32 · 5
2·3 24 2 · 13 22 · 32 2 · 23
7 17 33 37 47
2 · 32 22 · 7 2 · 19 24 · 3
19 29 3 · 13 72
5 6 7 8 9
2 · 52 22 · 3 · 5 2·5·7 24 · 5 2 · 32 · 5
3 · 17 61 71 34 7 · 13
22 · 13 2 · 31 23 · 32 2 · 41 22 · 23
53 32 · 7 73 83 3 · 31
2 · 33 26 2 · 37 22 · 3 · 7 2 · 47
5 · 11 5 · 13 3 · 52 5 · 17 5 · 19
23 · 7 2 · 3 · 11 22 · 19 2 · 43 25 · 3
3 · 19 67 7 · 11 3 · 29 97
2 · 29 22 · 17 2 · 3 · 13 23 · 11 2 · 72
59 3 · 23 79 89 32 · 11
10 11 12 13 14
22 · 52 2 · 5 · 11 23 · 3 · 5 2 · 5 · 13 22 · 5 · 7
101 3 · 37 112 131 3 · 47
2 · 3 · 17 24 · 7 2 · 61 22 · 3 · 11 2 · 71
103 113 3 · 41 7 · 19 11 · 13
23 · 13 2 · 3 · 19 22 · 31 2 · 67 24 · 32
3·5·7 5 · 23 53 33 · 5 5 · 29
2 · 53 22 · 29 2 · 32 · 7 23 · 17 2 · 73
107 32 · 13 127 137 3 · 72
22 · 33 2 · 59 27 2 · 3 · 23 22 · 37
109 7 · 17 3 · 43 139 149
15 16 17 18 19
2 · 3 · 52 25 · 5 2 · 5 · 17 22 · 32 · 5 2 · 5 · 19
151 7 · 23 32 · 19 181 191
23 · 19 2 · 34 22 · 43 2 · 7 · 13 26 · 3
32 · 17 163 173 3 · 61 193
2 · 7 · 11 22 · 41 2 · 3 · 29 23 · 23 2 · 97
5 · 31 3 · 5 · 11 52 · 7 5 · 37 3 · 5 · 13
22 · 3 · 13 2 · 83 24 · 11 2 · 3 · 31 22 · 72
157 167 3 · 59 11 · 17 197
2 · 79 23 · 3 · 7 2 · 89 22 · 47 2 · 32 · 11
3 · 53 132 179 33 · 7 199
20 21 22 23 24
23 · 52 2·3·5·7 22 · 5 · 11 2 · 5 · 23 24 · 3 · 5
3 · 67 211 13 · 17 3 · 7 · 11 241
2 · 101 22 · 53 2 · 3 · 37 23 · 29 2 · 112
7 · 29 3 · 71 223 233 35
22 · 3 · 17 2 · 107 25 · 7 2 · 32 · 13 22 · 61
5 · 41 5 · 43 32 · 52 5 · 47 5 · 72
2 · 103 23 · 33 2 · 113 22 · 59 2 · 3 · 41
32 · 23 7 · 31 227 3 · 79 13 · 19
24 · 13 2 · 109 22 · 3 · 19 2 · 7 · 17 23 · 31
11 · 19 3 · 73 229 239 3 · 83
©1996 CRC Press LLC
©1996 CRC Press LLC
0 · 53
1
2 22
· 32
3
4
5
6 28
7
8
9
25 26 27 28 29
2 22 · 5 · 13 2 · 33 · 5 23 · 5 · 7 2 · 5 · 29
251 32 · 29 271 281 3 · 97
·7 2 · 131 24 · 17 2 · 3 · 47 22 · 73
11 · 23 263 3 · 7 · 13 283 293
2 · 127 23 · 3 · 11 2 · 137 22 · 71 2 · 3 · 72
3 · 5 · 17 5 · 53 52 · 11 3 · 5 · 19 5 · 59
2 · 7 · 19 22 · 3 · 23 2 · 11 · 13 23 · 37
257 3 · 89 277 7 · 41 33 · 11
2 · 3 · 43 22 · 67 2 · 139 25 · 32 2 · 149
7 · 37 269 32 · 31 172 13 · 23
30 31 32 33 34
22 · 3 · 52 2 · 5 · 31 26 · 5 2 · 3 · 5 · 11 22 · 5 · 17
7 · 43 311 3 · 107 331 11 · 31
2 · 151 23 · 3 · 13 2 · 7 · 23 22 · 83 2 · 32 · 19
3 · 101 313 17 · 19 32 · 37 73
24 · 19 2 · 157 22 · 34 2 · 167 23 · 43
5 · 61 32 · 5 · 7 52 · 13 5 · 67 3 · 5 · 23
2 · 32 · 17 22 · 79 2 · 163 24 · 3 · 7 2 · 173
307 317 3 · 109 337 347
22 · 7 · 11 2 · 3 · 53 23 · 41 2 · 132 22 · 3 · 29
3 · 103 11 · 29 7 · 47 3 · 113 349
35 36 37 38 39
2 · 52 · 7 23 · 32 · 5 2 · 5 · 37 22 · 5 · 19 2 · 3 · 5 · 13
33 · 13 192 7 · 53 3 · 127 17 · 23
25 · 11 2 · 181 22 · 3 · 31 2 · 191 23 · 72
353 3 · 112 373 383 3 · 131
2 · 3 · 59 22 · 7 · 13 2 · 11 · 17 27 · 3 2 · 197
5 · 71 5 · 73 3 · 53 5 · 7 · 11 5 · 79
22 · 89 2 · 3 · 61 23 · 47 2 · 193 22 · 32 · 11
3 · 7 · 17 367 13 · 29 32 · 43 397
2 · 179 24 · 23 2 · 33 · 7 22 · 97 2 · 199
359 32 · 41 379 389 3 · 7 · 19
40 41 42 43 44 45
24 · 52 2 · 5 · 41 22 · 3 · 5 · 7 2 · 5 · 43 23 · 5 · 11 2 · 32 · 52
401 3 · 137 421 431 32 · 72 11 · 41
2 · 3 · 67 22 · 103 2 · 211 24 · 33 2 · 13 · 17 22 · 113
13 · 31 7 · 59 32 · 47 433 443 3 · 151
22 · 101 2 · 32 · 23 23 · 53 2 · 7 · 31 22 · 3 · 37 2 · 227
34 · 5 5 · 83 52 · 17 3 · 5 · 29 5 · 89 5 · 7 · 13
2 · 7 · 29 25 · 13 2 · 3 · 71 22 · 109 2 · 223 23 · 3 · 19
11 · 37 3 · 139 7 · 61 19 · 23 3 · 149 457
23 · 3 · 17 2 · 11 · 19 22 · 107 2 · 3 · 73 26 · 7 2 · 229
409 419 3 · 11 · 13 439 449 33 · 17
46 47 48 49
22 · 5 · 23 2 · 5 · 47 25 · 3 · 5 2 · 5 · 72
461 3 · 157 13 · 37 491
2 · 3 · 7 · 11 23 · 59 2 · 241 22 · 3 · 41
463 11 · 43 3 · 7 · 23 17 · 29
24 · 29 2 · 3 · 79 22 · 112 2 · 13 · 19
3 · 5 · 31 52 · 19 5 · 97 32 · 5 · 11
2 · 233 22 · 7 · 17 2 · 35 24 · 31
467 32 · 53 487 7 · 71
22 · 32 · 13 2 · 239 23 · 61 2 · 3 · 83
7 · 67 479 3 · 163 499
©1996 CRC Press LLC
©1996 CRC Press LLC
0 · 53
1
2
3
4 23
· 32
5
6
7
8
3 · 132
22
9
50 51 52 53 54
22
2 · 3 · 5 · 17 23 · 5 · 13 2 · 5 · 53 22 · 33 · 5
3 · 167 7 · 73 521 32 · 59 541
2 · 251 29 2 · 32 · 29 22 · 7 · 19 2 · 271
503 33 · 19 523 13 · 41 3 · 181
·7 2 · 257 22 · 131 2 · 3 · 89 25 · 17
5 · 101 5 · 103 3 · 52 · 7 5 · 107 5 · 109
2 · 11 · 23 22 · 3 · 43 2 · 263 23 · 67 2 · 3 · 7 · 13
11 · 47 17 · 31 3 · 179 547
· 127 2 · 7 · 37 24 · 3 · 11 2 · 269 22 · 137
509 3 · 173 232 72 · 11 32 · 61
55 56 57 58 59
2 · 52 · 11 24 · 5 · 7 2 · 3 · 5 · 19 22 · 5 · 29 2 · 5 · 59
19 · 29 3 · 11 · 17 571 7 · 83 3 · 197
23 · 3 · 23 2 · 281 22 · 11 · 13 2 · 3 · 97 24 · 37
7 · 79 563 3 · 191 11 · 53 593
2 · 277 22 · 3 · 47 2 · 7 · 41 23 · 73 2 · 33 · 11
3 · 5 · 37 5 · 113 52 · 23 32 · 5 · 13 5 · 7 · 17
22 · 139 2 · 283 26 · 32 2 · 293 22 · 149
557 34 · 7 577 587 3 · 199
2 · 32 · 31 23 · 71 2 · 172 22 · 3 · 72 2 · 13 · 23
13 · 43 569 3 · 193 19 · 31 599
60 61 62 63 64
23 · 3 · 52 2 · 5 · 61 22 · 5 · 31 2 · 32 · 5 · 7 27 · 5
601 13 · 47 33 · 23 631 641
2 · 7 · 43 22 · 32 · 17 2 · 311 23 · 79 2 · 3 · 107
32 · 67 613 7 · 89 3 · 211 643
22 · 151 2 · 307 24 · 3 · 13 2 · 317 22 · 7 · 23
5 · 112 3 · 5 · 41 54 5 · 127 3 · 5 · 43
2 · 3 · 101 23 · 7 · 11 2 · 313 22 · 3 · 53 2 · 17 · 19
607 617 3 · 11 · 19 72 · 13 647
25 · 19 2 · 3 · 103 22 · 157 2 · 11 · 29 23 · 34
3 · 7 · 29 619 17 · 37 32 · 71 11 · 59
65 66 67 68 69
2 · 52 · 13 22 · 3 · 5 · 11 2 · 5 · 67 23 · 5 · 17 2 · 3 · 5 · 23
3 · 7 · 31 661 11 · 61 3 · 227 691
22 · 163 2 · 331 25 · 3 · 7 2 · 11 · 31 22 · 173
653 3 · 13 · 17 673 683 32 · 7 · 11
2 · 3 · 109 23 · 83 2 · 337 22 · 32 · 19 2 · 347
5 · 131 5 · 7 · 19 33 · 52 5 · 137 5 · 139
24 · 41 2 · 32 · 37 22 · 132 2 · 73 23 · 3 · 29
32 · 73 23 · 29 677 3 · 229 17 · 41
2 · 7 · 47 22 · 167 2 · 3 · 113 24 · 43 2 · 349
659 3 · 223 7 · 97 13 · 53 3 · 233
70 71 72 73 74
22 · 52 · 7 2 · 5 · 71 24 · 32 · 5 2 · 5 · 73 22 · 5 · 37
701 32 · 79 7 · 103 17 · 43 3 · 13 · 19
2 · 33 · 13 23 · 89 2 · 192 22 · 3 · 61 2 · 7 · 53
19 · 37 23 · 31 3 · 241 733 743
26 · 11 2 · 3 · 7 · 17 22 · 181 2 · 367 23 · 3 · 31
3 · 5 · 47 5 · 11 · 13 52 · 29 3 · 5 · 72 5 · 149
2 · 353 22 · 179 2 · 3 · 112 25 · 23 2 · 373
7 · 101 3 · 239 727 11 · 67 32 · 83
22 · 3 · 59 2 · 359 23 · 7 · 13 2 · 32 · 41 22 · 11 · 17
709 719 36 739 7 · 107
©1996 CRC Press LLC
©1996 CRC Press LLC
0 75 76 77 78 79
· 3 · 53
2 23 · 5 · 19 2 · 5 · 7 · 11 22 · 3 · 5 · 13 2 · 5 · 79
80 81 82 83 84
1
2
3
4
5
7
8
9
·7 2 · 383 23 · 97 2 · 3 · 131 22 · 199
757 13 · 59 3 · 7 · 37 787 797
2 · 379 28 · 3 2 · 389 22 · 197 2 · 3 · 7 · 19
3 · 11 · 23 769 19 · 41 3 · 263 17 · 47
5 · 7 · 23 5 · 163 3 · 52 · 11 5 · 167 5 · 132
2 · 13 · 31 24 · 3 · 17 2 · 7 · 59 22 · 11 · 19 2 · 32 · 47
3 · 269 19 · 43 827 33 · 31 7 · 112
23 · 101 2 · 409 22 · 32 · 23 2 · 419 24 · 53
809 32 · 7 · 13 829 839 3 · 283
2 · 7 · 61 25 · 33 2 · 19 · 23 22 · 13 · 17 2 · 3 · 149
32 · 5 · 19 5 · 173 53 · 7 3 · 5 · 59 5 · 179
23 · 107 2 · 433 22 · 3 · 73 2 · 443 27 · 7
857 3 · 172 877 887 3 · 13 · 23
2 · 3 · 11 · 13 22 · 7 · 31 2 · 439 23 · 3 · 37 2 · 449
859 11 · 79 3 · 293 7 · 127 29 · 31
3 · 7 · 43 11 · 83 13 · 71 3 · 311 23 · 41
23 · 113 2 · 457 22 · 3 · 7 · 11 2 · 467 24 · 59
5 · 181 3 · 5 · 61 52 · 37 5 · 11 · 17 33 · 5 · 7
2 · 3 · 151 22 · 229 2 · 463 23 · 32 · 13 2 · 11 · 43
907 7 · 131 32 · 103 937 947
22 · 227 2 · 33 · 17 25 · 29 2 · 7 · 67 22 · 3 · 79
32 · 101 919 929 3 · 313 13 · 73
953 32 · 107 7 · 139 983 3 · 331
2 · 32 · 53 22 · 241 2 · 487 23 · 3 · 41 2 · 7 · 71
5 · 191 5 · 193 3 · 52 · 13 5 · 197 5 · 199
22 · 239 2 · 3 · 7 · 23 24 · 61 2 · 17 · 29 22 · 3 · 83
3 · 11 · 29 967 977 3 · 7 · 47 997
2 · 479 23 · 112 2 · 3 · 163 22 · 13 · 19 2 · 499
7 · 137 3 · 17 · 19 11 · 89 23 · 43 33 · 37
751 761 3 · 257 11 · 71 7 · 113
24
· 47 2 · 3 · 127 22 · 193 2 · 17 · 23 23 · 32 · 11
3 · 251 7 · 109 773 33 · 29 13 · 61
2 · 13 · 29 22 · 191 2 · 32 · 43 24 · 72 2 · 397
5 · 151 32 · 5 · 17 52 · 31 5 · 157 3 · 5 · 53
25 · 52 2 · 34 · 5 22 · 5 · 41 2 · 5 · 83 23 · 3 · 5 · 7
32 · 89 811 821 3 · 277 292
2 · 401 22 · 7 · 29 2 · 3 · 137 26 · 13 2 · 421
11 · 73 3 · 271 823 72 · 17 3 · 281
22 · 3 · 67 2 · 11 · 37 23 · 103 2 · 3 · 139 22 · 211
85 86 87 88 89
2 · 52 · 17 22 · 5 · 43 2 · 3 · 5 · 29 24 · 5 · 11 2 · 5 · 89
23 · 37 3 · 7 · 41 13 · 67 881 34 · 11
22 · 3 · 71 2 · 431 23 · 109 2 · 32 · 72 22 · 223
853 863 32 · 97 883 19 · 47
90 91 92 93 94
22 · 32 · 52 2 · 5 · 7 · 13 23 · 5 · 23 2 · 3 · 5 · 31 22 · 5 · 47
17 · 53 911 3 · 307 72 · 19 941
2 · 11 · 41 24 · 3 · 19 2 · 461 22 · 233 2 · 3 · 157
95 96 97 98 99
2 · 52 · 19 26 · 3 · 5 2 · 5 · 97 22 · 5 · 72 2 · 32 · 5 · 11
3 · 317 312 971 32 · 109 991
23 · 7 · 17 2 · 13 · 37 22 · 35 2 · 491 25 · 31
©1996 CRC Press LLC
6 22
· 33
2.3.14 FACTORIZATION OF 2m − 1 23 − 1 = 7 24 − 1 = 3 × 5 25 − 1 = 31 26 − 1 = 32 × 7 27 − 1 = 127
219 − 1 = 524287 220 − 1 = 3 × 52 × 11 × 31 × 41 221 − 1 = 72 × 127 × 337 222 − 1 = 3 × 23 × 89 × 683 223 − 1 = 47 × 178481
28 − 1 = 3 × 5 × 17 29 − 1 = 7 × 73 210 − 1 = 3 × 11 × 31 211 − 1 = 23 × 89 212 − 1 = 32 × 5 × 7 × 13
224 − 1 = 32 × 5 × 7 × 13 × 17 × 241 225 − 1 = 31 × 601 × 1801 226 − 1 = 3 × 2731 × 8191 227 − 1 = 7 × 73 × 262657 228 − 1 = 3 × 5 × 29 × 43 × 113 × 127
213 − 1 = 8191 214 − 1 = 3 × 43 × 127 215 − 1 = 7 × 31 × 151 216 − 1 = 3 × 5 × 17 × 257 217 − 1 = 131071
229 − 1 = 233 × 1103 × 2089 230 − 1 = 32 × 7 × 11 × 31 × 151 × 331 231 − 1 = 2147483647 232 − 1 = 3 × 5 × 17 × 257 × 65537 233 − 1 = 7 × 23 × 89 × 599479
218 − 1 = 33 × 7 × 19 × 73
234 − 1 = 3 × 43691 × 131071
2.3.15 MAGIC SQUARES A magic square is a square array of integers with the property that the sum of the integers in each row or column is the same. If (c, n) = (d, n) = (e, n) = (f, n) = (cf − en, n) = 1, then the array A = (aij ) will be magic (and use the n2 numbers 0, 1, . . . , n2 − 1) if aij = k with k k (mod n) and j ≡ dk + f (mod n) i ≡ ck + e n n For example, with c = 1, d = e = f = 2, and n = 3, a magic square is 6 2 4
1 3 8
5 7 0
2.3.16 TOTIENT FUNCTION Definitions • φ(n)
the totient function; is the number of integers not exceeding and relatively prime to n.
• σ (n)
is the sum of the divisors of n.
• τ (n)
is the number of divisors of n.
©1996 CRC Press LLC
Define σk (n) to be the k th divisor function, the sum of the k th powers of the divisors of n. Then φ(n) = σ0 (n) and σ (n) = σ1 (n). A perfect number n satisfies σ (n) = 2n.
Example The numbers less than 6 and relatively prime to 6 are {1, 5}. Hence φ(6) = 2. The divisors of 6 are {1, 2, 3, 6}. There are τ (6) = 4 divisors. The sum of these numbers is σ (6) = 1 + 2 + 3 + 6 = 12.
Properties 1. φ is a multiplicative function: if (n, m) = 1, then φ(nm) = φ(m)φ(n). 2. Gauss’s theorem states: n = d|n φ(d). 3. When n = i piαi . σk (n) =
dk =
d|n
pk(αi +1) − 1 i
i
pi − 1
4. Generating functions include ∞ σk (n) n=0 ∞ n=0
ns
= ζ (s)ζ (s − k)
φ(n) ζ (s − 1) = ns ζ (s)
5. The positive integer n is an even perfect number if, and only if, n = 2m−1 (2m −1), where m is a positive integer such that Mm = 2m − 1 is prime (Mm is called a Mersenne prime). The sequence of perfect numbers is {6, 28, 496, . . . } (corresponding to m = 2, 3, 5, . . . ).
©1996 CRC Press LLC
Table of Values
n φ(n) τ (n) σ (n) 1 0 1 1 2 1 2 3 3 2 2 4 4 2 3 7 5 4 2 6
n φ(n) τ (n) σ (n) 46 22 4 72 47 46 2 48 48 16 10 124 49 42 3 57 50 20 6 93
n φ(n) τ (n) σ (n) 91 72 4 112 92 44 6 168 93 60 4 128 94 46 4 144 95 72 4 120
n φ(n) τ (n) σ (n) 136 64 8 270 137 136 2 138 138 44 8 288 139 138 2 140 140 48 12 336
6 7 8 9 10
2 6 4 6 4
4 2 4 3 4
12 8 15 13 18
51 52 53 54 55
32 24 52 18 40
4 6 2 8 4
72 98 54 120 72
96 97 98 99 100
32 96 42 60 40
12 2 6 6 9
252 98 171 156 217
141 92 142 70 143 120 144 48 145 112
4 4 4 15 4
192 216 168 403 180
11 12 13 14 15
10 4 12 6 8
2 6 2 4 4
12 28 14 24 24
56 57 58 59 60
24 36 28 58 16
8 4 4 2 12
120 80 90 60 168
101 100 102 32 103 102 104 48 105 48
2 8 2 8 8
102 216 104 210 192
146 72 147 84 148 72 149 148 150 40
4 6 6 2 12
222 228 266 150 372
16 17 18 19 20
8 16 6 18 8
5 2 6 2 6
31 18 39 20 42
61 62 63 64 65
60 30 36 32 48
2 4 6 7 4
62 96 104 127 84
106 52 107 106 108 36 109 108 110 40
4 2 12 2 8
162 108 280 110 216
151 150 152 72 153 96 154 60 155 120
2 8 6 8 4
152 300 234 288 192
21 22 23 24 25
12 10 22 8 20
4 4 2 8 3
32 36 24 60 31
66 67 68 69 70
20 66 32 44 24
8 2 6 4 8
144 68 126 96 144
111 72 112 48 113 112 114 36 115 88
4 10 2 8 4
152 248 114 240 144
156 48 157 156 158 78 159 104 160 64
12 2 4 4 12
392 158 240 216 378
26 27 28 29 30
12 18 12 28 8
4 4 6 2 8
42 40 56 30 72
71 72 73 74 75
70 24 72 36 40
2 12 2 4 6
72 195 74 114 124
116 117 118 119 120
56 72 58 96 32
6 6 4 4 16
210 182 180 144 360
161 132 162 54 163 162 164 80 165 80
4 10 2 6 8
192 363 164 294 288
31 32 33 34 35
30 16 20 16 24
2 6 4 4 4
32 63 48 54 48
76 77 78 79 80
36 60 24 78 32
6 4 8 2 10
140 96 168 80 186
121 110 122 60 123 80 124 60 125 100
3 4 4 6 4
133 186 168 224 156
166 82 167 166 168 48 169 156 170 64
4 2 16 3 8
252 168 480 183 324
36 37 38 39 40
12 36 18 24 16
9 2 4 4 8
91 38 60 56 90
81 82 83 84 85
54 40 82 24 64
5 4 2 12 4
121 126 84 224 108
126 36 127 126 128 64 129 84 130 48
12 2 8 4 8
312 128 255 176 252
171 108 172 84 173 172 174 56 175 120
6 6 2 8 6
260 308 174 360 248
41 42 43 44 45
40 12 42 20 24
2 8 2 6 6
42 96 44 84 78
86 87 88 89 90
42 56 40 88 24
4 4 8 2 12
132 120 180 90 234
131 130 132 40 133 108 134 66 135 72
2 12 4 4 8
132 336 160 204 240
176 80 177 116 178 88 179 178 180 48
10 4 4 2 18
372 240 270 180 546
©1996 CRC Press LLC
n 181 182 183 184 185
φ(n) τ (n) σ (n) 180 2 182 72 8 336 120 4 248 88 8 360 144 4 228
n 226 227 228 229 230
φ(n) τ (n) σ (n) 112 4 342 226 2 228 72 12 560 228 2 230 88 8 432
n 271 272 273 274 275
φ(n) τ (n) σ (n) 270 2 272 128 10 558 144 8 448 136 4 414 200 6 372
186 60 187 160 188 92 189 108 190 72
8 4 6 8 8
384 216 336 320 360
231 232 233 234 235
120 112 232 72 184
8 8 2 12 4
384 450 234 546 288
276 88 277 276 278 138 279 180 280 96
191 190 192 64 193 192 194 96 195 96
2 14 2 4 8
192 508 194 294 336
236 116 237 156 238 96 239 238 240 64
6 4 8 2 20
420 320 432 240 744
281 282 283 284 285
196 84 197 196 198 60 199 198 200 80
9 2 12 2 12
399 198 468 200 465
241 242 243 244 245
240 110 162 120 168
2 6 6 6 6
242 399 364 434 342
201 202 203 204 205
132 100 168 64 160
4 4 4 12 4
272 306 240 504 252
246 247 248 249 250
80 216 120 164 100
8 4 8 4 8
206 102 207 132 208 96 209 180 210 48
4 6 10 4 16
312 312 434 240 576
251 252 253 254 255
250 72 220 126 128
210 104 140 106 168
2 6 4 4 4
212 378 288 324 264
216 72 217 180 218 108 219 144 220 80
16 4 4 4 12
221 192 222 72 223 222 224 96 225 120
4 8 2 12 9
211 212 213 214 215
n 316 317 318 319 320
φ(n) τ (n) σ (n) 156 6 560 316 2 318 104 8 648 280 4 360 128 14 762
12 2 4 6 16
672 278 420 416 720
321 322 323 324 325
212 132 288 108 240
4 8 4 15 6
432 576 360 847 434
280 92 282 140 144
2 8 2 6 8
282 576 284 504 480
326 327 328 329 330
162 216 160 276 80
4 4 8 4 16
492 440 630 384 864
286 287 288 289 290
120 240 96 272 112
8 4 18 3 8
504 336 819 307 540
331 332 333 334 335
330 164 216 166 264
2 6 6 4 4
332 588 494 504 408
504 280 480 336 468
291 292 293 294 295
192 144 292 84 232
4 6 2 12 4
392 518 294 684 360
336 337 338 339 340
96 336 156 224 128
20 2 6 4 12
992 338 549 456 756
2 18 4 4 8
252 728 288 384 432
296 297 298 299 300
144 180 148 264 80
8 8 4 4 18
570 480 450 336 868
341 342 343 344 345
300 108 294 168 176
4 12 4 8 8
384 780 400 660 576
256 128 257 256 258 84 259 216 260 96
9 2 8 4 12
511 258 528 304 588
301 302 303 304 305
252 150 200 144 240
4 4 4 10 4
352 456 408 620 372
346 347 348 349 350
172 346 112 348 120
4 2 12 2 12
522 348 840 350 744
600 256 330 296 504
261 262 263 264 265
168 130 262 80 208
6 4 2 16 4
390 396 264 720 324
306 307 308 309 310
96 306 120 204 120
12 2 12 4 8
702 308 672 416 576
351 352 353 354 355
216 160 352 116 280
8 12 2 8 4
560 756 354 720 432
252 456 224 504 403
266 267 268 269 270
108 176 132 268 72
8 4 6 2 16
480 360 476 270 720
311 312 313 314 315
310 96 312 156 144
2 16 2 4 12
312 840 314 474 624
356 357 358 359 360
176 192 178 358 96
6 630 8 576 4 540 2 360 24 1170
©1996 CRC Press LLC
n 361 362 363 364 365
φ(n) τ (n) σ (n) 342 3 381 180 4 546 220 6 532 144 12 784 288 4 444
n 406 407 408 409 410
φ(n) τ (n) σ (n) 168 8 720 360 4 456 128 16 1080 408 2 410 160 8 756
n 451 452 453 454 455
φ(n) τ (n) σ (n) 400 4 504 224 6 798 300 4 608 226 4 684 288 8 672
n 496 497 498 499 500
φ(n) τ (n) σ (n) 240 10 992 420 4 576 164 8 1008 498 2 500 200 12 1092
366 367 368 369 370
120 366 176 240 144
8 2 10 6 8
744 368 744 546 684
411 412 413 414 415
272 204 348 132 328
4 6 4 12 4
552 728 480 936 504
456 457 458 459 460
144 456 228 288 176
16 1200 2 458 4 690 8 720 12 1008
501 502 503 504 505
332 250 502 144 400
4 672 4 756 2 504 24 1560 4 612
371 372 373 374 375
312 120 372 160 200
4 12 2 8 8
432 896 374 648 624
416 417 418 419 420
192 276 180 418 96
12 882 4 560 8 720 2 420 24 1344
461 462 463 464 465
460 120 462 224 240
2 462 16 1152 2 464 10 930 8 768
506 507 508 509 510
220 312 252 508 128
8 864 6 732 6 896 2 510 16 1296
376 377 378 379 380
184 336 108 378 144
8 4 16 2 12
720 420 960 380 840
421 422 423 424 425
420 210 276 208 320
2 4 6 8 6
422 636 624 810 558
466 467 468 469 470
232 466 144 396 184
4 702 2 468 18 1274 4 544 8 864
511 512 513 514 515
432 256 324 256 408
4 592 10 1023 8 800 4 774 4 624
381 382 383 384 385
252 190 382 128 240
4 4 2 16 8
512 576 384 1020 576
426 427 428 429 430
140 360 212 240 168
8 4 6 8 8
864 496 756 672 792
471 472 473 474 475
312 232 420 156 360
632 900 528 960 620
516 517 518 519 520
168 460 216 344 192
12 1232 4 576 8 912 4 696 16 1260
386 387 388 389 390
192 252 192 388 96
4 6 6 2 16
582 572 686 390 1008
431 432 433 434 435
430 144 432 180 224
2 432 20 1240 2 434 8 768 8 720
476 477 478 479 480
192 312 238 478 128
12 1008 6 702 4 720 2 480 24 1512
521 522 523 524 525
520 168 522 260 240
2 522 12 1170 2 524 6 924 12 992
391 392 393 394 395
352 168 260 196 312
4 12 4 4 4
432 855 528 594 480
436 437 438 439 440
216 396 144 438 160
6 770 4 480 8 888 2 440 16 1080
481 482 483 484 485
432 240 264 220 384
532 726 768 931 588
526 527 528 529 530
262 480 160 506 208
4 792 4 576 20 1488 3 553 8 972
396 397 398 399 400
120 396 198 216 160
18 2 4 8 15
1092 398 600 640 961
441 442 443 444 445
252 192 442 144 352
9 741 8 756 2 444 12 1064 4 540
486 487 488 489 490
162 486 240 324 168
12 1092 2 488 8 930 4 656 12 1026
531 532 533 534 535
348 216 480 176 424
6 780 12 1120 4 588 8 1080 4 648
401 402 403 404 405
400 132 360 200 216
2 8 4 6 10
402 816 448 714 726
446 447 448 449 450
222 296 192 448 120
4 672 4 600 14 1016 2 450 18 1209
491 492 493 494 495
490 160 448 216 240
2 492 12 1176 4 540 8 840 12 936
536 537 538 539 540
264 356 268 420 144
8 1020 4 720 4 810 6 684 24 1680
©1996 CRC Press LLC
4 8 4 8 6
4 4 8 9 4
2.4 VECTOR ALGEBRA
2.4.1 NOTATION FOR VECTORS AND SCALARS A vector is an ordered n-tuple of values. A vector is usually represented by a lower case, bold faced letter, such as v. The individual components of a vector v are typically denoted by a lower case letter along with a subscript identifying the relative position of the component in the vector, such as v = [v1 , v2 , . . . , vn ]. In this case, the vector is said to be n-dimensional. If the n individual components of the vector are real numbers, then v ∈ Rn . Likewise, if the n components of v are complex, then v ∈ Cn . Two vectors, v1 and v2 , are said to be equal if they have exactly the same components. A negative vector, written as −v, is one that acts in direction opposite to v, but is of equal magnitude.
2.4.2 PHYSICAL VECTORS Any quantity that is completely determined by its magnitude is called a scalar. For example, mass, density, and temperature are scalars. Any quantity that is completely determined by its magnitude and direction is called, in physics, a vector. We often use a three-dimensional vector to represent a physical vector. Examples of physical vectors include velocity, acceleration, and force. A physical vector is represented by a directed line segment, the length of which represents the magnitude of the vector. Two vectors are said to be parallel if they have exactly the same direction, i.e., the angle between the two vectors equals zero.
2.4.3 FUNDAMENTAL DEFINITIONS 1. A row vector is a vector whose components are aligned horizontally. A column vector has its components aligned vertically. The transpose operator, denoted by the superscript T , switches the orientation of a vector between horizontal and vertical.
EXAMPLE 2.4.1
v = [1, 2, 3, 4], row vector
©1996 CRC Press LLC
1 2 T v = 3 , 4 column vector
(vT )T = [1, 2, 3, 4]. row vector
Vectors are traditionally written with either rounded braces or with square brackets. 2. Two vectors, vi and vj , are said to be orthogonal if vTi · vj = 0, where the operation “·” denotes an inner product. 3. A set of vectors {v1 , . . . , vn } is said to be orthogonal if vTi · vj = 0 for all i = j . 4. A set of orthogonal vectors {v1 , . . . , vm } is said to be orthonormal if, in addition to possessing the property of orthogonality, the set possesses the property that vTi · vi = 1 for all 1 ≤ i ≤ m.
2.4.4 LAWS OF VECTOR ALGEBRA 1. The vector sum of v1 and v2 , represented by v1 + v2 , results in another vector of the same dimension, and is calculated by simply adding corresponding vector components, e.g., if v1 , v2 ∈ Rn , then v1 + v2 = [v11 + v21 , . . . , v1n + v2n ]. 2. The vector subtraction of v2 from v1 , represented by v1 − v2 , is equivalent to the addition of v1 and −v2 . 3. If r is a scalar, then rv or vr represents a scaling of the vector v in the same direction as v, since the multiplicative scalar is distributed to each component of v. 4. If 0 ≤ r < 1, then the scalar multiplication rv shrinks the length of v, multiplication by r = 1 leaves v unchanged, and, if r > 1, then rv stretches the length of v. When r < 0, multiplication of v by r has the same effects on the magnitude, or length of v, but results in a vector oriented in the direction opposite to v.
EXAMPLE 2.4.2 1 4 4 0 = 0 , 3 12
1 −4 −4 0 = 0 . 3 −12
5. If r and s are scalars, and v1 , v2 , and v3 are vectors, the following rules of algebra are valid: v1 + v2 = v2 + v1 , (r + s)v1 = rv1 + sv1 = v1 r + v1 s = v1 (r + s), r(v1 + v2 ) = rv1 + rv2 , and v1 + (v2 + v3 ) = (v1 + v2 ) + v3 = v1 + v2 + v3 .
©1996 CRC Press LLC
(2.4.1)
2.4.5 VECTOR NORMS 1. A norm is the vector analog of absolute value for real scalars. Norms provide a distance measure for a vector space. 2. A vector norm applied to a vector v is denoted by a double bar notation v. 3. An n-dimensional real number space Rn together with a norm defines a metric space. 4. The properties of a vector norm are • For any vector x = 0, x > 0, • γ x = |γ | x, and • x + y ≤ x + y (triangle inequality). 5. The three most commonly used vector norms are • The L1 norm is defined as v1 = |v1 | + · · · + |vn | =
n
i=1
|vi |.
• The L2 norm (Euclidean norm) is defined as 2
2
2 1/2
v2 = (|v1 | + |v2 | + · · · + |vn | )
=
n i=1
1/2 vi2
.
(2.4.2)
• The L∞ norm is defined as v∞ = max |vi |. 1≤i≤n
6. In the absence of any subscript, the norm · is usually assumed to be the L2 norm. 7. A unit vector with respect to a particular norm · is a vector that satisfies the property that v = 1, and is sometimes denoted by vˆ .
2.4.6 DOT, SCALAR, OR INNER PRODUCT 1. The dot, scalar, or inner product of two vectors of the same dimension, represented by v1 · v2 or xT y, has two common definitions, depending upon the context in which this product is encountered. (a) In vector calculus and physics, the dot or scalar product is defined by v1 · v2 = v1 2 v2 2 cos θ,
(2.4.3)
where θ represents the angle determined by v1 and v2 (see Figure 2.4.1). (b) In optimization, linear algebra, and computer science, the inner product of two vectors, x and y, is equivalently defined as xT y =
n i=1
©1996 CRC Press LLC
xi yi = x1 y1 + · · · + xn yn .
(2.4.4)
From the first definition, it is apparent that the inner product of two perpendicular, or orthogonal, vectors is zero, since the cosine of 90◦ is zero. 2. The inner product of two parallel vectors (with v2 = rv1 ) is given by v1 · v2 = r v1 22 . For example, when r > 0, v1 · v2 = v1 2 v2 2 cos 0 = v1 2 v2 2 = r v1 22 .
(2.4.5)
3. The dot, scalar, or inner product is distributive, e.g., (v1 + v2 ) · v3 = v1 · v3 + v2 · v3 .
(2.4.6)
4. For a, b, c ∈ Rn with n > 1, aT b = aT c
⇒
b = c.
(2.4.7)
aT (b − c) = 0,
(2.4.8)
However, it is valid to conclude that aT b = aT c
⇒
and that the vector a is orthogonal to the vector (b − c).
FIGURE 2.4.1 Depiction of right-hand rule.
v1 × v2
v2 n
v1
2.4.7 VECTOR OR CROSS PRODUCT 1. The vector or cross product of two nonzero three-dimensional vectors v1 and v2 is defined as v1 × v2 = nˆ v1 2 v2 2 sin θ,
(2.4.9)
where nˆ is the unit normal vector perpendicular to both v1 and v2 in the direction adhering to the right-hand rule (see Figure 2.4.1) and θ is the angle between v1 and v2 .
©1996 CRC Press LLC
2. If v1 and v2 are parallel, then v1 × v2 = 0. 3. The quantity v1 2 v2 2 |sin θ| represents the area of the parallelogram determined by v1 and v2 . 4. The following rules apply for vector products: (γ v1 ) × (αv2 ) = (γ α)v1 × v2 , v1 × v2 = −v2 × v1 , v1 × (v2 + v3 ) = v1 × v2 + v1 × v3 , (v1 + v2 ) × v3 = v1 × v3 + v2 × v3 , v1 × (v2 × v3 ) = v2 (v3 · v1 ) − v3 (v1 · v2 ), (a × b) · (c × d) = (a · c)(b · d) − (a · d)(b · c),
(2.4.10)
(a × b) × (c × d) = [a · (b × d)]c − [a · (b × c)]d = [a · (c × d)]b − [b · (c × d)]a. ˆ corresponding to the 5. The pairwise cross products of the unit vectors ˆi, ˆj, and k, ˆ are given by directions of v = v1ˆi + v2 ˆj + v3 k, ˆi × ˆj = −(ˆj × ˆi) = k, ˆ ˆj × kˆ = −(kˆ × ˆj) = ˆi, ˆ = ˆj, and kˆ × ˆi = −(ˆi × k) ˆi × ˆi = ˆj × ˆj = kˆ × kˆ = 0. ˆ then 6. If v1 = a1ˆi + a2 ˆj + a3 kˆ and v2 = b1ˆi + b2 ˆj + b3 k, ˆi ˆj kˆ v1 × v2 = a1 a2 a3 , b1 b2 b3 ˆ = (a2 b3 − b2 a3 )ˆi + (a3 b1 − b3 a1 )ˆj + (a1 b2 − b1 a2 )k.
(2.4.11)
(2.4.12)
2.4.8 SCALAR AND VECTOR TRIPLE PRODUCTS 1. The scalar triple product involving three three-dimensional vectors a, b, and c, sometimes denotedby [abc] (not to be confused with a matrix containing three columns a b c ), can be computed using the determinant " ! b b3 ˆi − b1 b3 ˆj + b1 b2 kˆ [abc] = a · (b × c) = a · 2 c2 c3 c1 c3 c1 c2 b b3 − a2 b1 b3 + a3 b1 b2 = a1 2 c2 c3 c1 c3 c1 c2 a1 a2 a3 (2.4.13) = b1 b2 b3 c1 c2 c3 = a b c cos φ sin θ,
©1996 CRC Press LLC
where θ is the angle between b and c, and φ is the angle between a and the normal to the plane defined by b and c. 2. The absolute value of the triple scalar product calculates the volume of the parallelipiped determined by the three vectors. The answer you get is therefore independent of the order in which the triple product is taken. 3. Given three noncoplanar reference vectors v1 , v2 , and v3 , the reciprocal system is given by v∗1 , v∗2 , and v∗3 , where v∗1 = Note that
v2 × v3 , [v1 v2 v3 ]
v∗2 =
v3 × v1 , [v1 v2 v3 ]
# # 1 = v1 #v∗1 # = v2 # # and 0 = v1 #v∗2 # = v1
v∗3 =
# ∗# #v # = v3 # ∗2 # #v # = v2 3
v1 × v2 . [v1 v2 v3 ]
# ∗# #v # # 3∗ # #v # , etc. 1
(2.4.14)
(2.4.15)
The system ˆi, ˆj, kˆ is its own reciprocal. 4. The vector triple product involving three three-dimensional vectors a, b, and c, given by a × (b × c), results in a vector, perpendicular to a, lying in the plane of b and c, and is defined as a × (b × c) = (a · c)b − (a · b)c, ˆi ˆj a1 a2 = b2 b3 b3 b1 c2 c3 c3 c1
kˆ a 3 b1 b2 . c1 c2
(2.4.16)
5. (a × b) × (c × d) = [acd]b − [bcd]a = [abd]c − [abc]d
2.5 LINEAR AND MATRIX ALGEBRA
2.5.1 DEFINITIONS 1. An m×n matrix is a two-dimensional array of numbers consisting of m rows and n columns. By convention, a matrix is denoted by a capital letter emphasized with italics, as in A, B, D, or boldface, A, B, D, along with a subscript denoting the dimensions of the matrix, e.g., A2×3 . If A is a real n × m matrix, then we write A ∈ Rn×m . 2. Am×n is called rectangular if m = n. 3. Am×n is called square if m = n.
©1996 CRC Press LLC
4. A particular element or component of a matrix is denoted by the lower case letter of that which names the matrix, along with two subscripts corresponding to the row i and column j location of the component in the array, e.g.,
5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
16. 17. 18. 19.
20.
21.
Am×n
has components aij ;
Bm×n
has components bij .
For example, a23 is in the second row and third column of matrix A. Any component aij with i = j is called a diagonal element. Any component aij with i = j is called an off-diagonal element. Two matrices A and B are said to be equal if they have the same number of rows (m) and columns (n), and aij = bij for all 1 ≤ i ≤ m, 1 ≤ j ≤ n. An m × 1 dimensional matrix is called a column vector. Similarly, a 1 × n dimensional matrix is called a row vector. A column (row) vector with all components equal to zero is called a null vector and is usually denoted by 0. A column vector with all components equal to one is sometimes denoted by e. The analogous row vector is denoted by eT . The unit vectors of order n are usually {e1 , e2 , . . . , en } where ei is a n × 1 vector of all zeros, except for the i th component, which is one. The scalar xT x = ni=1 xi2 is the sum of squares of all components of the vector x. The weighted sum of squares is defined by xT Dw x = ni=1 wi xi2 , when x has n components and the diagonal matrix Dw is of dimension (n × n). If Q is a square matrix, then xT Qx is called a quadratic form. An n × n matrix A is called non-singular, or regular, if there exists an n × n matrix B such that AB = BA = I. The unique matrix B is called the inverse of A, and is denoted by A−1 . The scalar xT y = ni=1 xi yi is the sum of products of the components of x by those of y. The weighted sum of products is xT Dw y = ni=1 wi xi yi , when x and y have n components, and the diagonal matrix Dw is (n × n). The scalar xT Qy is called a bilinear form, where Q is a matrix of any appropriate dimension. The transpose of an m × n matrix A, denoted by AT , is an n × m matrix with rows and columns interchanged, so that the (i, j ) component of A is the (j, i) component of AT , and (AT )j i = Aij . The Hermitian conjugate of a matrix A, denoted by AH , is obtained by transposing A and replacing each element by its complex √ conjugate. Hence, if akl = ukl + ivkl , then (AH )kl = ulk − ivik , with i = −1. If Q is a square matrix, then xH Qx is called a Hermitian form.
©1996 CRC Press LLC
2.5.2 TYPES OF MATRICES 1. The diagonal alignment of components in a matrix extending from the upper left to the lower right is called the principal or main diagonal. 2. A square matrix with all components off the principal diagonal equal to zero is called a diagonal matrix, typically denoted by the letter D with a subscript indicating the typical element in the principal diagonal.
EXAMPLE 2.5.1
a11 Da = 0 0
0 a22 0
0 0 , a33
λ11 Dλ = 0 0
0 λ22 0
0 0 . λ33
3. The identity matrix, denoted by I, is the diagonal matrix with aij = 1 for all i = j , and aij = 0 for i = j . The n × n identity matrix is denoted In . 4. A matrix that has all main diagonal components equal to one and zeros everywhere else except a single nonzero component in location (i, j ) is called an elementary matrix, denoted by Eij .
EXAMPLE 2.5.2
E31
1 0 = −5 0
0 1 0 0
0 0 1 0
0 0 . 0 1
An elementary matrix can also be expressed as E = I − αuv T , where I is the identity matrix, α is a scalar, and u and v are vectors of the same dimension. In this context, the elementary matrix is referred to as a rank one modification of an identity matrix. T 5. Other elementary matrices are those of the form Eij = ei ej . Note that, using these matrices, A = i j aij Eij . 6. A matrix with all components above the principal diagonal equal to zero is called a lower triangular matrix.
©1996 CRC Press LLC
EXAMPLE 2.5.3
a11 L = a21 a31
0 a22 a32
0 0 a33
is lower triangular.
7. The transpose of a lower triangular matrix is called an upper triangular matrix. 8. A matrix whose components are arranged in m rows and a single column is called a column matrix, or column vector, and is typically denoted using bold face, lower case letters, e.g., a and b. 9. A matrix whose components are arranged in n columns and a single row is called a row matrix, or row vector, and is typically denoted as a transposed column vector, e.g., aT and bT . 10. A square matrix is called symmetric if A = AT . 11. A square matrix is called skew symmetric if AT = −A. 12. A square matrix A is called Hermitian if it equals its conjugate transpose, or A = AH . All real symmetric matrices are Hermitian. 13. A square matrix Q with orthonormal columns is said to be orthogonal.1 It follows directly that the rows of Q must also be orthonormal, so that QQT = QT Q = I, or QT = Q−1 . 14. A principal submatrix of a symmetric matrix A is formed by deleting rows and columns of A simultaneously, e.g., row 1 and column 1; row 9 and column 9, etc. 15. A zero, or null, matrix is one whose elements are all zero. 16. An m × n matrix A with orthonormal columns satisfies the property AT A = I, where I is an n × n identity matrix. 17. A square matrix, whose elements are constant along each diagonal, is called a Toeplitz matrix.
EXAMPLE 2.5.4
a A = b c
d a b
e d a
and
4 M = −11 3
0 4 −11
1 0 4
are Toeplitz matrices. Notice that Toeplitz matrices are symmetric about a diagonal extending from the upper right-hand corner element to the lower lefthand corner element. This type of symmetry is called persymmetry. 1 Note
the inconsistency in terminology that has persisted.
©1996 CRC Press LLC
18. A Vandermonde matrix is a square matrix V ∈ R(n+1)×(n+1) in which each column contains unit increasing powers of a single matrix value: 1 1 ··· 1 v1 v2 · · · v(n+1) 2 2 2 V = v1 v2 · · · v(n+1) . (2.5.1) .. .. .. . . . n v1n v2n · · · v(n+1) 19. A square matrix U is said to be in upper Hessenberg form if uij = 0 whenever i > j + 1. An upper Hessenberg matrix is essentially an upper triangular matrix with an extra nonzero element immediately below the main diagonal entry in each column of U. For example, u11 u12 u13 u14 b21 u22 u23 u24 U= is upper Hessenberg. 0 b32 u33 u34 0 0 b43 u44 20. If the sum of the components of each column of a matrix A ∈ Rn×n equals one, then A is called a Markov matrix. 21. A circulant matrix is an n × n matrix of the form c0 c1 c2 · · · cn−2 cn−1 cn−1 c0 c1 · · · cn−3 cn−2 cn−2 cn−1 c0 · · · cn−4 cn−3 C= (2.5.2) , .. .. . . c1
c2
c3
···
cn−1
c0
where the components cij are such that (j − i) = k mod n have the same value ck . These components comprise the k th stripe of C. 22. A matrix A is called graded across its rows if aij ≤ ai,j +1 , for all i, j ; graded up its columns if aij ≤ ai+1,j ; and doubly graded if both conditions apply.
2.5.3 CONFORMABILITY FOR ADDITION AND MULTIPLICATION 1. Two matrices A and B can be added (subtracted) if they are of the same dimension. The result is a matrix of the same dimension.
EXAMPLE 2.5.5 ! A2×3 + B2×3 =
©1996 CRC Press LLC
3 4
2 0
" ! −1 11 + 9 0
−2 1
" ! 3 14 = 1 4
0 1
" 2 . 10
2. Multiplication of a matrix or a vector by a scalar is achieved by multiplying each component by that scalar. If B = αA, then bij = αaij for all components. 3. The matrix multiplication AB is valid if the number of columns of A is equal to the number of rows of B. 4. The multiplication of two matrices Am×n and Bn×q results in a matrix Cm×q whose components are defined as cmq =
n
amk bkq .
(2.5.3)
k=1
Each cmq is the result of the inner (dot) product of the mth row of A with the q th column of B. This rule applies similarly for matrix multiplication involving more than two matrices, so that if ABCD = E, then eij = aik bkl clm dmj . (2.5.4) k
l
m
The second subscript for each matrix component must coincide with the first subscript of the next one.
EXAMPLE 2.5.6 !
2 −4
−1 1
"
5 3 2 4 −7
−3 2 1
! −3 −13 −1 = −46 5
−5 18
" 10 . 31
5. In general, matrix multiplication is not commutative: AB = BA. 6. Matrix multiplication is associative: A(BC) = (AB)C. 7. The distributive law of multiplication and addition holds: C(A+B) = CA+CB and (A + B)C = AC + BC. 8. Both the transpose operator and the Hermitian operator reverse the order of matrix multiplication: (ABC)T = CT BT AT and (ABC)H = CH BH AH . ! "! " ! " a11 a12 b11 b12 c11 c12 9. Strassen algorithm: The matrix product = a21 a22 b21 b22 c21 c22 can be computed in the following way: m1 = (a12 − a22 )(b21 + b22 ), m2 = (a11 + a22 )(b11 + b22 ), m3 = (a11 − a21 )(b11 + b12 ), m4 = (a11 + a12 )b22 , m5 = a11 (b12 − b22 ), m6 = a22 (b21 − b11 ), m7 = (a21 + a22 )b11 ,
©1996 CRC Press LLC
c11 c12 c21 c22
= m1 + m2 − m4 + m6 , = m4 + m5 , = m6 + m7 , and
(2.5.5) = m2 − m3 + m5 − m7 .
This computation uses 7 multiplications and 18 additions and subtractions. Using this formula recursively allows multiplication of two n × n matrices using O(nlog2 7 ) = O(n2.807... ) scalar multiplications. 10. The order in which matrices are grouped together for multiplication can change the number of scalar multiplications required. The straightforward number of scalar multiplications required to multiply matrix Xa×b by matrix Yb×c is abc, without using clever algorithms such as Strassen’s. For example, consider the matrix product P = A10×100 B100×5 C5×50 . The parenthesization P = ((AB)C) requires (10 × 100 × 5) + (10 × 5 × 50) = 7, 500 scalar multiplications. The parenthesization P = (A(BC)) requires (10 × 100 × 50) + (100 × 5 × 50) = 75,000 scalar multiplications.
2.5.4 DETERMINANTS AND PERMANENTS 1. The determinant of a square matrix A, denoted by |A| or det (A), is a scalar function of A defined as det(A) = sgn(σ )a1,σ (1) a2,σ (2) · · · an,σ (n) (2.5.6) σ
where the sum is taken over all permutations σ of {1, 2, . . . , n}. The signum function sgn(σ ) is the number of successive transpositions required to change the permutation σ to the identity permutation. Note these properties of determinants: |A| |B| = |AB| and |A| = |AT |. 2. For a 2 × 2 matrix, aa1121 aa1222 = a11 a22 − a12 a21 . For a 3 × 3 matrix, a11 a12 a13 a21 a22 a23 = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 a31 a32 a33 (2.5.7) − a13 a22 a31 − a11 a23 a32 − a12 a21 a33 . In general, for m = n, a11 a12 a21 a22 .. .. . . an1 an2
··· ··· .. . ···
a1n a2n (−1)δ a1i1 a2i2 · · · anin .. = . ann
(2.5.8)
where the sum is over all permutations i1 = i2 = · · · = in , and δ denotes the number of transpositions necessary to bring the sequence (i1 , i2 , . . . , in ) back into the natural order (1, 2, . . . , n). 3. Interchanging two rows (or columns) of a matrix changes the sign of its determinant. 4. A determinant does not change its value if a linear combination of other rows (or columns) is added to or subtracted from any given row (or column).
©1996 CRC Press LLC
5. Multiplying an entire row (or column) of A by a scalar γ causes the determinant to be multiplied by the same scalar γ . 6. For an n × n matrix A, |γ A| = γ n |A|. 7. If det (A) = 0, then A is singular; if det (A) = 0, then A is nonsingular or invertible. The determinant of the identity matrix is one. 8. det (A−1 ) = 1/ det (A). 9. The size of the determinant of a square matrix A is not related to the condition number of A. 10. When the edges of a parallelepiped P are defined by the rows (or columns) of A, the absolute value of the determinant of A measures the volume of P . Thus, if any row (or column) of A is dependent upon another row (or column) of A, the determinant of A equals zero. 11. The cofactor of a square matrix A, cofij (A), is the determinant of a submatrix obtained by striking the i th row and the j th column of A and choosing a positive (negative) sign if (i + j ) is even (odd).
EXAMPLE 2.5.7
2 cof23 6 −2
4 1 1
3 2+3 2 5 = (−1) −2 3
4 = −(2 + 8) = −10. 1
12. Let aij denote the components of A and a ij those of A−1 . Then, (2.5.9) a ij = cofj i (A)/|A|. $ % 13. Partitioning of determinants: Let A = B C . Assuming all inverses exist, DE then |A| = |E| B − CE−1 D) = |B| (E − DB−1 C) . (2.5.10) 14. Laplace development: The determinant of A is a combination of row i (column j ) and the cofactors of row i (column j ), i.e., |A|
= ai1 cofi1 (A) + ai2 cofi2 (A) + · · · + ain cofin (A), (2.5.11) = a1j cof1j (A) + a2j cof2j (A) + · · · + anj cofnj (A),
for any row i or any column j . 15. Omitting the signum function in Equation (2.5.6) yields the definition of permanent of A, given by per A = σ a1,σ (1) · · · an,σ (n) . Properties of the permanent include: (a) If A is an m × n matrix and B is an n × m matrix, then |per(AB)|2 ≤ per(AAH ) per(BH B).
©1996 CRC Press LLC
(2.5.12)
(b) If P and Q are permutation matrices, then per PAQ = per A. (c) If D and G are diagonal matrices, then per DAG = per D per A per G.
2.5.5 MATRIX NORMS 1. The mapping g : Rm×n ⇒ R is a matrix norm if g satisfies the same three properties as a vector norm: • g(A) ≥ 0 for A = 0 and g(A) = 0 if and only if A ≡ 0, so that (in norm notation) A > 0 for all nonzero A. • For two matrices A, B ∈ Rm×n , g(A + B) ≤ g(A) + g(B), so that A + B ≤ A + B. • g(rA) = |r|g(A), where r ∈ R, so that γ A = |γ | A. 2. The most common matrix norms are the Lp matrix norm and the Frobenius norm. The Lp norm of a matrix A is the number defined by Axp Ap = sup x=0 xp
(2.5.13)
where ·p represents one of the Lp (vector) norms with p = 1, 2, or ∞. 3. The matrix 1-norm of Am×n is defined as A1 = max1≤j ≤n m i=1 |aij |.
4. The matrix 2-norm of A is the square root of the greatest eigenvalue of AT A, i.e., A22 = λmax (AT A), which is the same as the largest singular value of A, A2 = σ1 (A). When A is symmetric, then A = max |λj |, where λj is the largest eigenvalue of A. 5. The L∞ norm is defined as A∞ = max1≤i≤m nj=1 |aij |.
6. The following properties hold: √ 1 √ A1 ≤ A2 ≤ n A1 , m √ max |aij | ≤ A2 ≤ mn max |aij |, and i,j
i,j
(2.5.14)
√ 1 √ A∞ ≤ A2 ≤ m A∞ . n 7. The matrix p norms satisfy the additional property of consistency, defined as ABp ≤ Ap Bp . 8. The Frobenius or Hilbert-Schmidt norm of a matrix A is a number defined by & ' n ' m AF = ( |aij |2 (2.5.15) i=1 j =1
©1996 CRC Press LLC
which satisfies A2F = trace(AT A). Since Rm×n is isomorphic to Rmn , the Frobenius norm can be interpreted as the L2 norm of an nm × 1 column vector in which each column of A is appended to the next in succession. See Section 2.5.20. 9. The Frobenius norm is compatible with the vector 2 norm, i.e., AxF ≤ AF x √2 . Additionally, the Frobenius norm satisfies the condition A2 ≤ AF ≤ n A2 .
2.5.6 SINGULARITY, RANK, AND INVERSES 1. An n × n matrix A is called singular if there exists a vector x = 0 such that Ax = 0 or AT x = 0. (Note that x = 0 means that all components of x are zero). If a matrix is not singular, it is called nonsingular. 2. (AB)−1 = B−1 A−1 , provided all inverses exist. 3. (A−1 )T = (AT )−1 . 4. (γ A)−1 = (1/γ )A−1 . 5. If Dw is a diagonal matrix, then D−1 w = D1/w . ! " B C 6. Partitioning: Let A = . Assuming that all inverses exist, then A−1 = D E $ % X Y , where ZU X = (B − CE−1 )−1 , −1
Y = −B CU,
U = (E − DB−1 C)−1 , −1
Z = −E DX.
(2.5.16) (2.5.17)
7. If A and B are both invertible, then −1 −1 −1 −1 (A + B)−1 = B−1 A−1 + B−1 A = A−1 A−1 + B−1 B . (2.5.18) 8. The row rank of a matrix A is defined as the number of linearly independent rows of A. Likewise, the column rank equals the number of linearly independent columns of A. 9. For any matrix, the row rank equals the column rank. 10. If A ∈ Rn×n has rank of n, then A is said to have full rank. 11. A square matrix is invertible if, and only if, it has full rank. 12. Rank(AB) ≤ min [rank(A), rank(B)]. 13. Rank(AT A) = rank(AAT ) = rank(A).
©1996 CRC Press LLC
2.5.7 SYSTEMS OF LINEAR EQUATIONS 1. Suppose that A is a matrix. Then Ax = b is a system of linear equations. If A is square and nonsingular, there exists a unique solution x = A−1 b. 2. For the linear system of equations involving n variables and m equations, written as Ax = c or a11 x1 + a12 x2 + · · · + a1n xn = c1 , a21 x1 + a22 x2 + · · · + a2n xn = c2 , .. .
(2.5.19)
am1 x1 + am2 x2 + · · · + amn xn = cm , three possible outcomes exist for the simultaneous determination of a solution: • No such solution exists and the system is called inconsistent. • A unique solution exists, and the system is called consistent. • Multiple solutions exist, the system has an infinite number of solutions, and the system is called undetermined. 3. For a system of linear equations Ax = b (for which A is nonsingular), the sensitivity of the solution x to pertubations in A and b is given in terms of the condition number of A defined by # # cond(A) = #A−1 # A .
(2.5.20)
where · is any of the p norms. In all cases, cond(A) ≥ 1. When cond(A) is equal to one, A is said to be perfectly conditioned. Matrices with small condition numbers are called well-conditioned. If cond(A) is large, then A is called ill-conditioned. 4. The size of the determinant of a square matrix A is not related to the condition number of A. 5. When A is singular, the definition of condition number is modified slightly, incorporating # # the generalized, or pseudo inverse, of A, and is defined by cond(A) = #A+ # A , where A+ represents the pseudo inverse of A (see Section 2.5.11). 6. Let A= (aij) be an n × n matrix. Using the L2 condition number, cond A = maxj λj (A) / mini |λi (A)|:
©1996 CRC Press LLC
Matrix An×n = (aij ) √ aij = 2/(n + 1) sin(ij π/(n + 1))
Condition number cond (A) = 1 cond (A) = 2 √ cond (A) = n + 1
aij = nδij + 1 aij = (i + j )/p, n = p − 1, p a prime The circulant whose first row is (1, 2, . . . , n) i/j if i ≤ j aij = j/i if i > j −2 if i = j aij = 1 if |i − j | = 1 0 if |i − j | ≥ 2 aij = 2 min(i, j ) − 1
cond (A) ∼ n cond (A) ∼ cn1+& , 0≤&≤1 cond (A) ∼ 4n2 /π 2 cond (A) ∼ 16n2 /π 2 log cond (A) ∼ Kn, K ≈ 3.5
aij = (i + j − 1)−1 (Hilbert matrix)
2.5.8 OTHER MATRIX TRANSFORMATIONS 1. A Householder transformation, or Householder reflection, is an n × n matrix H of the form H = I − (2uuT )/(uT u), where the Householder vector u ∈ Rn is nonzero. 2. A Givens rotation is defined as a rank two correction to the identity matrix given by
1 .. . 0 G(i, k, θ) = ... 0 . .. 0
··· .. . ···
0 .. .
···
−s .. .
···
c .. .
0
i
··· ··· .. . ··· ···
0 .. . s .. .
c .. .
0
··· ··· ··· .. . ···
0 .. . 0 i .. , . 0 k .. . 1
(2.5.21)
k
where c = cos θ and s = sin θ for some angle θ. Premultiplication by G(i, k, θ)T induces a counterclockwise rotation of θ radians in the (i, k) plane. For x ∈ Rn and y = G(i, k, θ)T x, the components of y are given by cxi − sxk , yj = sxi + cxk , xj ,
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for j = i for j = k for j = i, k.
(2.5.22)
2.5.9 LINEAR SPACES AND LINEAR MAPPINGS 1. The projection matrix, P, onto a subspace S of the nonzero m × n matrix A, denoted by PS , is the unique m × m matrix possessing the three properties: (a) PS = PTS ; (b) P2S = PS (the projection matrix is idempotent); (c) The vector bS lies in the subspace S if, and only if, bS = PS v for some m-dimensional vector v. In other words, bS can be written as a linear combination of the columns of PS . 2. Let R(A) and N(A) denote, respectively, the range space and null space of an m × n matrix A. They are defined by n. , R(A) = y | y = Ax; x ∈ R and
. N(A) = x ∈ Rn | Ax = 0 .
3. When the m×n matrix A has rank n, the projection of A onto the four subspaces of A is given by PR(A) = A(AT A)−1 AT , P
T R(A )
= I,
PN(A) = I, P
T N(A )
(2.5.23)
= I − A(AT A)−1 AT .
When A is of rank m, the projection of A onto the four subspaces of A is given by PR(A) = I, P
T R(A )
= AT (AAT )−1 A,
PN(A) = I − AT (AAT )−1 A, P
T N(A )
(2.5.24)
= I.
˜ satisfies the requirements for a 4. When A is not of full rank, the matrix AA ˜ projection matrix. The matrix A is the coefficient matrix of the system of ˜ generated by the least squares problem min b − Ax2 . equations x+ = Ab, 2 Thus, ˜ PR(A) = AA, ˜ P T = AA, R(A ) (2.5.25) ˜ PN(A) = I − AA, ˜ P T = I − AA. N(A )
©1996 CRC Press LLC
5. A matrix B ∈ Rn×n is called similar to a matrix A ∈ Rn×n if B = T−1 AT for some nonsingular matrix T. 6. If B is similar to A, then B has the same eigenvalues as A. 7. If B is similar to A and if x is an eigenvector of A, then y = T−1 x is an eigenvector of B corresponding to the same eigenvalue.
2.5.10 TRACES 1. The trace of an n × n matrix A, usually denoted as tr (A), is defined as the sum of the n diagonal components of A. 2. The trace of an n × n matrix A equals the sum of the n eigenvalues of A, i.e., tr A = a11 + a22 + · · · + ann = λ1 + λ2 + · · · + λn . 3. The trace of a 1 × 1 matrix, a scalar, is itself. 4. If A ∈ Rm×k and B ∈ Rk×m , then tr (AB) = tr (BA). 5. If A ∈ Rm×k , B ∈ Rk×r , and C ∈ Rr×m , then tr (ABC) = tr (BCA) = tr (CAB). For example, if B = b is a column vector and C = cT is a row vector, then tr (AbcT ) = tr (bcT A) = tr (cT Ab). 6. tr (A + γ B) = tr (A) + γ tr (B), where γ is a scalar. 7. tr (AB) = (Vec AT )T Vec B (see Section 2.5.20).
2.5.11 GENERALIZED INVERSES 1. Every matrix A (singular or nonsingular, rectangular or square) has a generalized inverse, or pseudoinverse, A+ defined by the Moore–Penrose conditions AA+ A = A, A+ AA+ = A+ , (AA+ )T = AA+ ,
(2.5.26)
(A+ A)T = A+ A. 2. Only if A is square and nonsingular, A+ will be unique and A+ = A−1 . Otherwise, there will exist infinitely many matrices A+ that will satisfy the defining relations. 3. If A is a rectangular m × n matrix of rank n, with m > n, then A+ is of order n × m and A+ A = I ∈ Rn×n . In this case A+ is called a left inverse, and AA+ = I. 4. If A is a rectangular m × n matrix of rank m, with m < n, then A+ is of order n × m and AA+ = I ∈ Rm×m . In this case A+ is called a right inverse, and A+ A = I. 5. For a square singular matrix A, AA+ = I, and A+ A = I.
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6. The least squares problem is to find the x that minimizes y − Ax. The x of least norm is x = A+ y. 7. A square matrix is called idempotent if AA = A2 = A. 8. AA+ and A+ A are idempotent. 9. Let A be of rank r; and select r rows and r columns which form a basis of A. Then, a pseudoinverse of A can be obtained as follows: invert the r × r matrix, place the inverse (without transposing) into the r rows corresponding to the column numbers and the r columns corresponding to the row numbers of the basis, and place zero into the remaining component positions. Thus, if A is of order 5 × 4 and rank 3, for example, and if rows 1, 2, 4 and columns 2, 3, 4 are selected as a basis, A+ of order 4 × 5 will contain the inverse components of the basis in rows 2, 3, 4 and column 1, 2, 4, and zeros elsewhere. 10. A pseudoinverse can also be computed for a general matrix Am×n using the singular value decomposition A = UEVT , where Um×m and Vn×n are orthogonal matrices. When A is of rank r > 0, Em×n will have exactly r positive singular values (σi ) along the main diagonal extending from the upper left-hand corner. The remaining components of E are zero. The pseudoinverse of A is then given by A+ = (UEVT )+ = (VT )+ E + U+ , with (VT )+ = V and U+ = UT because of the orthogonality of V and U. The components σi+ of the pseudoinverse of E (rectangular, in this case) are given by σi+
=
1/σi , 0,
if σi = 0; if σi = 0.
(2.5.27)
11. The pseudoinverse is ill-conditioned with respect to rank changing perturbations. For example !
1 2
" ! −1 1 +& −2 0
0 2
" +
=
! 1 −1 & 2 −1
1 2 1 2
" +
! 1 1 & 0
0 1 2
" .
2.5.12 EIGENSTRUCTURE 1. If A is a square n × n matrix, then the nth degree polynomial defined by det (A − λI) = 0 is called the characteristic polynomial, or characteristic equation of A. 2. The n roots (not necessarily distinct) of the characteristic polynomial are called the eigenvalues (or characteristic roots) of A. Therefore, the values, λi , i = 1, . . . , n, are eigenvalues if, and only if, |A − λi I| = 0.
©1996 CRC Press LLC
3. The characteristic polynomial det (A − λI) =
n
i i=0 ri λ
has the properties
rn = (−1)n rn−1 = −rn tr (A) 1 rn−2 = − rn−1 tr (A) + rn tr (A2 ) 2 1 rn−3 = − rn−2 tr (A) + rn−1 tr (A2 ) + rn tr (A3 ) 3 .. . 0 / n−1 1 j r0 = − rn−j tr (A ) n j =1 4. Each eigenvalue λ has a corresponding eigenvector x that solves the system Ax = λx, or (A − λI)x = 0. 5. If x solves Ax = λx, then so does γ x, where γ is an arbitrary scalar. 6. Cayley–Hamilton theorem: Any matrix A satisfies its own characteristic equation. 7. The eigenvalues of a triangular (or diagonal) matrix are the diagonal components of the matrix. 8. The eigenvalues of idempotent matrices are either zero or one. 9. If A is a real matrix with positive eigenvalues, then λmin (AAT ) ≤ [λmin (A)]2 ≤ [λmax (A)]2 ≤ λmax (AAT ),
(2.5.28)
where λmin denotes the smallest and λmax the largest eigenvalue. 10. If all the eigenvalues of a real symmetric matrix are distinct, then their associated eigenvectors are also distinct (linearly independent). 11. Symmetric and Hermitian matrices have real eigenvalues. 12. For an n×n matrix A with eigenvalues λ1 , λ2 , . . . , λn , the product λ1 λ2 · · · λn = det (A).
2.5.13 EIGENVALUE DIAGONALIZATION 1. If A ∈ Rn×n possesses n linearly independent eigenvectors x1 , . . . , xn , then A can be diagonalized as S−1 AS = F = diag(λ1 , . . . , λn ), where the eigenvectors of A are chosen to comprise the columns of S. 2. If A ∈ Rn×n can be diagonalized into S−1 AS = F, then Ak = SFk S−1 , or Fk = S−1 Ak S. 3. Spectral decomposition: Any real symmetric matrix A ∈ Rn×n can be diagonalized into the form A = UFUT , where F is the diagonal matrix of ordered eigenvalues of A such that λ1 ≥ λ2 ≥ · · · ≥ λn , and the columns of U are the corresponding n orthonormal eigenvectors of A.
©1996 CRC Press LLC
4. The spectral radius of a real symmetric matrix A, commonly denoted by ρ(A), is defined as ρ(A) = max1≤i≤n |λi (A)|. 5. If A ∈ Rn×n and B ∈ Rn×n are diagonalizable, then they share a common eigenvector matrix S if and only if AB = BA. (Not every eigenvector of A need be an eigenvector for B, e.g., A = I.) 6. Schur decomposition: If A ∈ Cn×n , then a unitary matrix Q ∈ Cn×n exists such that QH AQ = D + N, where D = diag(λ1 , . . . , λn ) and N ∈ Cn×n is strictly upper triangular. The matrix Q can be chosen so that the eigenvalues λi appear in any order along the diagonal. 7. If A ∈ Rn×n and symmetric, then a real orthogonal matrix Q exists such that QT AQ = diag(λ1 , . . . , λn ). 8. If A ∈ Rn×n possesses s ≤ n linearly independent eigenvectors, it is similar to a matrix with s Jordan blocks J1 · , J = M−1 AM = · Js
0
0
where each Jordan block Ji is an upper triangular matrix with only a single eigenvalue λi and a single eigenvector:
λi Ji =
0
1 ·
1 ·
0 1
(2.5.29)
λi
If Ji has m > 1, then λi repeats m times along the main diagonal, with (m − 1) 1’s appearing above the diagonal entries, and all other components are equal to zero.
2.5.14 MATRIX EXPONENTIALS 1. Matrix exponentiation is defined as (the series always converges): eAt = I + At +
(At)2 (At)3 + + ··· . 2! 3!
2. Common properties of matrix exponentials are
(a) eAs eAt = eA(s+t) ,
e−At = I, (b) eAt (c)
d At e dt
= AeAt , and
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(d) when A and B are square matrices, the commutator of A and B is C = [B, A] = BA − AB. Then e(A+B) = eA eB eC/2 provided that [C, A] = [C, B] = 0 (i.e., each of A and B commute with their commutator). n×n At 3. For a matrix A ∈ R , the determinant of e is given by det eAt = eλ1 t eλ2 t · · · eλn t = etrace(At) .
4. The diagonalization of eAt is given by eAt = SeDt S−1 where the columns of S consist of the eigenvectors of A, and the entries of the diagonal matrix D are the corresponding eigenvalues of A, that is, A = SDS−1 . 5. If A is skew-symmetric, then eAt is an orthogonal matrix.
2.5.15 QUADRATIC FORMS 1. For a symmetric matrix A, the product xT Ax is called a pure quadratic form. It has the form xT Ax =
n n i=1 j =1
aij xi xj = a11 x12 + a12 x1 x2 + a21 x2 x1 + · · · + ann xn2 .
2. For A symmetric, the gradient of xT Ax/xT x equals zero if, and only if, x is an eigenvector of A. Thus, the stationary values for this expression are the eigenvalues of A. 3. The ratio of two quadratic forms (B nonsingular) u(x) = (xT Ax)/(xT Bx) attains stationary values at the eigenvalues of B−1 A. In particular, umax = λmax (B−1 A),
umin = λmin (B−1 A).
and
4. A matrix A is positive definite if xT Ax > 0 for all x = 0. 5. A matrix A is positive semidefinite if xT Ax ≥ 0 for all x. 6. For a real, symmetric matrix A ∈ Rn×n , the following are necessary and sufficient conditions to establish the positive definiteness of A: (a) All eigenvalues, or characteristic roots, of A have λi and (b) The upper-left submatrices of A, defined by " ! a12 a A1 = a11 , , A2 = 11 a21 a22 a11 a12 · · · a21 a22 · · · An = . .. .. . an1
have det Ak > 0, for all k = 1, . . . , n.
©1996 CRC Press LLC
an2
···
> 0, for i = 1, . . . , n,
..., a1n a2n .. , . ann
7. If A is positive definite, then all of the principal submatrices of A are also positive definite. Additionally, all diagonal entries of A are positive. 8. For a real, symmetric matrix A ∈ Rn×n , the following are necessary and sufficient conditions to establish the positive semidefiniteness of A: • All eigenvalues, or characteristic roots, of A have λi ≥ 0, for i = 1, . . . , n, and • The principal submatrices of A have det Ak ≥ 0, for all k = 1, . . . , n. 9. If A is positive definite, then all of the principal submatrices of A are also positive definite. Additionally, all diagonal entries of A are nonnegative. 10. If xT denotes the radius vector (running coordinates [x, y, z]), and if a matrix Q is positive definite, then (xT − xT0 )Q−1 (x − x0 ) = 1 is the equation of an ellipsoid with its center at [x0 , y0 , z0 ] = xT0 and semiaxes equal to the square roots of the eigenvalues of Q.
2.5.16 MATRIX FACTORIZATIONS 1. Singular value decomposition (SVD): Any m × n matrix A can be written as the product A = UEVT , where U is an m × m orthogonal matrix, V is an n × n orthogonal matrix, and E = diag(σ1 , σ2 , . . . , σp ), with p = min (m, n) and σ1 ≥ σ2 ≥ · · · ≥ σp ≥ 0. The values σi , i = 1, . . . , p, are called the singular values of A. 2. When rank(A) = r > 0, A has exactly r positive singular values, and σr+1 = · · · = σp = 0. 3. When A is a symmetric n × n matrix, then σ1 = |λ1 |, · · · , σn = |λn |, where λ1 , λ2 , . . . , λn are the eigenvalues of A. 4. When A is an m × n matrix, if m ≥ n then the singular values of A are the square roots of the eigenvalues of AT A. Otherwise, they are the square roots of the eigenvalues of AAT . 5. Any m × n matrix A can be factored as PA = LU, where P is a permutation matrix, L is lower triangular, and U is an m × n matrix in echelon form. 6. QR factorization: If all the columns of A ∈ Rm×n are linearly independent, then A can be factored as A = QR, where Q ∈ Rm×n has orthonormal columns and R ∈ Rn×n is upper triangular and nonsingular. 7. If A ∈ Rn×n is symmetric positive definite, then T A = LDLT = LD1/2 D1/2 LT = LD1/2 LD1/2 = GGT (2.5.30) where L is a lower triangular matrix and D is a diagonal matrix. The factorization A = GGT is called the Cholesky factorization, and the matrix G is commonly referred to as the Cholesky triangle.
©1996 CRC Press LLC
2.5.17 THEOREMS 1. Frobenius–Perron theorem: If A > 0, then there exists a λ0 > 0 and x0 > 0 such that (a) Ax0 = λ0 x0 , (b) if λ is any other eigenvalue of A, λ = λ0 , then |λ| < λ0 , and (c) λ0 is an eigenvalue with geometric and algebraic multiplicity equal to one. 2. If A ≥ 0, and Ak > 0 for some positive integer k, then the results of the Frobenius–Perron theorem apply to A. 3. Courant–Fischer minimax theorem: If λi (A) denotes the i th largest eigenvalue of a matrix A = AT ∈ Rn×n , then λj (A) = max Sj
xT Ax min 0=x∈Sj xT x
j = 1, . . . , n
where x ∈ Rn and Sj is a j -dimensional subspace. 4. Cramer’s rule: The j th component of x = A−1 b is given by a11 a12 b1 det Bj .. .. .. , where Bj = . xj = . . det A an1 an2 bn
a1n .. . .
ann
(2.5.31)
The vector b = (b1 , . . . , bn )T replaces the j th column of the matrix A to form the matrix Bj . 5. Sylvester’s law of inertia: For a symmetric matrix A ∈ Rn×n , the congruence A ⇒ CT AC, for C nonsingular, has the same number of positive, negative, and zero eigenvalues. 6. Raleigh’s principle: The quotient R(x) = xT Ax/xT x is minimized by the eigenvector x = x1 corresponding to the smallest eigenvalue λ1 of A. The minimum of R(x) is λ1 , that is, min R(x) = min
xT λ1 x1 xT Ax1 xT Ax = R(x1 ) = 1 T = 1T = λ1 . T x x x1 x x1 x (2.5.32)
2.5.18 KRONECKER PRODUCTS OR TENSOR PRODUCTS If the matrix A = (aij ) has size m × n, and matrix B = (bij ) has size r × s, then the Kronecker product of these matrices, denoted A ⊗ B, is defined as the partitioned matrix a11 B a12 B . . . a1n B a21 B a22 B . . . a2n B A⊗B= . (2.5.33) .. .. . .. . . am1 B am2 B . . . amn B
©1996 CRC Press LLC
Hence, the A ⊗ B matrix !has size mr " × ns. ! b a11 a12 and B = 11 For example, if A = a21 a22 b21
" b12 , then b22
.. . a b a b b a b a 11 12 12 11 12 12 11 11 . " a11 b21 a11 b22 .. a12 b21 a12 b22 a12 B = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a22 B a21 b11 a21 b12 ... a22 b11 a22 b12 (2.5.34) .. a21 b21 a21 b22 . a22 b21 a22 b22
! A⊗B=
a11 B a21 B
The Kronecker product has the following properties: 1. If z and w are vectors of appropriate dimensions, then Az ⊗ Bw = (A ⊗ B)(z ⊗ w). 2. If α is a scalar, then A ⊗ (αB) = α(A ⊗ B). 3. The Kronecker product is distributive with respect to addition: • (A + B) ⊗ C = A ⊗ C + B ⊗ C, and • A ⊗ (B + C) = A ⊗ B + A ⊗ C. 4. The Kronecker product is associative: A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C. 5. (A ⊗ B)T = AT ⊗ BT . 6. The mixed product rule: If the dimensions of the matrices are such that the expressions following exist, then (A ⊗ B)(C ⊗ D) = AC ⊗ BD. 7. If the inverses exist, then (A ⊗ B)−1 = A−1 ⊗ B−1 . 8. If {λi } and {xi } are the eigenvalues and the corresponding eigenvectors for A, and {µj } and {yj } are the eigenvalues and the corresponding eigenvectors for B, then A ⊗ B has eigenvalues {λi µj } with corresponding eigenvectors {xi ⊗ yj }. 9. If matrix A has size n×n and B has size m×m, then det(A⊗B) = (det A)m (det B)n . 10. If f (z) is an analytic function and A has size n × n, then • f (In ⊗ A) = In ⊗ f (A), and • f (A ⊗ In ) = f (A) ⊗ In . 11. tr (A ⊗ B) = ( tr A)( tr B). 12. If A, B, C, and D are matrices with A similar to C and B similar to D, then A ⊗ B is similar to C ⊗ D. 13. If C(t) = A(t) ⊗ B(t), then ddtC = ddtA ⊗ B + A ⊗ ddtB .
©1996 CRC Press LLC
2.5.19 KRONECKER SUMS If the matrix A = (aij ) has size n × n and matrix B = (bij ) has size m × m, then the Kronecker sum of these matrices, denoted A ⊕ B, is defined2 as A ⊕ B = A ⊗ Im + In ⊗ B. The Kronecker sum has the following properties: 1. If A has eigenvalues {λi } and B has eigenvalues {µj }, then A⊕B has eigenvalues {λi + µj }. 2. The matrix equation AX + XB = C may be equivalently written as (BT ⊕ A) Vec X = Vec C, where Vec is defined in Section 2.5.20. 3. eA⊕B = eA ⊗ eB .
2.5.20 THE VECTOR OPERATION The matrix Am×n can be represented as a collection of m × 1 column vectors: A = a1 a2 . . . an . Define Vec A as the matrix of size nm × 1 by a1 a2 Vec A = . . .. an
(2.5.35)
This operator has the following properties: T 1. trAB = Vec AT Vec B. 2. The permutation matrix U that associates Vec X and Vec XT (that is, Vec XT = U Vec X) is given by: Ers ⊗ ETrs . U = Vec ET11 Vec ET21 . . . Vec ETn1 = r,s (2.5.36) 3. Vec(AYB) = (BT ⊗ A) Vec Y. 4. If A and B are both of size n × n, then (a) Vec AB = (In ⊗ A) Vec B. (b) Vec AB = (BT ⊗ A) Vec In .
! 2 Note
that A ⊕ B is also used to denote the (m + n) × (m + n) matrix
