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'k^ '•hen the parameters W+, w_, Pk^ 9^, 9-, ipk^ ^k of the product n-complex number u — u'u" are given by v^ = . / + < ,
(6.37)
Pk = p'kP'k,
(6-38)
forA; = l , . . . , [ ( n - l ) / 2 ] ,
tan 6'+ = 4= *^" ^+ **" ^+' v2 tan V"*; = tan V'^ tan ^j^',
(^-^^^ (6.40)
forA; = l , . . . , [ ( n - 3 ) / 2 ] , <^fe = '/'fe + C
(6.41)
for fc = 1,..., [(n - l)/2], and, if n is even, (6.42) tan ^ - = 4 = tan 6''_ tan O'L. \/2
(6.43)
207
Polar complex numbers in n dimensions
The Eqs. (6.37) and (6.42) can be checked directly, and Eqs. (6.38)-(6.41) and (6.43) are a consequence of the relations Vk = v'kv'k
v'kv'k, Vk
(6.44)
4^k + Wfc"lt>
and of the corresponding relations of definition. Then the product v in Eqs. (6.18) and (6.19) has the property that u = i^'u"
(6.45)
and, if v' > 0,i^" > 0, the amplitude p defined in Eq. (6.11) has the property that (6.46)
P'P"-
The fact that the ampUtude of the product is equal to the product of the amplitudes, as written in Eq. (6.46), can be demonstrated also by using a representation of the n-complex numbers by matrices, in which the n-complex number u = XQ + hixi + h2X2 + • • • + hn~iXn-i is represented by the matrix (
U =
X{)
Xi
X2
Xn-l
Xn-l
XQ
Xi
Xn~2
Xn-2
Xn-\
XQ
^n-3
Xi
X2
^3
^0
\
\
(6.47) )
The product u — u^u" is represented by the matrix multiplication U = U^U". The relation (6.45) is then a consequence of the fact the determinant of the product of matrices is equal to the product of the determinants of the factor matrices. The use of the representation of the n-complex numbers with matrices provides an alternative demonstration of the fact that the product of n-complex numbers is associative, as stated in Eq. (6.5). According to Eqs. (6.37, (6.38), (6.42), (6.28) and (6.29), the modulus of the product uv! is, for even n. n/2-l
K P = l(^^.;)2 + i(^_,'_)2 + i Y. iPkP'k?, n
n
n
1
9 {n-l)/2
(6.48)
f-^.
and for odd n (6.49) jfc=i
208
Commutative Complex Numbers in n Dimensions
Thus, if the product of two n-complex numbers is zero, uu' — 0, then v^v\ = 0, pkp'k = 0,fc= 1,..., [(n - l)/2] and, if n is even, ?;_?;'_ = 0. This means that either n = 0, or ?i' = 0, or ?/, ?/' belong to orthogonal hypersurfaces in such a way that the afore-mentioned products of components should be equal to zero. 6.1.3
The polar n-dimensional cosexponential functions
The exponential function of a hypercomplex variable u and the addition theorem for the exponential function have been written in Eqs. (1.35)(1.36). If w = a;o + h\X\ + h2X2 -t- • • • -f /i^i-i^^n-i, then expu can be calculated as expt^ = exp^o • exp(/iixi) • • •exp(/i^_ia::yi_i). It can be seen with the aid of the representation in Fig. 6.1 that hl^v ^hl,p
integer,
(6.50)
for A; = 1, ...,n — 1. Then e^^^ can be written as n-l
g/iA.2/ ^ ^ hp-n[kpln]9np{y)^
(6.51)
p=Q
where the expression of the functions Qnk^ which will be called polar cosexponential functions in n dimensions, is oo
Sn;t(y) = E y ' " ' ' " / ( ^ + p n ) ! ,
(6.52)
forfc= 0,1, ...,n - 1. If n is even, the polar cosexponential functions of even index k are even functions, gn,2p{-y) = gn,2p{y)^ P = 0,1,...,n/2 - 1, and the polar cosexponential functions of odd index are odd functions, gn^2p-^i{—y) ~ —gn,2p+i{y)^ p = 0,1, ...,n/2 - 1. For odd values of n, the polar cosexponential functions do not have a definite parity. It can be checked that
J29nk{y)=ey
(6.53)
and, for even n, E(-'^)''Snk(y) = e-y.
(6.54)
The expression of the polar n-dimensional cosexponential functions is , ^
l^ip^
r
/27r/\1
r
. /2nl\
2nkl]
(6.55)
Polar complex numbers in n dimensions
209
for fc = 0, l,...,n - 1. In order to check that the function in Eq. (6.55) has the series expansion written in Eq. (6.52), the right-hand side of Eq. (6.55) will be written as
1 ^"^
f
r/
I.cos
9nk{y) = - J ^ R e ^ e x p
27r/ . . 27r/\ ,2Trkl h I Sin — y — % }, n n ) n
(6.56)
for A: = 0,1,..., n — 1, where Re{a + ib) = a, with a and b real numbers. The part of the exponential depending on y can be expanded in a series, oo n - 1
(6.57)
9nk{y) = - £ X) ^*^ 1 ~f ^^P r~^P - ^) U" f' for A; = 0,1,..., n — 1. The expression of gnkiv) becomes
(6.58) for A: = 0,1,..., n — 1 and, since 1 n—1
27rl,
,,
— y ^ cos ^^^(jt? — /;;) =
I 1, if p —fcis a multiple of n, 0, otherwise,
(6.59)
this yields indeed the expansion in Eq. (6.52). It can be shown from Eq. (6.55) that n—1
1 n—\
12y cos
n
(?)
(6.60)
It can be seen that the right-hand side of Eq. (6.60) does not contain oscillatory terms. If n is a multiple of 4, it can be shown by replacing y by iy in Eq, (6.60) that n/4-1
E(-i)'5^.(y) A:=0
1 H- COS 2y +
^
'2Td\ COS 2y cos (
n
nJ
, (6.61)
which does not contain exponential terms. Addition theorems for the polar n-dimensional cosexponential functions can be obtained from the relation exphi{y + z) = exphiy • exp/ii^, by substituting the expression of the exponentials as given in Eq. (6.51) for fc = 1, e^'y = gnoiy) + higniiy) + • • • + hn-ign,n-i{y),
9nk{y + z)=^ gno{y)9nk{z) + gni{y)gn,k-i{z) + • • • + gnk{y)9no{z) +9nMliy)9n,n-l{z)
+9n.n--l{y)9nMl(^)^
+ ffn,fc+2(?/)ffn,n-2(^) + * * '
(6-62)
210
Commutative Complex Nmnhers in n Dimensions
where fc = 0,1,..., 72 - 1. For y = z the relations (6.62) take the form
gnki'^y) = gno{y)9nk{y) + 9n\{y)9n,k-\{y) + • • • + 9nk{y)gnoiy) +9nMliy)9n,n-l{y)
+ 9nM2{y)9n,n~2{y)
+ '"
+ffn,n-l(y)9n,it+l(y),
(6.63)
where A; = 0,1, ...,n - 1. For y = -z the relations (6.62) and (6.52) yield 9n^{y)9nQ{-y)
+ 9n\{y)9n,n-\{-y)
+9n,n~l{y)9nl{-y)
+ 9n2{y)9n,n-2{-y)
+ "\.
= I.
(6.64)
9no{y)9nk{-y) + 9ni{y)9n,k~i{-y) H- • •' + 9nk{y)9noi-y) +9nMliy)9n,n~l{-y)
+ 9nM2{y)9n,n-2{-y)
+9n,n-i{y)9nMii-y)
= 0,
+ '" (6.65)
for fe = 1, ...,n — 1. From Eq. (6.51) it can be shown, for natural numbers /, that
(
n-l
\'
n-1
Yl hkp-n[kp/n]9np{y) j = Yl ^kp~n[kp/n]9np{ly), p=0 J p=0
(6.66)
where A; = 0,1, ...,n - 1. For A: = 1 the relation (6.66) is {gnoiy) + hiQnliy) + • • • 4- hn-l9n,n-l{y)Y (ly).
= 9no{ly) + hiQulily) + V' (6.67)
If
«fe = 23 s'np(y) cos ( ~ ^ ) '
^^'^^^
for fc = 0,1,..., n — 1, and n-l
(6.69)
tfc = XI 9np{y) sin ( -^!-^ ) p=o \ n /
for A; = 1, ...,n — 1, where gnkiv) are the polar cosexponential functions in Eq. (6.55), it can be shown that ^27rA:
f2TTk\
cos ysm
ak = exp
n
)i
(6.70)
where A; = 0,1,..., n — 1, /27rA:\
bk = exp
sm
. nnk\]
(6.71)
Polar complex
numbers
in n dimensions
211
where A; = 1,..., n — 1. If Gl = al-^bl
(6.72)
for A; = 1, ...,n - 1, then from Eqs. (6.70) and (6.71) it results that f2nk\ ^2 Gk = exp 2 y c o s ^ — j j
(6.73)
where fc = 1,..., n — 1. If G-f = 5n0 + fl'nl H
+
ffn,n-l,
(6.74)
from Eq. (6.68) it results t h a t G^ = ao, so that G^ = e^, and, in an even number of dimensions n, if G- = QnO- 9nl-^
^• 9n,n-2 - 9 n , n - b
(6.75)
from Eq. (6.68) it results t h a t G- = a^/2, so that G„/2 = ^~^- T h e n with the aid of Eq. (6.26) applied for p = 1 it can be shown t h a t the polar n-dimensional cosexponential functions have the property that, for even n, n/2-l G+G_ n Gl==h k=l
(6.76)
and in an odd number of dimensions, with the aid of Eq. (6.27) it can be shown t h a t (n-l)/2 G+ n Gl = l. k=l
(6.77)
T h e polar n-dimensional cosexponential functions are solutions of the n*^-order differential equation
This equation has solutions of the form ( ( u ) = Aognoi"^) + ^iS'nil'^) + • • • +An-ign,n~i{u). It cau be checked that the derivatives of the polar cosexponential functions are related by ddnO _ dgnl _ du du ^gn^n — - = du
dgn,n-2 du
_
ffn,n-2.
(6.79)
212
Commutative
6.1.4
Complex Numbers
in n
Dimensions
Exponential and trigonometric forms of polar n-complex numbers
In order to obtain the exponential and trigonometric forms of n-complex numbers, a canonical base e 4 - , e _ , e i , e i , . . . , e ^ / 2 - i 7 ^ n / 2 - i for ^he polar ncomplex numbers will be introduced for even n by the relations
\
(
2 n
efc
h
2 n^^^
0
2trfc n
fsin^ n
(
1
n
2 ^.qg 27r(n-2)fc
1
L i n ? ^ ^
i Sin 2^^(l^^:i)i
n
n
n
co^^Jl^lz})k n
\ (6.80)
\ K-\ I where k — l , 2 , . . . , n / 2 — 1. For odd n, the canonical base e-f-,ei,ei,... ^(n-i)/2^ ^ ( T I - I ) / 2 for ^he polar n-complex numbers will be introduced by the relations
i
!cos^
0
n n ^ sill V n n
Icos^ 0 ^sin^
V
i
n
2 n
n 2
cos •
n 27r(n-l) n ^ 2n(n-l)
\ /
1 \ hi h2
f sm—^-—^ n 2
, (6.81)
n 27r(n-l)A;
- cos —^ ^ ^ n 2 n
:
n 27r(y?,-1)fc n
/ V hn-\ I
where A; = 0 , 1 , - , (n - l ) / 2 . T h e multipHcation relations for the new bases are, for even n, 2
2
e ; = e+, e_ =
4 = efe, 4
e_, e+e_ = 0, e+Cfc = 0, e+e<: = 0, e^ek - 0, e_efc = 0, = -Cfe- Ckf-'k = Cfc-fiA-e/= 0, ekh = 0, efeC; = 0,
ki^l, (6.82)
Polar complex numbers in n dimensions
213
where fc, / = 1,..., n/2 — 1. For odd n the multipUcation relations are e\ = e^, e^Ck = 0, e+e/^ = 0, el = ek, el = -e^, e^h = h^ e^e/ = 0, ekh = 0, hh = 0, k^l,
(6,83)
where fc, / = 1,..., (n — l ) / 2 . The moduH of the new bases are
It can be shown that, for even n, n/2-l
(6.85) and for odd n (n-l)/2
XQ + hiXi-\
+ hn~iXn-i == e^v^ +
^
(ekVk + ekVk)'
(6.86)
The relations (6.85),(6.86) give the canonical form of a polar n-complex number. Using the properties of the bases in Eqs. (6.82) and (6.83) it can be shown that exp{ek(t)k) = l-ek
+ Ck COS^A, -f e/t sin>A;,
(6.87)
exp(eA: In PA:) = 1 - eA; 4- eA;pA:,
(6.88)
exp(e4. \nv^) = 1 - e_^ + e+'^-h
(6.89)
and, for even n, exp(e_ ln?;_) = 1 — e_ + e-V-..
(6.90)
In Eq. (6.89), lnt;-f exists as a real function provided that v^ ~ xo •{- x\ -^ • • • + Xn-i > 0, which means that 0 < 9+ < 7r/2, and for even n, Infexists in Eq. (6.90) as a real function provided that v^ = XQ — x\ -\- - - -\' Xn~'2 — Xn-i > 0, which means that 0 < 0_ < 7r/2. By multiplying the relations (6.87)-(6.90) it results, for even n, that r n/2-l ] exp e-|_ lnt;+ + e_ lni;_ + ^2 (^^ ^^/^^ ~^ ^kM "= e^v^ + e_T;_ L '^^i J n/2-l
+ 1 ] (ekVk + ekh),
(6.91)
214
Commutative Complex Numbers in n Dimensions
where the fact has ben used that n/2-l
(6.92) the latter relation being a consequence of Eqs. (6.80) and (6.26). Similarly, by multiplying the relations (6.87)-(6.89) it results, for odd n, that (n-l)/2
exp I e-i- ln^4. +
J2
i^k ^^^Pk +
hM
e^v+
k=^i
(n-l)/2
+
Y.
i^^kl^k +
(6.93)
hh),
k=l
where the fact has ben used that (n~l)/2
(6.94) the latter relation being a consequence of Eqs. (6.81) and (6.27). By comparing Eqs. (6.85) and (6.91), it can be seen that, for even n. xo -f- h]Xi H
-f hn-\Xn-i
= exp [e^_ \nv^ + e_ lnt;_
n/2-l
+ Y
(6.95)
{^k^npk-hekMl
k=\
and by comparing Eqs. (6.86) and (6.93), it can be seen that, for odd n. Xo -\-hiXi + ••• ^hn-\Xn~i
= exp[e+In?;^.
(n-l)/2
+
Y
(6.96)
{ek^npk-i-ekMl
k=]
Using the expression of the bases in Eqs. (6.80) and (6.81) yields, for even values of n, the exponential form of the n-complex number u = xo + hiX] + h hn-\Xn-i as
{
n-l
Y,hp
1 v/2 ^ ( - 1 ) ^ x/2 - In —I In tan 6n tan 6.^ n 77./2-1
—
>
" t2
cos
(6.97)
lntan'0;t-i
^ "^
k=\
Polar complex numbers in n dimensions
215
where p is the ampUtude defined in Eq. according to Eq. (6.18) the expression p = (V-^V-PI
' • • PI/2-I)
(6.11), which for even n has
.
(6.98)
For odd values of n, the exponential form of the n-complex number u is
{
n-l
I N / 2 n tan 6^4.
2 ^""Jl^^ f2nkp\ , > cos ) Intant/^it.i n k=2 T^ \ n J
5 1 Zip - I n ; — p=l
(n-l)/2
+ E
^
^k(f>k\.
(6.99)
where for odd n, p has according to Eq. (6.19) the expression p={v+pl---pl_,y^y^\
(6.100)
It can be checked with the aid of Eq. (6.87) that the n-complex number u can also be written, for even n, as /
xo + hixi H
1- hn-iXn-i
=
n/2-l
e+v+ + e-V- + ^ \
/n/2-l
exp
e^pk
k=\
\
Y. ^k4>k J ,
(6.101)
and for odd n, as /
XQ + hlXl^
(n-l)/2
1-/j„-ix„_i = I e+t;++ /(n-l)/2
exp(
^
\
e/tpfe j
\
5 3 efc(/)fcj.
(6.102)
Writing in Eqs. (6.101) and (6.102) the radius pi, Eqs. (6.35) and (6.36), as a factor and expressing the variables in terms of the polar and planar angles with the aid of Eqs. (6.31)-(6.33) yields the trigonometric form of the n-complex number it, for even n, as _ ''"
/ny/2/_J__ U J
1
Vtan2^+"^tan2^_^
1
•^tan2^i • ^ t a n 2 ^ 2 " ^ " '
__l__V'^V±t^
e_v^
tan21/)„/2-2 /
tan^_
\tan0+
1
"^^ ^
ek
tanV'fc-i
216
Commutative Complex Numbers in n Dimensions
(6.103) and for odd n as
" = ^(i)
1 tan^ t/^i
( 74.^
1 tan^ ^2
+
••
-1/2
+ tan^V(n-3)/2, e+ v/2 tan ^+
(n-l)/2
+ ei+
^A:
E
tail'0A:-l
;!:=2
an-l)/2
exp I Y.
^^^^ I • (6-104)
k=\ In Eqs. (6.103) and 6.104), the n-complex number u, written in trigonometric form, is the product of the modulus d, of a part depending on the polar and planar angles ^^_, ^_, ^ i , . . . , 'ip[(n-3)/2] ? ^iid of a factor depending on the azimuthal angles <;^i, ...,0[(n~i)/2]- Although the modulus of a product of n-complex numbers is not equal in general to the product of the moduli of the factors, it can be checked that the modulus of the factor in Eq. (6.103) is n/2-l e+\/2 e-v/2 ^ ("k tan ^4tan0_ ~! tanV^/c_ k=2
_ py/y \nj 1 + --T-,
1
1
\ tan-^ 0-^ \'/' '
^
tan^ 6-
1
1
tan-^ ipi
tan-^ ip2 (6105)
tan'' il>n/2-21
and the modulus of the factor in Eq. (6.104) is (n-l)/2
e+ v/2 tan^ +
fc=2
2y/2/ n)
efc tanV^fc_i
1 tan'-^ e^
1 + tan'^ : V'(n-;j)/2
+ 1 +tan^ ij)\ + tan'^ •^j +
N 1/2
(6.106) /
Moreover, it can be checked t h a t r(("-i)/2! = 1.
exp fc=i
(6.107)
Polar complex numbers in n dimensions
217
The modulus d in Eqs. (6.103) and (6.104) can be expressed in terms of the amplitude p, for even n, as 2(n-2)/2n
d= p
xl/n
-;=— (tan0-1- t a n ^ - tan ^ i • • • tan ipn/2-2)
Vtan^'^9^
+ -^—^-:r~-f-l + : — T - T - + :—o~7-+ tan^0_ tan^i/?i tan'^t/j2
1
+r-Ti
V^'
'
(6-108)
tan^^^/2-2/
and for odd n as 2(n-l)/2n
d= p
1/^
j=— (tan^-f tan i/?i • • • tan ip(n-3)/2)
/ 1 Itan-^^-f.
,
1 tan^'i/'i
1 tan^i/'2
1 ^^^^i^in-^)/!/
1/2
(6.109) 6.1.5
E l e m e n t a r y functions of a polar n - c o m p l e x variable
The logarithm ui of the n-complex number n, wi = Intx, can be defined as the solution of the equation u^e^^K
(6.110)
For even n the relation (6.91) shows that Inti exists as an n-complex function with real components if i^-f = XQ + xi + - — + Xn-i > 0 and i;_ =^ XQ~XI + " ' + Xn-2 ~Xn-i > 0, which means that 0 < ^.f < 7r/2,0 < 0_ < 7r/2. For odd n the relation (6.93) shows that Intz exists as an ncomplex function with real components ii v+ = xo + xi -^ -- - + Xn-i > 0, which means that 0 < 9-^ < IT/2. The expression of the logarithm, obtained from Eqs. (6.95) and (6.96), is, for even n, n/2-l
lnu = e^lnv^
+ e-lnv-+
Y^ {eklnpk + hh)^
(6.111)
and for odd n the expression is (n-l)/2
lnu = e+lnv^+
JZ
{eklnpk + ekM-
(6.112)
218
Commutative Complex Numbers in n Dimensions
An expression of the logarithm depending on the ampHtude p can be obtained from the exponential forms in Eqs. (6.97) and (6.99), for even n, as n-l
\nu == Inp + y^ hp
n
tan 0.^
tan 9-
n
2"^' /27rfep\ 22 COS ( I In tan ipk^ i n k=2
n/2-l
+ Yl ^^^^'
(6.113)
k=l
and for odd n as (n-l)/2
n-l
1, V2 \nu = Inp-\- ^ hp — In n
tan^-i-
n
rr^
^\ ""^ n J'
(n-l)/2
(6.114) k=l
The function In u is multivalued because of the presence of the terms ekcpk- It can be inferred from Eqs. (6.37)-(6.43) and (6.46) that (6.115)
l n ( W ) = Int/ -h Inn', up to integer multiples of 27re;t^fc= 1,..., [(n — l)/2]. The power function u^ can be defined for real values of m as
(6.116) Using the expression of Inix in Eqs. (6.111) and (6.112) yields, for even values of n, n/2-l
u"^ = e^v"^ + e-v"^-^
J2 pf(ekCosm(l)k + €k^inm(f)k),
(6.117)
k=:i
and for odd values of n (n-l)/2
u ^ = e-f ?;![' +
^
p'kiek cos rn(f)k + e/, s\nm(j)k).
(6.118)
For integer values of m, the relations (6.117) and (6.118) are valid for any xo, ...,Xn-i. The power function is multivalued unless m is an integer. For integer m, it can be inferred from Eq. (6.115) that i\m = u"' „,m u.irn {uuT
(6.119)
Polar complex numbers in n dimensions
219
The trigonometric functions of the hypercomplex variable u and the addition theorems for these functions have been written in Eqs. (1-57)(1.60). In order to obtain expressions for the trigonometric functions of n-complex variables, these will be expressed with the aid of the imaginary unit i as 1 _ 1 cosu - -{e^^ + e ^^), s'mu = —(e^ 2 2z
(6.120)
*).
The imaginary unit i is used for the convenience of notations, and it does not appear in the final results. The validity of Eq. (6.120) can be checked by comparing the series for the two sides of the relations. Since the expression of the exponential function e^^^ in terms of the units 1, /ii, .../in-i given in Eq. (6.51) depends on the polar cosexponential functions gnp{y)j the expression of the trigonometric functions will depend on the functions gptiy) = {i/2)[9np{iy) + 9np{-w)] and^J,!^(y) = {l/2i)[gnp{iy) -gnp{-iy)l n-l
(6.121)
cos{hky) = Yl ^kp-n[kp/n]9pi{y)^ p=0
n-l
(6.122)
Sm{hky) = Yl ^kp~n[kp/n]9p-{y)^ p=0
where
9piiy) "^ ;^H {^^^ \y^^^(~)
(2'Klp\
'27d\
^^^^\y^^^(n J
2TXI\ . /27r/^ r /27r sinh ysm[ — sm y cos — L Vn
)].„(?^)}, ,.m,
(^) 1 !ili (r /27r/\l , r . /27r/\l 9pl{y) = - 2 ^ | s i n ycos ( — 1 cosh ysin ( — 1 l-Q
+ cos ycos
(v)j
27r/\ sinh y s3m m Il -— j sm
(2TTlp\
^"^ \rir-)
(^)}
(6.124)
The hyperbolic functions of the hypercomplex variable u and the addition theorems for these functions have been written in Eqs. (1.62)-(1.65). In order to obtain expressions for the hyperbolic functions of n-complex variables, these will be expressed as coshn = -(e"" + e"^), sinhw = -(e"" - e""").
(6.125)
220
Commutative Complex Nmnbers in n Dimensions
The validity of Eq. (6.125) can be checked by comparing the series for the two sides of the relations. Since the expression of the exponential function ^hkv [^ terms of the units l,/ii, .../i„_i given in Eq. (6.51) depends on the polar cosexponential functions gnpiv), the expression of the hyperbolic functions will depend on the even part gp-^{y) = {l/2)[gnp(y) -f gnp{-y)] and on the odd part gp^(y) = {l/2)[gnp(y) - gnp{-y)] of gnp, n-l
(6.126)
cosh{hky) = Yl^kp-n[kp/n]9p+{y)^ p=0 n-l
(6.127)
smhihky) = Y. f^kp-n[kp/n]9p-{y)^ p=0
where 9p^iy) = l t
{<^°«h [ycos ( ? ^ ) ] cos [ysin ( ^ ) ] cos ( ? ^ )
+ sinh ycos ( — )
sin ysin f — j sin f ^ ^ ^ j I ,
9p-{y) = - 5 Z jsinh ycosf — I cos + cosh
. /27rl\ ysm(^—j
COS
(6.128)
m
.^'°' (v)] '•" h'" (v)]''" (^)} • (^-^^^^
The exponential, trigonometric and hyperbolic functions can also be expressed with the aid of the bases introduced in Eqs. (6.80) and (6.81). Using the expression of the n-complex number in Eq. (6.85), for even n, yields for the exponential of the n-complex variable u n/2~l
e^ = e+e^++e_e^~ + J ] e^'(^fccosi)^ + e/fcsint)^).
(6.130)
k=l
For odd n, the expression of the n-complex variable in Eq. (6.86) yileds for the exponential {n-l)/2
e^ = e+e'^^ +
^
e^'^ (e^ cos Vk + h sin % ) ,
(6.131)
k=i
The trigonometric functions can be obtained from Eqs. (6.130) and (6.131 with the aid of Eqs. (6.120). The trigonometric functions of the
Polar complex numbers in n dimensions
221
n-complex variable u are, for even n, cos u = e^_ cos v^ -f e_ cos v^ n/2~-l
+ ^
(eit COS-Ufc cosh lifc-e^ sin i;itsinht;fc),
(6.132)
sinu = e-f sini;^ + e_ sint;_ n/2-l
+ ^
(ejfc sin-Ufc cosh t)/c + efc COS'Ufcsinh'y/c),
(6.133)
and for odd n the trigonometric functions are (n-l)/2
cosw = e+cost;_f.+ ^
(e^ cos T;/^ cosh t;^ — e^tsini^jtsinhiJjt), (6.134)
(n-l)/2
sinu = e^-sin?;^. +
^
(e^ sin 1;^^ cosht)jt + efccost^jtsinht)^) .(6.135)
A:=l
The hyperbohc functions can be obtained from Eqs. (6.130) and (6.131 with the aid of Eqs. (6.125). The hyperbohc functions of the n-complex variable u are, for even n, cosh u = CJ^ cosh i;_)_ + e_ cosh vn/2-1
4- 2Z (^A: cosh T;^ cos t;^ H-ejtsinht;^ sin {;/fc),
(6.136)
k=i
sinh u = e^ sinh v^ 4- e_ sinh Vn/2-1
-f ^
{eic sinhVk cosVfc + ejc coshVksinvfc),
(6.137)
fe=i
and for odd n the hyperbolic functions are cosh ti = e-j- cosh 1;+ (n-l)/2
+
^
(e/fc cosh-^fc COS i5it + Cfc sinh t;^ sint;^),
(6.138)
A:=l
sinhu = 64-sinh ^4(n-l)/2
4- ^2 fc=i
(efcsinhT;fcCOS?)fc 4-ejfccosh?;/fcsin0jfc).
(6.139)
222
Commutative Complex Nmnbers in n Dimensions
6.1.6
P o w e r series of polar n-complex n u m b e r s
An n-compIex series is an infinite sum of the form ao -f ai + a2 -f • • • 4- an + • • •,
(6.140)
where the coefficients a^ are n-complex numbers. The convergence of the series (6.140) can be defined in terms of the convergence of its n real components. The convergence of a n-complex series can also be studied using n-complex variables. The main criterion for absolute convergence remains the comparison theorem, but this requires a number of inequalities which will be discussed further. The modulus d = \u\ oi an n-complex number u has been defined in Eq. (6.20). Since |a:o| < |u|,|3;i| < |t£|,..., |xn-i| < |^|, a property of absolute convergence established via a comparison theorem based on the modulus of the series (6.140) will ensure the absolute convergence of each real component of that series. The modulus of the sum wi + U2 of the n-complex numbers wi, U2 fulfils the inequality \\u'\ - |w"|| < |ii' + u"| < |u'| -f \u'\.
(6.141)
For the product, the relation is IwV'l < v/^|u'||7i"|,
(6.142)
as can be shown from Eqs. (6.28) and (6.29). The relation (6.142) replaces the relation of equality extant between 2-dimensional regular complex numbers. The equality in Eq. (6.142) takes place for pip[ = 0, ...,p[(n_i)/2] From Eq. (6.142) it results, for u — u', that lu^l < v^|ti|2.
(6.143)
The relation in Eq. (6.143) becomes an equality for pi = 0, ...,/^[(n-i)/2] = 0 and, for even n^ v^ = 0 or v.. = 0. The inequality in Eq. (6.142) implies that |n'|
(6.144)
where / is a natural number. From Eqs. (6.142) and (6.144) it results that \au^\
(6.145)
A power series of the n-complex variable w is a series of the form ao + aiu + a2U^ 4- • • • + aiii^ -f • • •.
(6.146)
223
Polar complex numbers in n dimensions Since
(6.147) a sufficient condition for the absolute convergence of this series is that
lim V5i^!±ilM < 1.
(6.148)
/->oo
Thus the series is absolutely convergent for \u\ < c,
(6.149)
where C=
i i m -7=r
(6.150)
r.
/->oo ^yn\a^^l\ The convergence of the series (6.146) can be also studied with the aid of the formulas (6.117), (6.118) which for integer values of m are valid for any values of XQ, ...,:i:n-i, as mentioned previously. If a/ = YlpZo hpaip^ and n~l
(6.151) n-1
2'Kkp ^Ik = Yl ^^P ^^^ n p-0 n-l
Alk = Yl ^IP sm p=0
(6.152)
2'Kkp
(6.153)
n
for fc = 1,..., [(n — l)/2], and for even n n~l
yl,_ = X ( - l ) % p ,
(6.154)
p=0
the series (6.146) can be written, for even n, as n/2-1
E /=0
k=i
(6.155)
224
Commutative Complex Numbers in n Dimensions
and for odd n as (n~l)/2
e+Ai^v^^ + 1=0
J2
i^kAik + ekAik)(ekVk +
hhY
(6.156)
k=l
The series in Eq. (6.146) is absolutely convergent for |^;+| < C4., |?;_| < c_, pk < Ck,
(6.157)
for k = 1,..., [(n - l)/2], where c^ = hm —
c, = ^hn
r, c_ = lim 7-
r,
^^ ^''^ ^^,. VA + 1,A: "^ ^/ + l,A:j
(6.158)
The relations (6.157) show that the region of convergence of the series (6.146) is an n-dimensional cylinder. It can be shown that, for even n, c = (l/y/n) min(c-f ,c_,ci, ...,c„/2-i)^ and for odd n c = (l/y/n) min(c-{-,ci, ...,C(^_])/2)^ where min designates the smallest of the numbers in the argument of this function. Using the expression of \u\ in Eqs. (6.28) or (6.29), it can be seen that the spherical region of convergence defined in Eqs. (6.149), (6.150) is a subset of the cylindrical region of convergence defined in Eqs. (6.157) and (6.158). 6.1.7
A n a l y t i c functions of polar n-complex variables
The analytic functions of the hypercomplex variable u and the series expansion of functions have been discussed in Eqs. (1.85)-(1.93). If the n-complex function f{u) of the n-complex variable u is written in terms of the real functions Pk{xo,..',Xn-\),k = 0,1,...,n - 1 of the real variables X{),Xi^
..., Xfi-l
Q-S n-l
/(^) = 5Z^^^^(^0'-'^^-i)'
(6.159)
then relations of equality exist between the partial derivatives of the functions Pk' The derivative of the function / can be written as
l'i"n 4- E (hf^ E l^^-^'l '
(6.160)
Polar complex numbers in n dimensions
225
where n-l
Ai/= J2^i^^i'
(6.161)
A:=0
The relations between the partials derivatives of the functions P^t ^^e obtained by setting successively in Eq. (6.160) Aw = /i/Ax/, for / = 0,1,..., n— 1, and equating the resulting expressions. The relations are dPk ^ dPk-,1 ^ dxo dx\
^
dPn-i dxn-k-i
^
dPo ^ dxn-k
^ dPk-i dxn-i'
(gjg2)
for A: = 0,1, ...,n -^ 1. The relations (6.162) are analogous to the Riemann relations for the real and imaginary components of a complex function. It can be shown from Eqs. (6.162) that the components P^ fulfil the secondorder equations
dxodxi
dxidxi-i
dxm2]dxi_m2]
dxi+ldXn-\
= ^
dxi+2dXn-2
^
>
(6-163)
for kj = 0,1, ...,n — 1. 6,1.8
Integrals of polar n - c o m p l e x functions
The singularities of n-complex functions arise from terms of the form l/{u— uo)^, with n > 0. Functions containing such terms are singular not only at u = uo^ but also at all points of the hypersurfaces passing through the pole Uo and which are parallel to the nodal hypersurfaces. The integral of an n-complex function between two points A, B along a path situated in a region free of singularities is independent of path, which means that the integral of an analytic function along a loop situated in a region free of singularities is zero, i f{u)du = 0,
(6.164)
where it is supposed that a surface E spanning the closed loop F is not intersected by any of the hypersurfaces associated with the singularities of
226
Commutative Complex Numbers in n Dimensions
the function f{u). Using the expression, Eq. (6.159), for f{u) and the fact that n-l
du = ^
hkdxk,
(6.165)
A;=0
the exphcit form of the integral in Eq. (6.164) is - n- l
-
* f{u)du
71-1 n-l
= j> Yi ^''k Yl A;=0
(6.166)
PldXk-(-tn[(n-k-l-\-l)/n]-
1=0
If the functions Pk are regular on a surface S spanning the loop F, the integral along the loop F can be transformed in an integral over the surface E 0ftermS0fthef0rm5P//aTjt_^+n[(n-)t+m-l)/n]-5^m/c>Xfc-Hn[(n-A:+/-l)/n]-
These terms are equal to zero by Eqs. (6.162), and this proves Eq. (6.164). The integral of the function (?/ - UQ)^ on a closed loop F is equal to zero for m a positive or negative integer not equal to -1, /.
{u - uo)^du = 0, m integer, m ^
-I.
(6.167)
This is due to the fact that j{u — U{))^du — {u — UQ)^'^^/{m + 1), and to the fact that the function {u — UQY^'^^ is singlevalued for m an integer. The integral §^du/{u — UQ) can be calculated using the exponential form, Eqs. (6.97) and (6.99), for the difference u — UQ^ which for even n is fn-l II — no = pexp
n
yp=\
2""^^ ''
t a n dj^
tan 6-
n/2-l
/2nkp\,
> cos fc=2 V n
n
(6.168)
lntanV^jt-1
;
and for odd n is
{
u — uo
n-l
1, v/2 m n tan 6j^
p=l
(n-l)/2
—
2^
COS I
n
/
) Intant/^A:-i fe=:l (6.169)
Thus for even n the quantity rf?i/(n - UQ) is du
1
n-l
-din r- H n tanW-t-
rfln n
-—-— tant/_
227
Polar complex numbers in n dimensions n/2-l T)
-^-—'
COS
n fc=2
n/2-l
r27rkp\ „ I 1 alntan^fc-i
(6.170)
and for odd n du
I
n—1
w-wo
P
j ^
n
tan 6^
(n-l)/2
—
(n-l)/2
>
^
cos
fc=2
(6.171)
alntani/'fe_i ^
"^
^
Since p, ln(\/2/tan^4.),ln(\/2/tan^-),ln(tanV^/j_i) are singlevalued variables, it follows that ^ dpjp = 0, ^ d(ln \ / 2 / tan 0^.) = 0, ^ d{\n yflj tan 0-) = 0, ^ (i(ln tani/?jt-i) = 0. On the other hand since, (/)fc are cyclic variables, they may give contributions to the integral around the closed loop F. The expression of ^ dujiu — WQ) can be written with the aid of a functional which will be called int(M, C), defined for a point M and a closed curve C in a two-dimensional plane, such that int(M,C) =
1 if M is an interior point of C, 0 if M is exterior to C.
(6.172)
With this notation the result of the integration on a closed path F can be written as
i
du
lU-UQ
[(n~l)/2]
=
Y.
27reA:int(uoi,^,,F^,;yJ,
(6.173)
k-l
where WO^AT/A; ^^^ ^^km ^^^ respectively the projections of the point UQ and of the loop F on the plane defined by the axes ^k and rj^^ as shown in Fig. 6.3. If f{u) is an analytic n-complex function which can be expanded in a series as written in Eq. (1.89), and the expansion holds on the curve F and on a surface spanning F, then from Eqs. (6.167) and (6.173) it follows that f{u)du / .r u - iiQ
[(n-l)/2]
= 27r/(no)
J2
^fc in^(^o?,r?,,r^,r?J-
(6.174)
A:=l
Substituting in the right-hand side of Eq. (6.174) the expression of f{u) in terms of the real components F^, Eq. (6.159), yields
i
f{u)du
r n~uo
, [(n~l)/2] n - 1
5 E E". sm A:=l
int(iXo?fc77fc,r^fcr?J-
27r(/ - m)k n
PmM (6.175)
228
Commutative Complex Numbers in n Dimensions
Figure 6.3: Integration path F and pole WQ, and their projections F^^^^^ and u^^^^rik ^n ^he plane ikrikIt the integral in Eq. (6.175) is written as
f{u)du /.
n-l
E^'^<'
(6.176)
it can be checked that n-l
(6.177) 1=0
If f{u) can be expanded as written in Eq. (1.89) on F and on a surface spanning F, then from Eqs. (6.167) and (6.173) it also results that [("-l)/2]
/(r.)du_^^^(„,^^^^ / .r {u — UQ)^'^^
n\
E
^fcint(uo,,,„r^,,J,
(6.178)
k=i
where the fact has been used that the derivative f^^^uo) is related to the expansion coefficient in Eq. (1.89) according to Eq. (1.93). If a function f{u) is expanded in positive and negative powers of w — i//, where ui are n-complex constants, / being an index, the integral of / on a
Polar complex numbers in n dimensions
229
closed loop r is determined by the terms in the expansion of / which are of the form r//(w — w/),
Then the integral of / on a closed loop V is J, [(n~l)/2] (b /(n)du = 2 7 r ^ ^ CA; int(w/^^.^^,,r^^;;Jn. -^^
6,1.9
/
(6.180)
k=i
Factorization of polar n - c o m p l e x p o l y n o m i a l s
A polynomial of degree m of the n-complex variable u has the form Pm{u) = u'^ + aiu"^-^ +'-+
am-iu + am,
(6.181)
where a/, for / = l,...,m, are in general n-complex constants. If a/ = J2p=o hpaip, and with the notations of Eqs. (6,151)-(6.154) applied for / = 1, • • •, m, the polynomial Pm{u) can be written, for even n, as
n/2-l
+E A:=l
(6.182) where the constants Ai^,Ai^,Aik,Aik expression of the polynomial is
are real numbers. For odd n the
Pm^e^L^ + f^Ai^v^A ( n - l ) / 2 |A:=l
m 1=1
(6.183) The polynomials of degree m in efc'yfc + ^k^k in Eqs. (6.182) and (6.183) can always be written as a product of linear factors of the form ek{vk — '^kp) + ^ki^k ~" ^kp)^ where the constants v^p, v^p are real. The polynomials of degree m with real coefficients in Eqs. (6.182) and (6.183) which are multiplied by CJ^ and e_ can be written as a product of linear or quadratic factors with real coefficients, or as a product of linear factors which, if
230
Commutative Complex Numbers in n Dimensions
imaginary, appear always in complex conjugate pairs. Using the latter form for the simplicity of notations, the polynomial Pm can be written, for even n, as m
m
n/2-l m
+ Yl TL {^k{Vk - Vkp) -h hih - Vkp)} 1
(6.184)
A:=l p = l
where the quantities Vp^ appear always in complex conjugate pairs, and the quantities Vp^ appear always in complex conjugate pairs. For odd n the polynomial can be written as m
Pm=e^]J
(n-l)/2 m
(?^+ - Vp^) + p=\
X ] n i^kiVk - Vkp) + hih - Vkp)} , k=\
p~l
(6.185) where the quantities Vp^ appear always in complex conjugate pairs. Due to the relations (6.82),(6.83), the polynomial Pm{u) can be written, for even n, as a product of factors of the form m Pmiu)
= n {^+0^+ - ^^4-) + e^{v^ p:=l n/2-1
+ Yl {^kin - ^^kp) + ikih
-
Vp^) >
- Vkp)} > '
(6.186)
J
A;=l
For odd n, the polynomial Pm{'^^) can be written as the product m
Pm{u)= n {e+("+-'^p+) p=l (n-l)/2
+
^
J2 i^ki'^'k - ""kp) + h{Vk - f'kp)} > • fc=l J
(6.187)
These relations can be written witii the aid of Eqs. (6.85) and (6.86) as m Prniu)=ll{u-Up), p=\
(6.188)
where, for even n, n/2-l Up = e-|-i;p+ + e-.Vp-
+ ^
(eA:t;j^^p + h^^kp),
(6.189)
Polar complex numbers in n dimensions
231
and for odd n (n-l)/2
Up = e^Vpj^ +
Yl
(^kVkp + hvkp),
(6.190)
/c=l
for p = 1,..., m. The roots Vpj^^ the roots Vp- and, for a given fc, the roots ek'^ki + ^fc^fci, '"^ekVkm -I- eA;{;A;m defined in Eqs. (6.184) or (6.185) may be ordered arbitrarily. This means that Eqs. (6.189) or (6.190) give sets of m roots ?ii,...,Wrn of the polynomial Pm{u)^ corresponding to the various ways in which the roots Vp^^Vp^^ e^Vkp + ^k^kp ^^^ ordered according to p in each group. Thus, while the n-complex components in Eq. (6.183) taken separately have unique factorizations, the polynomial Pm{'^) can be written in many different ways as a product of linear factors. If P{u) = V? — I, the degree is m = 2, the coefficients of the polynomial are a\ = 0, a2 = —1, the n-complex components of a2 are a2o = —1,021 = 0, ...,a2,n-i = 0, the components A2^',A2-^A2k^^2k calculated according to Eqs. (6.151)-(6.154) are ^2+ = -1,>12- = -l,^2fc = -l^Mk = 0,fc = 1,..., [{n— l)/2]. The expression oiP{u) for even n, Eq. (6.182), is ej^{v\ — l) + e_(?;?.-l)-f-Xifc=r {(^fc^fc + ^A;^fc)^""^fc}i ^^d Eq. (6.184) has the form n 2 ~ l = 6H.(^+ + l)(^+-l)-f-e-(^_ + l ) ( ^ _ - l ) + E ; ? S " ' {^k{vk + 1) + hvk] {^ki'^k ~ 1) +h^k}' For odd n, the expression of P{u), Eq. (6.183), is e+{vX — l)+Ylk=i {(^A:'^A:+^fc^fc)^-e/fc}, andEq. (6.185) has the form u ^ 1 = ej^{v^ + l){v^ - 1) + EtlV^'^^ {ek{vk + 1) + hh} {ek{vk - 1) + ekVk}. The factorization in Eq. (6.188) is u^ — 1 = {u — ui){u — 1^2), where for even n, u\ = ±6+ ± e_ ± ei ± 62 ± • • • ± e?i/2-i?^2 = —^i, so that there are 2^*/^ independent sets of roots wi, W2 of w*^ — 1. It can be checked that ( ± e + ± e _ ±ei ±e2 ± • • • ±e^/2-i)^ = e+-f e_ -fei +62 + he^/2_i = 1. For odd n, ui — ±64. ib ei ± e2 ± • • • ± G{n-i)/2y'^'^2 = —'^1? so that there are 2(^-i)/2 independent sets of roots wi,U2 of u^ — 1. It can be checked that (±6+ ± ei ± 62 ± • • • ± e(„_i)/2)^ == e+ + ei + 62 + • • • + e(^_i)/2 = 1.
6.1.10
Representation of polar n-complex numbers by irreducible matrices
If the unitary matrix written in Eq. (6.21), for even n, is called Tg, and the unitary matrix written in Eq. (6.22), for odd n, is called To, it can be
232
Commutative Complex Numbers in n Dimensions
shown that, for even n, the matrix TgUT^ ' has the form
TeUT,- 1
( v^ 0 0
0 v_ 0
0 0 Vi
V 0
0
0
0 0 0
•••
(6.191)
F„/2_i /
and, for odd n, the matrix ToUT~^ has the form
ToUZ
-1
/ t;+ 0 0
0 0 1^1 0 0 F2
V 0
0
0
0 0 0
\
^(n-l)/2
/
(6.192)
where U is the matrix in Eq. (6.47) used to represent the n-complex number u. In Eqs. (6.191) and (6.192), I4 are the matrices Vk =
~Vk
Vk
(6.193)
for k = 1,..., [{n — l)/2], where Vk'>Vk ^^^ ^he variables introduced in Eqs. (6.14) and (6.15), and the symbols 0 denote, according to the case, the real number zero, or one of the matrices or
0 0 0 0
(6.194)
The relations between the variables Vk, Vk for the multiplication of n-complex numbers have been written in Eq. (6.44). The matrices TeUT~^ and ToUT~^ provide an irreducible representation [7] of the n-complex numbers u in terms of matrices with real coefficients.
6.2 6.2.1
P l a n a r complex numbers in even n dimensions O p e r a t i o n s w i t h planar n - c o m p l e x n u m b e r s
A hypercomplex number in n dimensions is determined by its n components (xo,a;i, ...,,Xn-i)- The planar n-complex numbers and their operations discussed in this section can be represented by writing the n-complex
Planar complex numbers in even n dimensions
233
Figure 6.4: Representation of the hypercompiex bases l^hi,.,.,hn-i by points on a circle at the angles a^ == nk/n. The product hjhk will be represented by the point of the circle at the angle 7r(j H- k)/2n, j^k = 0,1, ...,n — 1. If TT < 7r{j + k)/2n < 27r, the point is opposite to the ba.sis hi of angle a/ = 7r{j + k)/n — TT. number {xo^xi, .,.^Xn-i) as w = XQ + hix\ + /i2^2 H (" hn-iXn-i, hi,h2r " 1 hfi-i Sire bases for which the multiplication rules are hjhk = (-l)[(^'+^)/^l/i/, l=j
+ k- n[{i + k)/nl
where
(6.195)
for j , A;, / = 0,1,..., n - 1, where ho = 1. In Eq. (6.195), [{j + k)/n] denotes the integer part of {j + k)/n, the integer part being defined as [a] < a < [a] + 1, so that 0 < jf -h fc — n[{j -f- k)/n] < n — I. As already mentioned, brackets larger than the regular brackets [ ] do not have the meaning of integer part. The significance of the composition laws in Eq. (6.195) can be understood by representing the bases hj, h^ by points on a circle at the angles aj = nj/n,ak = nk/n^ as shown in Fig. 6.4, and the product hjh^ by the point of the circle at the angle 7r(j 4- k)/n. If TT < 7r(j + fc)/n < 27r, the point is opposite to the basis hi of angle a/ = n{j -f- k)/n — TT. In an odd number of dimensions n, a transformation of coordinates according to ^21 = x[,X2m-l
= -^(n-l)/24-m'
(6.196)
234
Commutative Complex Nmnhers in n Dimensions
and of the bases according to h2l = h[,h2m~l = -h[n-l)/2-hm^
(6.197)
where / = 0,..., (n — l)/2, m = 1,..., (n - l)/2, leaves the expression of an n-complex number unchanged, n~l
n-1
YihkXk = Yih'kxl
(6.198)
and the products of the bases /ij^ are h'jh'„ = hll=j^k-
n[{j -f k)/n],
(6.199)
for j^kj = 0,1,...,n — 1. Thus, the n-complex numbers with the rules (6.195) are equivalent in an odd number of dimensions to the polar ncomplex numbers described in the previous chapter. Therefore, in this section it will be supposed that n is an even number, unless otherwise stated. Two n-complex numbers n = XQ + hixi + h2X2 H -f- hn-iXn~i, u' — XQ + h\x\ + /i2^2 + • • • + hn~ix'^_i are equal if and only if Xj = x'^.j = 0,1, ...,n — 1. The sum of the n-complex numbers u and v! is w + z/' = xo 4- 4 -^ ^iC^'i + rr'i) -f • • • + hn^i{xn-x + < _ i ) .
(6.200)
The product of the numbers u^u' is UU' = XOX'Q - Xix'^_i
- X2X^ri~2 " ^ 3 ^ n - 3
-\-hx{:x^x\ -f x\x'^ - a:2<_i - X'^x\^2 + /l2(^0^2 + ^lA + ^2^0 - ^3a:'„„i
Xn~\x\
^n-i4) Xn~\x!'^)
(6.201)
+ /l^_l ( a : o < _ i + Xxx'^_2 + ^ 2 < _ 3 + ^ 3 < - 4 + • • • + ^ n - l 4 ) -
The product uv! can be written as
w=x:\fc x:(-i)[("-'=-i+"/"ix,x;_,^„((„_,_.^,,/„,.
(6.202)
If 1/, w', 1/" are n-complex numbers, the multiplication is associative (nn')t/' = u{y!v!')
(6.203)
and commutative mi = n't/,
(6.204)
Planar complex numbers in even n dimensions
235
because the product of the bases, defined in Eq. (6.195), is associative and commutative. The fact that the multiphcation is commutative can be seen also directly from Eq. (6.201). The n-complex zero is 0-f/ii-OH \-hn-i'0, denoted simply 0, and the n-complex unity is 1 + /ii • 0 + • • • 4- hn-i • 0, denoted simply 1. The inverse of the n-complex number u = XQ -^ hiXi -h h2X2 -+•••• + hn~iXn-i is the n-complex number u' ~ XQ-}- hix[ + /^2^2 "• ^ ^n-i^n_i having the property that W = 1.
(6.205)
Written on components, the condition, Eq. (6.205), is - xix'^_i - X2.<_2 - a;3a:'„_3 Xox[ + XIXQ - X2<_i - ^ 3 < - 2 xox'2 + xix[ -f X2x'o - a;3<_i
XOXQ
Xn-ix\ = 1, ^ n - l 4 = 0^ Xn~\x'^ = 0,
(6.206)
The system (6.206) has a solution provided that the determinant of the system, (6.207)
ly = det(A),
is not equal to zero, z>' 7^ 0, where /
A = {
^0
—Xji—\
x\
XQ
X2
Xi
X^ _ 1
X^
-Xn-l
''
X^
\
-X2
-x^
Xo
o
—X\
—Xn—2
1
Xn
(6.208) )
It will be shown that i/ > 0, and the quantity (6.209) will be called amplitude of the n-complex number u = XQ -^ hixi + h2X2 + h/^n-i^n-i- The quantity u can be written as a product of linear factors 71
i^=Y[
[^O + ^kXl +elx2 + '." +€^~^Xn-l),
(6.210)
A:=l
where e^ = e^^^'^^ ^^/^, k = l,...,n, and i being the imaginary unit. The factors appearing in Eq. (6.210) are of the form ^0 + ^kXl + 4^2 + • • • + e)b ^^n-1 ^Vk + iVk,
(6.211)
236
Commutative Complex Numbers in n Dimensions
where ^' n(2k ^k = 22 ^P*^^^
^^ . Vk = l^XpSin—
-l)p ,
7T{2k-l)p ^,
(6.212)
(6.213)
for fc = l,...,n. The variables Vk^v^.k ~ l,...,n/2 will be called canonical polar n-complex variables. It can be seen that v^ = t'n-A:+i?^fc = ~^n-k-\-\^ for A; = 1, ..,^n/2. Therefore, the factors appear in Eq. (6.210) in complexconjugate pairs of the form Vk-\-Wk and Vn-k-\-\ +i^n-A:+i = '^^k — i^k^ where k = l,...n/2, so that the determinant ly is a real and positive quantity, /y > 0 , n/2 ly =
- n PI
(6.214)
k=\
where 2 , -2 Pk
•
(6.215)
Thus, an n-complex number has an inverse unless it lies on one of the nodal hypersurfaces pi — 0, or p2 = 0, or ... or p^j^ — 0.
6,2.2
Geometric representation of planar n-complex numbers
The n-complex number x^-\-h\X\ 4-/^2^2H h/^n-i-^n-i ^*an be represented by the point A of coordinates (.xo,.ri,.... j:n-i)- If O is the origin of the n-dimensional space, the distance from the origin O to the point A of coordinates {;x^^x\,.,.,Xn-\) has the expression
d^ = a;^ + 2'? + . " + 4 - i -
(6.216)
The quantity d will be called modulus of the n-complex number ii = J:O 4h\X\ + h2X2 H V hri-iXfi-i. The modulus of an n-complex number u will be designated by d = \u\. The exponential and trigonometric forms of the n-complex number u can be obtained conveniently in a rotated system of axes defined by a
237
Planar complex numbers in even n dimensions transformation which has the form /
: \ Vk / fl
0
TIcOS^M^n
v^sin^^i^
...
/I,^3M2fc-lMn-2)
/ I cos M2fc-l)(n-I)
T f sin - ( ^ f e - y » - ^ ) Yn n
/ l s i „ ^ ( 2 f e - l ) ( nZil -l Yn n
a:o (6.217) \ Xn-i J where k = 1,2, ...,n/2. The hnes of the matrices in Eq. (6.217) give the components of the n vectors of the new basis system of axes. These vectors have unit length and are orthogonal to each other. By comparing Eqs. (6.212)-(6.213) and (6.217) it can be seen that (6.218) i.e. the two sets of variables differ only by a scale factor. The sum of the squares of the variables vi^^Vk is
T.(^Uvl) = '^d\
(6.219)
The relation (6.219) has been obtained with the aid of the relation n/2
^ _ ir{2k - l)p 7 cos = 0, n k=i
(6.220)
for p = 1, ...,n — 1. From Eq. (6.219) it results that (6.221) n
238
Commutative Complex Numbers in n Dimensions
The relation (6.221) shows that the square of the distance d, Eq. (6.216), is equal to the sum of the squares of the projections pk\/2/n. This is consistent with the fact that the transformation in Eq. (6.217) is unitary. The position of the point A of coordinates (.xo^^i, -.-^Xn-i) can be also described with the aid of the distance d, Eq. (6.216), and of n — 1 angles defined further. Thus, in the plane of the axes Vk^v^^ the azimuthal angle 0jt can be introduced by the relations cos(l)k = Vk/Pk^ sin(/)it = Vk/Pk^
(6.222)
where 0 < 0/^ < 27r, fe = 1, ...,n/2, so that there are n/2 azimuthal angles. The radial distance p^ in the plane of the axes Vfc^v^ has been defined in Eq. (6.215). If the projection of the point A on the plane of the axes Vk, Vk is i4fc, and the projection of the point A on the 4-dimensional space defined by the axes vi^vii'^^ki'^^k is A\fc, the angle ipk-i between the line OAik and the 2-dimensional plane defined by the axes Vk^Vk is t'dni/jk-i ^ PilPk,
(6.223)
where 0
(6.224)
tant/jA;-!
for fe = 2, ...,n/2, where 9 n(f / 1 1 1 \ //> r,^r\ /9? = - — 1 4- — 2 — 4- - - 2 — + • • • + — ^ . (6.225) 2 y tan^V^i tan^i/^2 ^^^^ rn/2-i J If U' =^ X'Q + hix[ + /?24 + • • • + /ln-l<-l,^^" = 4 ' + / M 4 ' + /l24' + ••• + /?,„_i4_i ^rc n-complex numbers of parameters p]^,V^j[.,0'^ and respectively pl,'^k^(i>l, then the parameters v^,pk,ipki(pk of the product ncomplex number u — u'u'^ are given by Pk = p'kPl
(6-226)
for k — l,...,/z/2, tan^/e = tan^^ tan^^',
(6.227)
Planar complex numbers in even n dimensions
A.k
\
239
k
P.
X
—y
\ / \ ' \k
O
0)
p,
'
Figure 6.5: Radial distance pic and azimuthal angle (f)^ in the plane of the axes VkiV^j and planar angle tpk-i between the line OAik and the 2dimensional plane defined by the axes Vk^Vk- ^k is the projection of the point A on the plane of the axes Vk^vi^ and A\jc is the projection of the point A on the 4-diinensional space defined by the axes vi^vi^Vk^Vk^ for k = l,...,n/2 ~ 1, (6.228)
^k = (f>'k + ^Jfc^
for k = l,...,n/2. relations
The Eqs.
(6.226)-(6.228) are a consequence of the
n = ^k4 - ^k^L ^k = v'k^k + ^'k^'L
(6.229)
and of the corresponding relations of definition. Then the product v in Eq. (6.214) has the property that u = u'ur
(6.230)
and the amplitude p defined in Eq. (6.209) has the property that p=PP"
(6.231)
240
Commutative Complex Numbers in n Dimensions
The fact that the ampHtude of the product is equal to the product of the ampHtudes, as written in Eq. (6.231), can be demonstrated also by using a representation of the n-complex numbers by matrices, in which the n-complex number u = XQ + h\Xx + h2X2 H + hn~\Xn-\ is represented by the matrix /
u
XQ
Xi
X2
Xn~\
^n — l
Xo
Xi
Xn-2
^n-2
~^n—1
XQ
^n—3
-xi
-X2
\
-xz
Xo
\
(6.232) )
The product u = tx'u" is be represented by the matrix multiplication JJ =z U'U". The relation (6.230) is then a consequence of the fact the determinant of the product of matrices is equal to the product of the determinants of the factor matrices. The use of the representation of the n-complex numbers with matrices provides an alternative demonstration of the fact that the product of n-complex numbers is associative, as stated in Eq. (6.203). According to Eqs. (6.219 and (6.215), the modulus of the product W is given by n/2
\^^f--X^pkP'k?'
(6.233)
k=\
Thus, if the product of two n-complex numbers is zero, uu' = 0, then PkP^k ~ ^•>k — l,...,n/2. This means that either ii, = 0, or u' — 0, or u,u' belong to orthogonal hypersurfaces in such a way that the afore-mentioned products of components should be equal to zero.
6.2.3
The planar n-dimensional cosexponential functions
The exponential function of a hypercomplex variable n and the addition theorem for the exponential function have been written in Eqs. (1.35)(1.36). It can be seen with the aid of the representation in Fig. 6.4 that /i^+^ = (-1)^/1^, p integer,
(6.234)
where A: = 1, ...,n ~ 1. For k even, e^^^ can be written as n-l
(6.235) p=0
Planar complex numbers in even n dimensions
241
where /lo = 1, and where gnp are the polar n-dimensional cosexponential functions. For odd fc, e^^^ is e^^y = J2(-'^)^'''"K-n[kp/n]fnpiy),
(6.236)
p=0
where the functions fnk, which will be called planar cosexponential functions in n dimensions, are y
k+pn
for A; = 0,1, ...,n — 1. The planar cosexponential functions of even index k are even functions, fn,2i{-'y) = fn,2i{y)^ ^^^ the planar cosexponential functions of odd index are odd functions, /n,2/+i(~y) = -/n,2/+i(y), I = 0,...,n/2 - 1 . The planar n-dimensional cosexponential function fnkiv) is related to the polar n-dimensional cosexponential function gnkiv) discussed in the previous chapter by the relation fnkiy) = e - - ^ / - 9nk . (e-/-y) ,
(6.238)
for A: = 0, ...,n — 1. The expression of the planar n-dimensional cosexponential functions is then n
fnkiv) = - J ^ e x p
ycosf
7r{2l-iy
J cos y sm (
I
n 7r(2/ -
l)k
(6.239) n for k = 0, l,...,n ~ 1. The planar cosexponential function defined in Eq. (6.237) has the expression given in Eq. (6.239) for any natural value of n, this result not being restricted to even values of n. In order to check that the function in Eq. (6.239) has the series expansion written in Eq. (6.237), the right-hand side of Eq. (6.239) will be written as
fnkiy) = - X ] ^ ^ | ^ ^ P ('cos
7r(2/-l) n
. . 7r(2/-l) h ^ sm > n
.7rfc(2/-l)]1
(6.240)
for fc = 0,1, ...,n— 1, where Re(a + i6) = a, with a and b real numbers. The part of the exponential depending on y can be expanded in a series,
1
fnkiy) =
(I
-Y.T.^^\-,^''P
•.7r(2/-l)
n
-ip-k)
V• } •
(6.241)
242
Commutative Complex Numbers in n Dimensions
forfc= 0, l,...,n fnkiy) =
1. The expression of fnkiv) becomes "7r(2/-l) f 1 -ip cos n
1 p
wherefc= 0,1, 1 n
E
-A:)]/},
(6.242)
1 and, since n{2l - 1)
COS
n
(p-fc)
1, if p —fcis an even multiple of n, = < 1 , if p — k is an odd multiple of n, 0, otherwise, (6.243)
this yields indeed the expansion in Eq. (6.237). It can be shown from Eq. (6.239) that
i: fnkiy) = ^E-^P f2^^os ( ^ ^ ^ ^ ) 1 .
(6.244)
It can be seen that the right-hand side of Eq. (6.244) does not contain oscillatory terms. If n is a multiple of 4, it can be shown by replacing y by iy in Eq. (6.244) that n/4
n-l
h-(^^)
(6.245)
which does not contain exponential terms. For odd n, the planar n-dimensional cosexponential function fnkiy) ^^ related to the n-dimensional cosexponential function gnkiy) discussed in the previous chapter also by the relation fnkiy) = i-D'onki-y),
(6.246)
as can be seen by comparing the series for the two classes of functions. For values of the form n = 4p + 2, p = 0,1,2,..., the planar n-dimensional cosexponential function fnkiy) is related to the n-dimensional cosexponential function gnkiy) by the relation (6.247)
fnkiy) = e-'^'^l'^gnkiiy).
Addition theorems for the planar n-dirnensional cosexponential functions can be obtained from the relation exp/ji(j/ + z) = erxjphi'ij • exp/iiz, by substituting the expression of the exponentials as given in Eq. (6.236) for A; = 1, e^^y = fnoiy) + hfniiy) + ••• + K-ifn,n-iiy), fnkiy
+ fnliy)fn,k-\iz)
+ Z) = fnoiy)fnkiz) -fn,k+liy)fn,n-liz)
+ ••• + fnkiy)
fnoi^)
- fn,k+2iy)fn,n~2iz)
-/n,n-l(y)/n,fe+l(^),
(6-248)
Planar complex numbers in even n dimensions
243
where fc = 0, l,...,n — 1. For y = z the relations (6.248) take the form
fnk{2y) = fnQ{y)fnk{y) + fni{y)fn,k-i{y) 4- • • • + fnk{y)fno{y) -fnMliy)fn,n~l{y)
- fnM2{y)fn,n-2{y)
-/n,n-l(y)/n,fc+l(2/),
(6.249)
where /e = 0,1,..., n - 1. For y = -z the relations (6.248) and (6.237) yield fno{y)fno{-y)
- fnl{y)fn,n-l{-y)
-fn.n-l{y)fnl{~y)
" fn2{y)fn,n-2{-y)
= h
(6.250)
fno{y)fnk{-y) + fn\{y)fn,k-i{-y) + "• + fnk{y)fno{-y) - / n , f c + l ( y ) / n , n - l ( - y ) - fn,k-^2{y)fn,n-2{-y)
-/n,n-l(y)/n,^+l(--J/)=0,
(6.251)
where fc = 1, ...,n — 1. From Eq. (6.235) it can be shown, for even k and natural numbers /, that ^n-l
\'
n-1
E(-l)'''^''^^P-[^p/n]^np(y)
=
E(-l)^''^"'^^P-r^[^p/n]^np(/?/),
(6.252) where k = 0,1,...,n — 1. For odd k and natural numbers /, Eq. (6.236) implies
(
n-l
y
n-1
E(-l)f'^/"l/^fcp-n[fcp/nl/np(y) P=0 /
= E (-l)'''^"'^^P-n[^p/n]/np(/?/), p=:0
(6.253) where A; = 0,1,..., n — 1. For k = 1 the relation (6.253) is {/no(y) 4- hifnliy)
+ ••• + /ln-l/n,n-l(y)}'
= /no(iy) + hifnlily)
+ '" + '^n—l/n,n—I (/y).
(6.254)
If ak = E /np(y) COS (^^^tLlk) p=o \ n
,
(6.255)
ft. = E fnpiy) sin ( ! [ l ^ i z l ) ^ ) , p=o \ n y
(6.256)
/
and
244
Commutative Complex Numbers in n Dimensions
for A; = 1,..., 71, where fnp(y) are the planar cosexponential functions in Eq. (6.239), it can be shown that /7r(2A;-l)y
ttk = exp
fn(2k-l)\]
h = exp
r . cos y sin I \ .sin
/ir(2k-l)\ n J
.
fn{2k-l)\]
(6.257)
(6.258)
for fc = l,...,n. If (6.259)
(^k = H + h^ from Eqs. (6.257) and (6.258) it results that r /7r(2A;-l) 'Gl = exp 2 y c o 8 ( - ^
)]
(6.260)
1
for k = l,...,n. Then the planar n-dimensional cosexponential functions have the property that n/2
(6.261) p=l
The planar n-diniensional cosexponential functions are solutions of the n^"-orcler differential equation (6.262)
-c.
This equation has solutions of the form ((?/) = Aofno(u) + A]fni{u) -f • • • +An-}fn,n-}{'^^)' It Can be checked that the derivatives of the planar cosexponential functions are related by dfnO _ r — ~Jn,n~\i
du
du
6,2.4
~
dfn\ _ . dfn,n~2 j _ -~ JnO^ •••• ;_
an
du
_ ~
, Jn,n-3')
Jn,n-2'
(6.263)
Exponential and trigonometric forms of planar n-complex numbers
In order to obtain the exponential and trigonometric forms of n-complex numbers, a canonical base ei, ei,..., e^,y2, e^/2 f^^^ ^^^ planar n-complex num-
Planar complex numbers in even n dimensions
245
bers will be introduced by the relations /
: \ efc
( : 2 n
2 ;r(2fc-l) n^^^ n
0
Isin2l2^
/
1
2 cos '^(2fc-l)(n-2)
2 ^.^g
2^ n
2 : n
n{2k-l){n-2) n
n{2k-\){n-l) ^(2fc-lV-l) n
\
hi
(6.264)
\ hn-l J where A: = 1,2, ...,n/2. The multiplication relations for the bases ek^e^ are
(6.265) where fc, / = 1,..., n/2. The moduh of the bases e^, e^ are (6.266) It can be shown that n/2 Xo + hiXi
+ • • • + hn-\Xn~l
= ^{ekVk
+ e^f^A;).
(6.267)
The relation (6.267 gives the canonical form of a planar n-complex number. Using the properties of the bases in Eqs. (6.264) it can be shown that expiekM
= 1 - e)t + e^: cos(l)k + h sm(t)k,
exp(efc Inpk) = 1 ~ e^ + Ckph-
(6.268) (6.269)
By multiplying the relations (6.268), (6.269) it results that n/2
'n/2
exp Y^icklnpk k=i
(6.270)
+ ekM k=l
246
Commutative Complex Numbers in n Dimensions
where the fact has ben used that n/2
(6.271) the latter relation being a consequence of Eqs. (6.264) and (6.220). By comparing Eqs. (6.267) and (6.270), it can be seen that 'n/2
xo + hiXi -\
h hn-iXn-i
(6.272)
= exp Y^ieklupk + hM k=\
Using the expression of the bases in Eq. (6.264) yields the exponential form of the n-complex number u = XQ + hiXi H + hn-\Xn~.\ as
{
u
n-l
2"^ /7r(2A:-~l)p\, — > cos lntan'0A:-i n k^2 n
Y,hp ^
n/2
(6.273) where p is the amplitude defined in Eq. (6.209), and has according to Eq. (6.214) the expression l/n
(6.274)
P = (pi • • • pl/2)
It can be checked with the aid of Eq. (6.268) that the n-complex number u can also be written as /n/2
^0 + hixx + • • • -f hn-\Xn~\ =
\
/n/2
\
JZ ^^Pk ] ^^P [Yl ^^^^ ik={
'
(6-275)
ik^l
Writing in Eq. (6.275) the radius pi, Eq. (6.225), as a factor and expressing the variables in terms of the planar angles with the aid of Eq. (6.223) yields the trigonometric form of the n-complex nmnber w as
2)
\
tan^V^i
X -1/2
n/2
n/2
«> + E r.
tan^V^2
1 ^^^^^ tan^^^/2-i
ek
(6.276)
In Eq. (6.276), the n-complex inimbcr w, written in trigonometric form, is the product of the moduhis d, of a part depending on the planar angles
Planar complex
numbers
in even n
247
dimensions
?/?!,..., ^n/2-i5 ^^d of a factor depending on the azimuthal angles >i,..., (f>n/2Although the modulus of a product of n-complex numbers is not equal in general to the product of the moduli of the factors, it can be checked that the modulus of the factors in Eq. (6.276) are n/2
^tan^/^it-^i 2^l/2/
=©
1 1 1+ . o . + tan^ rpi ta,n^ ip2
1/2
+ ••• + t a n 2 ^ ^ / 2 - i (6.277)
and n/2
exp I J ^ ek(t)k
(6.278)
= 1.
/c=l
T h e modulus d in Eqs. (6.276) can be expressed in terms of the amplit u d e p as d = p
2(n-2)/2n 2/n n • •''•tan^ y=— (^tanV'i -tan-0^/2-ij \/n
1+ 6.2.5
1 1 4tan^^ t/)i tan"^ -02
1
+ ••• + tan^'0„/2-i
1/2
(6.279)
Elementary functions of a planar n-complex variable
T h e logarithm ui of the n-complex number ti, ui = Inn, can be defined as the solution of the equation w= e
Ui
(6.280)
T h e relation (6.270) shows t h a t I n n exists as an n-complex function with real components for all values of XQ, ...,a;n-i for which p 7^ 0. T h e expression of the logarithm, obtained from Eq. (6.272) is n/2
l n n = ^{eklnpk
+
ekM-
(6.281)
248
Commutative Complex Numbers in n Dimensions
An expression of the logarithm depending on the amplitude p can be obtained from the exponential forms in Eq. (6.273) as 2 "^ (Tx{2k - \)p\ , In u = In p + y ^ /ip — 2^ cos I J In tan ipk-i n/2
+ Y.^k(t>k'
(6.282)
k-\
The function Inu is multivalued because of the presence of the terms e^0fc. It can be inferred from Eqs. (6.226)-(6.228) and (6.231) that l n ( W ) = Inn + Ini/,
(6.283)
up to integer multiples of 27rejt, k = 1,..., n/2. The power function u^ can be defined for real values of m as ^^m^gmlnu
^Q 284)
Using the expression of In?/ in Eq. (6.281) yields Tl/2
^"^ = Ylpf
{ek COS m(l)k -i-eksimncpk)'
(6.285)
The power function is multivalued unless m is an integer. For integer m, it can be inferred from Eq. (6.283) that ( W ) ^ = ti^^tx'^^.
(6.286)
The trigonometric functions of the hypercomplex variable u and the addition theorems for these functions have been written in Eqs. (1.57)(1.60). In order to obtain expressions for the trigonometric functions of n-complex variables, these will be expressed with the aid of the imaginary unit i as cosix = ^(e^^ + e-"^), sinu = ~{e''' ~ e"^^).
(6.287)
The imaginary unit i is used for the convenience of notations, and it does not appear in the final results. The validity of Eq. (6.287) can be checked by comparing the series for the two sides of the relations. Since the expression of the exponential function e/^^'^ in terms of the units 1, hi, .../in-i given in Eq. (6.236) depends on the planar cosexponential functions fnpiy)^
249
Planar complex numbers in even n dimensions
the expression of the trigonometric functions will depend on the functions fi%) = {im[fnp{iy) + fnp{-iy)] a n d / i l ^ y ) = {I/2i)[fnp{iy)fnp{-iy)], (6.288)
COS(hky) = E(-l)^'''^"''»^P-n[/tp/nl4+(?/)' p=0 n-l
sinihky) =
(6.289)
J2(-^)^''^"K-n[kp/n]fi%), p=0
where
f>)='-t{-os
7r(2/ - 1) ycos
n f7r{2l -
. /7r(2/-l)\l
cosh
— sin ycos
l)p\
n 7r(2/ - 1) n
-{'^M'^^)}^
sinh ysin
Acl \ y ) = i | : {sin [ycos ( ^ ^ ( ^ ) . (•n{2l-\)W
cosh r
(•K{2l-l)p\
(T^(2i-i)\]
cos sinh
.
/7r(2/-l)\l
.
(Tr{2l-\)p\\
(6.291)
The hyperbolic functions of the hypercomplex variable u and the addition theorems for these functions have been written in Eqs. (1.62)-(1.65). In order to obtain expressions for the hyperbolic functions of n-complex variables, these will be expressed as cosh It = - ( e " + e""), sinht* = - ( e " - e~").
(6.292)
The validity of Eq. (6.292) can be checked by comparing the series for the two sides of the relations. Since the expression of the exponential function ^kV in terms of the units 1, hi^.,.hn~i given in Eq. (6.236) depends on the planar cosexponential functions fnpiv)^ the expression of the hyperbolic
250
Commutative Complex Numbers in n Dimensions
functions will depend on the even part fp+{y) = (l/2)[/„p(y) + /„p(-y)] and on the odd part /p_(y) = (l/2)[/„p(?/) - /„p(-y)] of /„p, n-l
cosh(/ifcy) =
Y^i-lf^^^'Kp-nikpMfp+iy),
(6.293)
p=0 n-l
(6.294) p=0
where
/p+(y) = -
Z
n (2/-l)=l
)]-(^^^)
cos ysm
inh w cos + sinh L
H"^)
{c^^h
\
n
J (6.295)
. r . fM^l-m]
.
/p-(y) - L f ' n h ^ycos (^ cos=I ysm
fn(2l-l)p\\
j
/7r(2^-l)\] + cosh r . /7r(2^-l)\l . /7r(2[-l)p\1 Sin I?/sin I —^ 11 sin I —^ ^ ) ^.
(6.296)
The exponential, trigonometric and hyperbolic functions can also be expressed with the aid of the bases introduced in Eq. (6.264). Using the expression of the n-complex inimber in Eq. (6.267) yields for the exponential of the n-complex variable n n/2
^u ^ Y^ ^vk (g^ ^Qg ^^ ^ g^ gjni;^). (6.297) A;=l The trigonometric functions can be obtained from Eq. (6.297) with the aid of Eqs. (6.287). The trigonometric functions of the n-complex variable u are n/2
COS w = ] ^ (e^ cos Vk cosh Vk — ek sin v^ sinh % ) , A:=l
(6.298)
Planar complex numbers in even n dimensions
251
n/2
s i n n = y ^ (efc sin Vk cosh Vk + e^ cos t?jb sinh Vk).
(6.299)
The hyperbohc functions can be obtained from Eq. (6.297) with the aid of Eqs. (6.292). The hyperbohc functions of the n-complex variable u are n/2
cosh u — Y^i^k cosh Vk COS Vk + ejt sinhi;jt sin t)jt),
(6.300)
k=l n/2
sinhw = y ^ (efc sinh v^ cos Vk + ^k cosh ^^ sin Vk) -
(6.301)
k=l
6.2.6
Power series of planar n-complex numbers
An n-complex series is an infinite sum of the form ao + ai + a2 -i
\- an-]
,
(6.302)
where the coefficients a-n are n-complex numbers. The convergence of the series (6.302) can be defined in terms of the convergence of its n real components. The convergence of a n-complex series can also be studied using n-complex variables. The main criterion for absolute convergence remains the comparison theorem, but this requires a number of inequalities which will be discussed further. The modulus d = \u\ of an n-complex number u has been defined in Eq. (6.216). Since \xo\ < \u\,\xi\ < \u\^.,.^\xn-i\ < |w|, a property of absolute convergence established via a comparison theorem based on the modulus of the series (6.302) will ensure the absolute convergence of each real component of that series. The modulus of the sum ui -f- U2 of the n-complex numbers ui, U2 fulfils the inequality \\u'\ -- |w''|| < |n' -f u"| < \u'\ + |i/"i.
(6.303)
For the product, the relation is \u'u"\ < y | | u ' | | u " | ,
(6.304)
as can be shown from Eq. (6.221). The relation (6.304) replaces the relation of equality extant between 2-dimensional regular complex numbers.
252
Commutative Complex Numbers in n Dimensions For u~u'
Eq. (6.304) becomes
l^'l < y ^ h f ,
(6.305)
and in general W\ < (^2 j
l"i '
(6-306)
where / is a natural number. From Eqs. (6.304) and (6.306) it results that \au'\ < (^-j
|a||«|'.
(6.307)
A power series of the n-complex variable u is a series of the form ao + aiu + a^v? 4- • • • + aivt + • • •.
(6.308)
Since
< Yi(nl2)'l^\a,\\ut
(6.309)
a sufficient condition for the absolute convergence of this series is that lim ^ '
7^^' ' < 1.
(6.310)
\ai\
i-^oo
Thus the series is absolutely convergent for 1^1 < c ,
(6.311)
where
c = Iim-^4^;J
-.
(6.312)
/->oo i/n/2|a/^_i| The convergence of the series (6.308) can be also studied with the aid of the formula (6.285) which is valid for any values of XQ, ..., Xn-i, as mentioned previously. If ai = YlpZo hpaip, and ^ "i^ 7r{2k - l)p ^ik = 2^ Clip cos , p=o ^ . ^-^ . 7r(2fc-l)p Aik = V Clip sm , p=0
^..._^ (6.313)
(6.314)
Planar complex numbers in even n dimensions
253
where A; = 1,..., n/2, the series (6.308) can be written as "n/2
E
Y,{ekAik + ekAik){ekVk +
hh)
(6.315)
The series in Eq. (6.315) can be regarded as the sum of the n/2 series obtained from each value of fc, so that the series in Eq. (6.308) is absolutely convergent for Pk < Ck,
(6.316)
for A; = 1,..., n/2, where
Ck = lim — L i ^
'-^
-J.
(6.317)
The relations (6.316) show that the region of convergence of the series (6,308) is an n-dimensional cylinder. It can be shown that c = \f2jn min(c^,c_,ci, ...,0^^2-1)7 where min designates the smallest of the numbers in the argument of this function. Using the expression of \u\ in Eq. (6.221), it can be seen that the spherical region of convergence defined in Eqs. (6.311), (6.312) is a subset of the cylindrical region of convergence defined in Eqs. (6.316) and (6.317). 6.2.7
A n a l y t i c f u n c t i o n s of planar n - c o m p l e x variables
The analytic functions of the hypercomplex variable u and the series expansion of functions have been discussed in Eqs. (1.85)-(1.93). If the n-complex function j{u) of the n-complex variable u is written in terms of the real functions P^(3;o, ...,a:n-i),fc = 0,1,...,n — 1 of the real variables n-1
f[u) = 5]/ifcPfc(:ro,...,:rn-i),
(6.318)
where /IQ = 1, then relations of equality exist between the partial derivatives of the functions F;f The derivative of the function / can be written as
li™ ^ E ^fc E l ^ ^ ^ ' l '
(6-319)
254
Commutative Complex Numbers in n Dimensions
where n-]
A w = Y^hiAxi.
(6.320)
The relations between the partials derivatives of the functions P^ are obtained by setting successively in Eq. (6.319) An = hiAxi, for / = 0,1,..., n 1, and equating the resulting expressions. The relations are dPk ^ dP^ dxo dxi
^
^
dPn-i dxn~k-\
^
9P0 ^ dxn~k
^
dPk-i dxn-i'
for A: = 0,1, ...,n — 1. The relations (6.321) are analogous to the Riemann relations for the real and imaginary components of a complex function. It can be shown from Eqs. (6.321) that the components Pk fulfil the secondorder equations d'Pk dxodxi
d'Pk dxidxi^i
d'^Pk ^•^[//2]^^/-[;/2l
dxi^]dXn~l
^
dxi^2dXn^2
d^
(6.322)
dXi^]^l(^n-l-2)/2]9Xn-l-[{n-l-2)/2]' for A:,/ = 0 , 1 , ...,n — 1.
6.2.8
Integrals of planar n - c o m p l e x functions
The singularities of n-complex functions arise from terms of the form l/(n — UQY^, with m > 0. Functions containing such terms are singular not only at w = uo, but also at all points of the hypersurfaces passing through the pole UQ and which are parallel to the nodal hypersurfaces. The integral of an n-complex function between two points A, B along a path situated in a region free of singularities is independent of path, which means that the integral of an analytic function along a loop situated in a region free of singularities is zero, i f{u)du = 0,
(6.323)
where it is supposed that a surface E spanning the closed loop F is not intersected by any of the hypersinfaces associated with the singularities of
Planar complex numbers in even n dimensions
255
the function f{u). Using the expression, Eq. (6.318), for /(n) and the fact that n-l
du = ^
(6.324)
hkdxk,
the explicit form of the integral in Eq. (6.323) is
/ f{u)du=/ x^ /ifc i:(-i)'("-'-'+')/"iPidxfc_i+„[(„_fe_i+i)/„]. (6.325) If the functions Pk are regular on a surface E spanning the loop F, the integral along r can be transformed in an integral over the surface E of terms of t h e form dPi/dXk-m->rn[{n-k-\-m-\)ln]
~
~l-^n[{n-k+l-l)/n]'>
where s = [{n — k + m — 1)/n] — [{n — k +1 — 1)/n]. These terms are equal to zero by Eqs. (6.321), and this proves Eq. (6.323). The integral of the function {u — uo)^ on a closed loop F is equal to zero for m a positive or negative integer not equal to -1, (6.326)
j> {u — uo)^du = 0, m integer, rn ^ —1.
This is due to the fact that J{u - uo)'^du = (u - wo)^"^V(^ + 1)^ ^^^ to the fact that the function {u - txo)"^"^^ is singlevalued for n an integer. The integral §j.du/{u — UQ) can be calculated using the exponential form, Eq. (6.273), for the difference u ~ UQ,
i
n-l
y cos (
I lntant/^fc_i
p=l
n/2
(6.327)
+ Yl ^^^^ k=l
Thus the quantity du/{u — UQ) is du
dp
v-^ .
—- = — + E ^
2^
/2nkp\
— y cos alntan7/?^_i "fc=2 V n y
n/2
+ E hd(t>kfc=i
(6.328)
256
Commutcitive Complex Numbers in n Dimensions
Sincep and ln(tanipk-i) are singlevalued variables, it follows that ^ dp/p = 0, and ^d{lnta.mpk_i) = 0. On the other hand, since (pk are cyclic variables, they may give contributions to the integral around the closed loop
r. The expression of ^ du/{u — WQ) can be written with the aid of a functional which will be called int(M, C), defined for a point M and a closed curve C in a two-dimensional plane, such that ' .rn^ ^\ I 1 if M is an interior point of C, mt(M,C) = < ^ .. , . . . . . ^ ^ ^ 1 0 if M IS exterior to C
,^ ^^^, (6.329) ^ ^
With this notation the result of the integration on a closed path F can be written as du /.
""' = Y\ 27reA; int(wo^^^^,r^^^ ),
(6.330)
where ^^0Cfc7?fc and r^;^^^ ^^^ respectively the projections of the point u^ and of the loop r on the plane defined by the axes (^k and r/)t, as shown in Fig. 6.6. If f{u) is an analytic n-complex function which can be expanded in a series as written in Eq. (1.89), and the expansion holds on the curve F and on a surface spanning F, then from Eqs. (6.326) and (6.330) it follows that
i
rn-uo
= 2nf{uo) J2 ^fc •"t("oe^'?;t' ^^A')J^^,
(6.331)
Substituting in the right-hand side of Eq. (6.331) the (expression of f{u) in terms of the real components Pk, Eq. (6.318), yields
i
j{u)du ^ 2 yv ^ ^^ "-*
'n{2k-l)p
5:(-l)[('-'')Hsin P ^ ^ ^ - ^
int(Mo?,;„,r^,;,J.
P„-p+,-nl(n-p+0/n](«0)
(6.332)
It the integral in Eq. (6.332) is written as
i
i^^^ ru-uo
= y lull, fr'o
(6.333)
Planar complex numbers in even n dimensions
257
Figure 6.6: Integration path T and pole UQ, and their projections T^^^^ and uo^j^.r)k Oil ^he plane ^km-
it can be checked that 72-1
(6.334) /==o
If f{u) can be expanded as written in Eq. (1.89) on T and on a surface spanning F, then from Eqs. (6.326) and (6.330) it also results that
i
f(
W
9
^
r {u-~ Wo)
[(^-i)/2]
= ;^/^"nno) ni
E
e,int(«oc..„rc,,J,
(6.335)
fc=l
where the fact has been used that the derivative /^^H^o) is related to the expansion coefficient in Eq. (1.89) according to Eq. (1.93). If a function f{u) is expanded in positive and negative powers ofu — ui^ where ui are n-complex constants, / being an index, the integral of / on a closed loop F is determined by the terms in the expansion of / which are of the form n/(w — w/), /(w) = . . . + 5 ] - ! L . + . . . .
(6.336)
u-ui Then the integral of / on a closed loop F is
i
f{u)du -=2nJ2Yl^k I
fc=i
int(w/^,;^^,F^.^Jn-
(6.337)
258
Commutative Complex Numbers in n Dimensions
6.2.9
Factorization of planar n - c o m p l e x p o l y n o m i a l s
A polynomial of degree m of the n-complex variable u has the form Pm{u) ^u"^ + aiu"^-^ + • • • -f am-\u + am:
(6.338)
where a;, for / = l,...,m, are in general n-complex constants. If a/ = YlpZo hpaip, and with the notations of Eqs. (6.313), (6.314) applied for / = 1, • • •, m, the polynomial PmM ^an be written as n/2 r
m
\
k^\ I
1=1
J (6.339)
where the constants The polynomials written as a product where the constants
Aik.Aik are real numbers. of degree m in ekVk-VekVk in Eq. (6.339) can always be of linear factors of the form ek{vk — Vkp) + ek{vk — Vkp)-, v^p-^ v^p are real, m
{ckVk + ekh)'^ + Y^i^kAik + ekAik){ekVk +
hh)"^"^
m
= n {^k(^f^ " '^kp) + ^kih - Vfcp)} .
(6.340)
Then the polynomial Pm can be written as n/2
m
^^"="Y1]1
{^k[vk - Vkp) + hivk - ^kp)] '
(6.341)
k=lp=\
Due to the relations (6.265), the polynomial Pm(w) can be written as a product of factors of the form m
(n/2
^m(^) = n { Z ] {^ki'^^k - Vkp) + hih
\
- ^A;p)} > •
(6.342)
This relation can be written with the aid of Eq. (6.267) as m
Pm(r/)=n(^^~M'
(6-343)
P=:l
where n/2
^P = H {^k^p + eA:^A:p) , A;=l
(6.344)
Planar complex numbers in even n dimensions
259
for p = 1,..., m. For a given fc, the roots ekVki + ek^ku-i ekVkm + h^km defined in Eq. (6.340) may be ordered arbitrarily. This means that Eq. (6.344) gives sets of m roots ui^.,.^Um of the polynomial Pm{y)^ corresponding to the various ways in which the roots ekVkp + ekVkp ^^e ordered according to p for each value of k. Thus, while the n-complex components in Eq. (6.340) taken separately have unique factorizations, the polynomial ^m(^) can be written in many different ways as a product of linear factors. If P(u) = 1^^ + 1, the degree is m = 2, the coefficients of the polynomial are «i = 0, a2 = 1? ^he n-complex components of ^2 ^^^ <^20 = l,fl2i = 0, ...,a2,n~i = 0, the components A2kyA2k calculated according to Eqs. (6.313), (6.314) are A2k = 1,^2^ = 0,k = l,...,n/2. The left-hand side of Eq. (6.340) has the form {ekVk + ekVk)^ + ^ki stud since ek = ~e^, the right-hand side of Eq. (6.340) is [ekVk + hi^k + 1)} {^k'^k + h{h - l)}^ so that Vkp = ^,Vkp = ±l,fc = l,...,n/2,p = 1,2. Then Eq. (6.341) has the form n^ + 1 = TH'^I {^k'^k + h{h + 1)} {^k'^k + h{^k - !)}• The factorization in Eq. (6.343) is w'^ -f 1 = {u — u\)[u — U2)'> where wi = ±ei ± 62 ± • • • ± eyj/2,n2 = —u\, so that there are 2^/^~^ independent sets of roots u\^U2 of vr + 1. It can be checked that (±ei ± 82 ± • • • ± eri/2)'^ = -61-62
6.2.10
e^/2 = - 1 -
Representation of planar n-compIex numbers by irreducible matrices
If the unitary matrix written in Eq. (6.217) is called T, it can be shown that the matrix TUT^^ has the form
TUT'^
=
( Vx 0 . . . 0 F2 •••
V 0
0
...
0 0
\ (6.345)
v;/2 ;
where U is the matrix in Eq. (6.232) used to represent the n-complex number u. In Eq. (6.345), the matrices Vk are the matrices
V, = { "". ' 0 ,
(6.346)
for fc = 1, ...,n/2, where Vk,Vk are the variables introduced in Eqs. (6.212) and (6.213), and the symbols 0 denote the matrix (6.347)
260
Commutative Complex Numbers in n Dimensions
The relations between the variables ?;^, v^ for the multiplication of n-complex numbers have been written in Eq. (6.229). The matrix TUT~^ provides an irreducible representation [7] of the n-complex number u in terms of matrices with real coefficients. For n = 2, Eqs. (6.212) and (6.213) give vi = ^0,^1 = ^1, and Eq. (6.264) gives ei = l,ei = hi, where according to Eq. (6.195) h'l = ~ 1 , so that the matrix Ki, Eq. (6.346), is
.1 == f "^ " 0 '
(6.348)
\^ ~Xi Xo J which shows that, for n = 2, the hypercomplex numbers XQ + hiXi are identical to the usual 2-dimensional complex numbers x + iy.
Bibliography [1] G. Birkhoff and S. MacLane, Modem Algebra (Macmillan, New York, Third Edition 1965), p. 222. [2] B. L. van der Waerden, Modern Algebra (F. Ungar, New York, Third Edition 1950), vol. II, p. 133. [3] O. Taussky, Algebra, in Handbook of Physics, edited by E. U. Condon and H. Odishaw (McGraw-Hill, New York, Second Edition 1958), p. 1-22. [4] D. Kaledin, arXiv:aIg-geom/9612016; K. Scheicher, R. F. Tichy, and K. W. Tomantschger, Anzeiger Abt. II 134, 3 (1997); S. De Leo and P. Rotelli, arXiv:funct-an/9701004, 9703002; M. Verbitsky, arXiv:alggeom/9703016; S. De Leo, arXiv:physics/9703033; J. D. E. Grant and I. A. B. Strachan, arXiv:solv-int/9808019; D. M. J. Calderbank and P. Tod, arXiv:math.DG/9911121; L. Ornea and P. Piccinni, arXiv:math.DG/0001066. [5] S. Olariu, arXiv:math.OA/0007180, math.CV/0008119-0008125; Int. J. Math. Math. Sci. vol. 25, p. 429 (2001). [6] E. T. Whittaker and G. N. Watson A Course of Modern Analysis, (Cambridge University Press, Fourth Edition 1958), p. 83. [7] E. Wigner, Group Theory (Academic Press, New York, 1959), p. 73.
261
This Page Intentionally Left Blank
Index polar 5-complex, 159 polar 6-complex, 175 polar fourcomplex, 127 polar n-complex, 212 tricomplex, 40 twocomplex, 5 canonical form circular fourcomplex, 67 hyperbolic fourcomplex, 81 planar 6-complex, 189 planar fourcomplex, H I planar n-complex, 245 polar 5-complex, 160 polar 6-complex, 175 polar fourcomplex, 127 polar n-complex, 213 twocomplex, 5 canonical variables circular fourcomplex, 56 hyperbolic fourcomplex, 79 planar 6-complex, 183 planar fourcomplex, 95 planar n-complex, 236 polar 5-complex, 151 polar 6-complex, 169 polar fourcomplex, 125 polar n-complex, 201 tricomplex, 40 twocomplex, 3 complex units circular fourcomplex, 54 hyperbolic fourcomplex, 77 planar 6-complex, 181
amplitude circular fourcomplex, 55 hyperbolic fourcomplex, 79 planar 6-complex, 184 planar fourcomplex, 95 planar n-complex, 235 polar 5-complex, 153 polar 6-complex, 171 polar n-complex, 200 tricomplex, 25 twocomplex, 4 analytic function, hypercomplex definition, 12 expansion in series, 12 power series, 12 azimuthal angle polar fourcomplex, 125 azimuthal angle, tricomplex, 21 azimuthal angles circular fourcomplex, 56 planar 6-complex, 184 planar fourcomplex, 96 planar n-complex, 238 polar 5-complex, 152 polar 6-complex, 170 polar n-complex, 204 canonical base circular fourcomplex, 67 hyperbolic fourcomplex, 81 planar 6-complex, 188 planar fourcomplex, 112 planar n-complex, 245 263
264 planar fourcomplex, 93 planar n-complex, 233 polar 5-coniplex, 150 polar 6-complex, 168 polar fourcomplex, 122 polar n-complex, 198 tricomplex, 19 twocomplex, 2 complex units, powers of circular fourcomplex, 59 hyperbolic fourcomplex, 83 planar fourcomplex, 100 planar n-complex, 240 polar fourcomplex, 128 polar n-complex, 208 tricomplex, 27 twocomplex, 6 convergence of power series circular fourcomplex, 66 hyperbolic fourcomplex, 88 planar 6-complex, 190 planar fourcomplex. 111 planar n-complex, 252 polar 5-complex, 162 polar 6-complex, 177 polar fourcomplex, 140 polar n-complex, 223 tricomplex, 40 twocomplex, 11 convergence, region of circular fourcomplex, 68 hyperbolic fourcomplex, 89 planar 6-complex, 191 planar fourcomplex, 112 planar n-complex, 253 polar 5-complex, 162 polar 6-complex, 178 polar fourcomplex, 141 p)olar n-complex, 224 tricomplex, 42
Index twocomplex, 11 cosexponential function, polar 5complex definition, 154 cosexponential function, polar fourcomplex definitions, 128 cosexponential function, polar ncomplex definition, 208 cosexponential functions, planar 6complex addition theorems, 186 definition, 185 differential equations, 188 expressions, 185 parity, 185 cosexponential functions, planar fourcomplex addition theorems, 100 definitions, 100 differential equations, 103 expressions, 103 parity, 100 cosexponential functions, planar ncomplex addition theorems, 243 definition, 241 differential equations, 244 expressions, 241 parity, 241 cosexponential functions, polar 5complex addition theorems, 159 differential equations, 159 expressions, 154, 156 cosexponential functions, polar 6complex addition theorems, 173 definitions, 172
Index differential equations, 175 expressions, 172 parity, 172 cosexponential functions, polar fourcomplex addition theorems, 129 differential equations, 133 expressions, 131 parity, 129 cosexponential functions, polar ncomplex addition theorems, 210 differential equations, 211 expressions, 209 parity, 208 cosexponential functions, tricomplex addition theorems, 28 definitions, 28 differential equations, 31 expressions, 30 derivative, hypercomplex definition, 12 of power function, 12 derivative, independence of direction fourcomplex, 68 planar n-complex, 253 polar n-complex, 224 tricomplex, 43 twocomplex, 13 distance circular fourcomplex, 56 hyperbolic fourcomplex, 80 planar 6-complex, 183 planar fourcomplex, 95 planar n-complex, 236 polar 5-complex, 152 polar 6-complex, 170 polar fourcomplex, 124
265 polar n-complex, 202 tricomplex, 23 twocomplex, 4 distance, canonical variables polar fourcomplex, 125 divisors of zero hyperbolic fourcomplex, 79 planar n-complex, 240 polar n-complex, 208 tricomplex, 20 twocomplex, 4 exponential form circular fourcomplex, 62 hyperbolic fourcomplex, 84 planar 6-complex, 189 planar fourcomplex, 107 planar n-complex, 246 polar 5-complex, 160 polar 6-complex, 176 polar fourcomplex, 134 polar n-complex, 215 tricomplex, 34 twocomplex, 7 exponential function, hypercomplex addition theorem, 6 definition, 6 exponential, expression twocomplex, 7 exponential, expressions circular fourcomplex, 60 hyperbolic fourcomplex, 83 planar 6-complex, 185, 190 planar fourcomplex, 100 planar n-complex, 241 polar 5-complex, 154, 161 polar 6-complex, 172, 177 polar fourcomplex, 128 polar n-complex, 208, 220 tricomplex, 28
266 functions, real components circular fourcomplex, 68 planar 6-complex, 191 planar n-complex, 253 polar 5-complex, 163 polar n-complex, 224 tricomplex, 42 twocomplex, 13 hyperbolic functions, expressions circular fourcomplex, 65 hyperbolic fourcomplex, 86 planar 6-complex, 190 planar fourcomplex, 109 planar n-complex, 250, 251 polar 5-complex, 161 polar 6-complex, 177 polar fourcomplex, 138 polar n-complex, 220, 221 tricomplex, 37 twocomplex, 9 hyperbolic functions, hypercomplex addition theorems, 9 definitions, 9 integrals, path circular fourcomplex, 70 hyperbolic fourcomplex, 90 planar 6-complex, 192 planar fourcomplex, 114 planar n-complex, 255 polar 5-complex, 163 polar 6-complex, 179 polar fourcomplex, 142 polar n-complex, 226 tricomplex, 45 twocomplex, 14 inverse circular fourcomplex, 55 hyperbolic fourcomplex, 78 planar fourcomplex, 94
Index planar n-complex, 235 polar fourcomplex, 122 polar n-complex, 200 tricomplex, 20 twocomplex, 3 inverse, determinant hyperbolic fourcomplex, 78 planar fourcomplex, 95 planar n-complex, 235, 236 polar fourcomplex, 123 polar n-complex, 200, 201 tricomplex, 20, 26 twocomplex, 3 logarithm circular fourcomplex, 64 hyperbolic fourcomplex, 85 planar 6-complex, 189 planar fourcomplex, 109 planar n-complex, 247 polar 5-complex, 161 polar 6-complex, 176 polar fourcomplex, 137 polar n-complex, 217, 218 tricomplex, 35 twocomplex, 8 matrix representation circular fourcomplex, 59 hyperbolic fourcomplex, 82 planar 6-complex, 184 planar fourcomplex, 99 planar n-complex, 240 polar 5-complex, 153 polar 6-complex, 171 polar fourcomplex, 128 polar n-complex, 207 tricomplex, 26 twocomplex, 6 modulus hyperbolic fourcomplex, 80
267
Index modulus, canonical variables circular fourcomplex, 67 hyperbolic fourcomplex, 82 planar 6-complex, 184 planar fourcomplex, 112 planar n-complex, 237 polar 6-complex, 171 polar n-complex, 203 tricomplex, 41 twocomplex, 4 modulus, definition circular fourcomplex, 65 hyperbolic fourcomplex, 87 planar fourcomplex, 110 planar n-complex, 236 polar fourcomplex, 139 polar n-complex, 202 tricomplex, 39 twocomplex, 10 modulus, inequalities circular fourcomplex, 66 hyperbolic fourcomplex, 87 planar 6-complex, 190 planar fourcomplex, 110 planar n-complex, 251 polar 5-complex, 152, 162 polar 6-complex, 177 polar fourcomplex, 139 polar n-complex, 222 tricomplex, 39 twocomplex, 10 nodal hyperplanes circular fourcomplex, 56 planar fourcomplex, 95 polar fourcomplex, 123 nodal hypersurfaces planar n-complex, 236 polar n-complex, 201 nodal line, tricomplex, 20 nodal lines
twocomplex, 4 nodal plane, tricomplex, 20 planar angle circular fourcomplex, 56 planar fourcomplex, 96 polar 5-complex, 152 polar 6-complex, 170 planar angles planar 6-complex, 184 planar n-complex, 238 polar n-complex, 204 polar angle polar 5-complex, 152 polar angle, tricomplex, 23 polar angles polar 6-complex, 171 polar fourcomplex, 125 polar n-complex, 204 poles and residues circular fourcomplex, 71 planar 6-complex, 192 planar fourcomplex, 116 planar n-complex, 256 polar 5-complex, 163 polar 6-complex, 179 polar fourcomplex, 144 polar n-complex, 227 tricomplex, 45 polynomial, canonical variables circular fourcomplex, 75 hyperbolic fourcomplex, 91 planar 6-complex, 192 planar fourcomplex, 119 planar n-complex, 258 polar 5-complex, 164 polar fourcomplex, 146 polar n-complex, 229 tricomplex, 48 twocomplex, 15 polynomial, factorization
268 circular fourcomplex, 76 hyperbolic fourcomplex, 92 planar 6-complex, 192 planar fourcomplex, 120 planar n-complex, 258 polar 5-complex, 164 polar fourcomplex, 146 polar n-complex, 230 tricomplex, 49 twocomplex, 15 polynomials, canonical variables polar 6-complex, 180 polynomials, factorization polar 6-complex, 180 power function circular fourcomplex, 64 hyperbolic fourcomplex, 85, 86 planar 6-complex, 189 planar fourcomplex, 109 planar n-complex, 248 polar 5-complex, 161 polar 6-complex, 176 polar fourcomplex, 137 polar n-complex, 218 tricomplex, 35 twocomplex, 8 power series circular fourcomplex, 66 hyperbolic fourcomplex, 88 planar 6-complex, 190 planar fourcomplex. I l l planar n-complex. 252 polar 5-complex, 162 polar 6-complex, 177 polar fourcomplex, 140 polar n-compl(!X, 223 tricomplex, 40 twocomplex, 10 product circular fourcomplex, 54
Index hyperbolic fourcomplex, 77 planar 6-complex, 182 planar fourcomplex, 93 planar n-complex, 234 polar 5-complex, 150 polar 6-complex, 168 polar fourcomplex, 121, 122 polar n-complex, 199 tricomplex, 19 twocomplex, 2, 3 relations between partial derivatives circular fourcomplex, 69 hyperbolic fourcomplex, 89 planar 6-complex, 191 planar fourcomplex, 113 planar n-complex, 254 polar 5-complex, 163 polar 6-complex, 178 polar fourcomplex, 141 polar n-complex, 225 tricomplex, 44 twocomplex, 14 relations between second-order deriva tives circular fourcomplex, 69 hyperbolic fourcomplex, 90 planar 6-complex, 191 planar fourcomplex, 113 planar n-complex, 254 polar 5-complex, 163 polar 6-complex, 179 polar fourcomplex, 141 polar n-complex, 225 tricomplex, 44 twocomplex, 14 rei)resentation by irreducible matrices, 50 circular fourcomi)lex, 76 hyperbolic fourcomplex, 93
Index planar 6-complex, 193 planar fourcomplex, 121 planar n-complex, 259 polar 5-complex, 165 polar 6-complex, 181 polar fourcomplex, 147 polar n-complex, 232 twocomplex, 16 roots circular fourcomplex, 73 planar fourcomplex, 118 rules for derivation and integration, 13 series circular fourcomplex, 65 hyperbolic fourcomplex, 87 planar fourcomplex, 110 planar n-complex, 251 polar fourcomplex, 139 polar n-complex, 222 tricomplex, 38 twocomplex, 10 sum circular fourcomplex, 54 hyperbolic fourcomplex, 77 planar 6-complex, 182 planar fourcomplex, 93 planar n-complex, 234 polar 5-complex, 150 polar 6-complex, 168 polar fourcomplex, 121, 122 polar n-complex, 198 tricomplex, 19 twocomplex, 2, 3 transformation of variables circular fourcomplex, 58, 67 hyperbolic fourcomplex, 80 planar 6-complex, 184
269 planar fourcomplex, 97, 108, 112 planar n-complex, 238 polar 5-complex, 153 polar 6-complex, 171 polar fourcomplex, 124, 125, 134, 136 polar n-complex, 206 tricomplex, 23, 25, 42 twocomplex, 5 trigonometric form circular fourcomplex, 63 expression, 7 hyperbolic fourcomplex, 84 planar 6-complex, 189 planar fourcomplex, 107 planar n-complex, 246 polar 5-complex, 160 polar 6-complex, 176 polar fourcomplex, 135 polar n-complex, 216 tricomplex, 34 trigonometric functions, expressions circular fourcomplex, 64 hyperbolic fourcomplex, 86 planar 6-complex, 190 planar fourcomplex, 109 planar n-complex, 249, 250 polar 5-complex, 161 polar 6-complex, 177 polar fourcomplex, 138 polar n-complex, 219, 221 tricomplex, 36 twocomplex, 9 trigonometric functions, hypercomplex addition theorems, 9 definitions, 9 trisector line, 21
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