Conformal Representation

-~ tl.k_ -~) 1 - k (h + tl.h)

...... (39"3)

must be satisfied identically. If we now assume that

cp'(h)=1~h2' and hence, by integration, that ¢(h)=! log~~~.

. ..... (39"4)

§§ 39-42]

21

ANGLE AND DISTANCE

Direct substitution in (39"3) then verifies that that equation is satisfied identically.

40. If z 1 , z2 are two arbitrary points of the non-Euclidean plane, the non-Euclidean motion (§ 35) z-z~

y=---

1-ZtZ

transforms z 1 into the point 0, and z 2 into the point (z2 -z1)/(l- z,z.). By a rotation about 0 this last point can now be transformed into Iz.-z,l/l1-z,z21, so that, by (39"4),

D (z,' z.) = ! log

I~ =-;' z"i ~ I~· =~~~ ;} Nj

Z2

..,2

N}

• •• • •

•< 40"1)

I z, I < 1, I z2l < 1. 41. In the case when the orthogonal circle coincides with the real axis the non-Euclidean distance between two points z., z2 in the upper halfplane can be deduced as follows: the Mobius' transformation ...... (41"1)

transforms the real axis of the z-plane into the cirde I w I = 1, the point z1 into the point w1 = 0, and the point z2 into the point w 2= (z2 - z1)/(zl- z2). In this way we obtain w)-11 lz,-z:l+lz.-z.!.} ])(z.,""2 -~ o g - _ - - - - - - , ...... ( 41"2) lz,-z.l-1 z,-z.l I (zt) > 0, I (z,) > 0.

42. The triangle theorem. It is easy to show by direct calculation from (40"1) that, for any three points of the non-Euclidean plane, D (::1 , ::,) ~ D (z1 , z2 ) + D (z2 , z3 ), •••••• (42·1) where the Pign of equality holds if and only if z2 lies on the non-Euclidean segment which joins z1 and z3 • But the following proof of (42"1) is much more instructive. All the axioms which Euclid postulates at the beginning of his work, excepting only the parallel postulate, hold also in the non-Euclidean plane. Hence, since the parallel postulate is not used in the earlier theorems, we can apply the first sixteen propositions of Euclid's first book without change to our figures; in particular, theorems concerning the congruence of triangles, the theorem that the greatest side of a triangle is opposite to the greatest angle, and, lastly, the triangle theorem expressed by (42"1).

22

NON-EUCLIDEAN GEOMETRY

(CHAP. II

43. Non-Euclidean length of a curve. The conception of the length of a curve can at once be extended in the sense of this geometry to curves in the non-Euclidean plane. We suppose that the curve is given in terms of a parameter. On the curve take a finite set of points arranged in order so that the parameter always increases (or always decreases) from one point to the next. Let these points in order be taken as the corners of a polygon whose sides are non-Euclidean straight lines, and define the length of the polygon as the sum of the lengths of its sides; the upper limit (when finite) of the lengths of all such inscribed polygons now defines the length of the given curve. Replacing z1 , z2 in (40·1) by z, z + az, we obtain, as az tends to zero, I azl

D (z, z + az) ...... I= I :"I , and it follows that the non-Euclidean length of an arc of the curve which is given by the complex function z (t) is expressed by the integral

<«U _ at·

r~o I

h

1-lz"(t)l

The analogous formula for the case in which the orthogonal circle is the real axis is obtained in a similar way from (41"2); this formula is

I z' I -;1 [, ---dt. z t, (z-z) 2

44. Geodesic curvature. It has been shown by P. Finsler (10) that, with the· most general metric, the geodesic curvature of a curve Cat a point P on it may be obtained as follows: consider any family of curves which contains C, and construct that geodtsic line of the metric which touches Cat P; let R be any point of this tangent geodesic, and lets denote the extremal distance P R and() the angle of intersection* at R of the geodesic P R with that curve of the family which passes through R; then the geodesic curvature at P is k=

lim~ ....... (44·1)

....... o $

It can be shown that the number k so obtained is independent of the particular choice of the family of curves. * It is assumed that the

me~tsure

Fig. 8 of angles is suitably defined in the metric.

§§ 43-45]

23

ROTATIONS AND TRANSLATIONS

Applying the above definition of curvature to our non-Euclidean geometry, we see that curvature is invariant for non-Euclidean motions. Further, if the orthogonal circle is lzl = 1 and if, according to §39, t/J' (0) = 1, the curvature of a curve at the centre of this circle is the same as its ordinary Euclidean curvature, since the non-Euclidean straight lines through this point are ordinary straight lines and the value of angles is the same in both geometries. The non-Euclidean curvature' of a curve z (t) at a point Zo may therefore be calculated by writing Z-Zo

(

• )

'(t)= 1- z ()-' ...... 442 t z0 and then determining the ordinary curvature of' (t) at '=0. But this is known to be lc = ""f:r'- - - - (f." -. . ..••• (44:3) 2i <'t)i , and consequently, from {44"2), we have, on replacing z0 by z, the final formula k -- ¥z'z' (zz'- zz') + (1- zz) (z'z"- z'z"). . ..... (44"4) 2i (z'z')i 45. Non-Euclidean motions. It was shown in§ 32 that every non-Euclidean motion can be obtained by two successive inversions with respect to non-Euclidean straight lines; there are three essentially distinct types of non-Euclidean motions, corresponding to different relative dispositions of these two straight lines. Suppose first that the inversions are with respect to two non-Euclidean straight lines which intersect at a point P, where they :tilake an angle !8 with o~e another; the motion is in this case a non-Euclidean rotation about P, the angle of rotation being 9. By keeping P fixed and letting () vary we obtain a one-parameter group of non-Euclidean rotations. Let P 1 be the point inverse toP with respect to the orthogonal circle ; then every circle of the elliptic pencil which has the points P and P 1 as limiting points is transformed into itself by every operation of this group, so that the circles of this pencil, in so far as they lie inside the non-Euclidean plane, are non-Euclidean circles with common non-Euclidean centre P. If C is any one of these circles, all points on Care at the same non-Euclidean distance from P; this Fig. 9 distance is the non-Euclidean radius of C.

c

24

NON-EUCLIDEAN GEOMETRY

(CHAP.

n

46. Suppose, secondly, that the two inversions to which the n-otion i!l reducible are with respect to two non-Euclidean straight lines which neither intersect nor are parallel in the sense of Lobatschewsky. The conjugate pencil, consisting of all circles which are transformed into themselves by the motion, is hyperbolic; its common points M 1 , M, lie on the orthogonal c.arcle. Among the circles of this pencil there is exactly one non-Euclidean straight line. It intersects the nonEuclidean straight lines with respect M2 to which the inversions are performed at the points P~o P,. If the non-Euclidean distance P 1P 1 is denoted by !k, the motion can be regarded as a translation of the plane through distance It along the non-Euclidean straight line Fig. IO ~P1 P,M,. By keeping the non-Euclidean straight line ~Ms fixed and letting It vary we obtain a one-parameter group of motions, such that every motion of the group transforms into itself every circular arc which lies inside the non-Euclidean plane and joins the points lrL, M 2 • These circular arcs are called kypercycles; a hypercycle can also be defined as the locus of a point whose non-Euclidean distance from the non-Euclidean straight line ~M, is constant. Finally, if the two successive inversions are with respect to parallel non-Euclidean straight lines (§ 37), whose common end-point is L, we obtain a non-Euclidean motion which is called a limit-rotation ( Grenzdrekung).

Given two pairs of parallel non-Euclidean straight lines LJYI, LN and L1Mu L 1N 1, there is always a nonEuclidean motion which transforms each pair into the other pair, so that the points L, Jf, X are transformed into the points LlJ Ml> N 1 or L1, N1, fl'/1. It follows that from the point of view of nonL1 Euclidean geometry all limit-rotations are equivalent, or rather that they can differ only in the sense of the rotation. For the limit-rotation obtained by inversions with respect to LM and L.!I.T M L is the only fixed point. Circles inside Fig. 11

§§ 46--48]

PARALLEL CURVES

25

the non-Euclidean plane which touch the orthogonal circle at L are transformed into themselves; these circles are called oricycles.

47. Every ordinary circle which lies wholly within the non-Euclidean plane is also a non-Euclidean circle, and it is easy to find its nonEuclidean centre. Similarly every circle which touches the orthogonal circle is an oricycle, and every circular arc whose end-points lie on the orthogonal circle is a hypercycle. The curves of these three types are the only curves of the non-Euclidean plane which have a constant nonzero curvature. By § 44 the curvature· of the oricycle is (disregarding sign) equal to 2, while that of a hypercycle is less than 2, and of a nonEuclidean circle greater than 2. If k denotes the curvature and r the non-Euclidean radius of one of these circles, we obtain the relation e"" +e_,. k=2 e"" -t:r=2coth2r.* ...... (47·1) -e

48. Parallel curves. Consider the aggregate of non-Euclidean circles with given nonEuclidean radius r whose centres are at the points of an arbitrary set A of points of the non-Euclidean plane. These circles cover a set of points B (r), whose frontier, if it exists, contains all points of the non-Euclidean plane which are at a distance r from A. If we take as the set A a curve 0 of finite curvature, and if we let r vary while remaining less than a sufficiently small ~pper bound, we obtain a family of parallel curves (in the sense of the JlOn-Euclidean metric). It Ca.n be proved in just the same way as with other similar problems of the calculus of variationst that the orthogonal trajectories of such a family of equidistant curves are non-Euclidean straight lines. Conversely, if a portion of the non-Euclidean plane is simply covered by a family of non-Euclidean straight lines, the orthogonal trajectories of the family are parallel curves in the sense defined above. Thus the simplest examples of families of parallel curves are the following : (a) non-Euclidean circles with a common centre; (b) oricycles which touch the orthogonal circle at a common point; (c) hypercycleshaving the same end-points M;, M 2 (see Fig. 10). 11"

• For r< 2 we ha.ve

1

4

16

k=:;.+ar-45r3+ .... t See Frank a.nd v. Mises. Die Differential- und Integralgleichungen der Mechanik und Physik. Vol. 1, ch. 5.

CHAPTER ill

ELEMENTARY TRANSFORMATIONS 49. The ezponentlal function. The function

...... (49•1)

W=e"

gives rise to two important special transformations. On introducing

rectangular coordinates a:, yin the z-plane and polar coordinates p, 4> in the w-plane, i.e. writing z =a:+ iy and w = pt!+, we replace (49·1) by the two equations p = e", 4> = y. A horizontal strip of the z-plane hounded by the lines y = y, and '!J = y,, where I y,-y2 1< 2r, is transformed into a wedge-shaped region of the

0

z-plane Fig.12

w-plane, the angle of the wedge being ex= lt/>2- t/J, I=I y,- !It 1- The representation is conformal throughout the interior of these regions, since the derivative of e" is never zero. As a special case, if IJ/2 - y,l = r (e.g., y1 = 0, '!J2 = '~~"), the wedge hecomes a half-plane. The restriction on the width of the strip, namely that I'!J1 - y2 1< 2w, may be dropped. If, for example, y, = 0 and y2 = 21r, the strip is repre-

,...-- , .......

/ 27T( \

\

' __ ,. .......

27T I

/

z-plane

w-plane Fig.13

sented on the w-plane cut along the positive real axis; and if IY1 -y" I> 21r the wedge obtained covers part of the ~v-plane multiply.

§§ 49-52]

THE WEDGE AND THE STRIP

27

The cases when 1!/1- '!hi is an integral multiple of 2,. are of particular importance; for the strip is then transformed into a Riemann surface with an algebraic branch-point, and cut along one sheet.

50. We obtain the second special transformation furnished by (49"1) by considering an arbitrary vertical strip bounded by the lines z = z., z = z 2 • This strip is represented on a Riemann surface which covers the annular region P1
...... (51"1) gives, of course, on interchanging the z-plane and w-plane, precisely the same conformal transformations as (49"1). A number of problems of conformal representation, important in the proofs of general theorems, can be solved by combining the above transformations with the Mobius' transformations discussed in Chapter I. 52. Representation of a rectilinear strip on a circle. The strip IJ!t(w)l
·2/o.-

1 +tz .

=u=1-iz'

28

ELEMENTARY TRANSFORMATIONS

[cHAP. m

and hence 2h 1 + iz w = ..,--- log 1- - . . Z7r - ZZ

...... (52"1)

This gives 4k dw ...... (52"2) dz = r (1 + z2) ' which is positive when z = 0, as the conditions of the problem required. If the point z describes the unit-circle z = e;8 , then dw 2ki d() = 71" cos () ' a pure imaginary as the circumstances require. If the point z describes the circle Iz I= r < 1, w describes a curve which lies inside a finite circle and whose form is easily determined; for this curve, by (52·1), 2h

1 +r

r

1-r

li(w)l~--log

---.

. ..... (52·3)

The inequality (52"3) has many applications. 53. An arbitrary wedge, whose angle we denote by ?TOt, and whose vertex we suppose to be at w = 0, can be transformed into a half-plane. For, on introducing polar coordinates z=re18 , w=pe"~> ...... {53"1) in the relation W=Z"=e>wgz, ...... (53"2) this relation takes the form p=,..., cp =OC8; ..•... (53"3) hence (53"2) effects the desired transforma- 0 Fig. 16 tion •. Hence, by combination with a Mobius' transformation, a wedge can be transformed into the interior of a circle. The relations (53"3) show that the representation of a wedge on a half-plane (or on a circular area) is conformal at all points inside or on the boundary of the wedge except at its vertex w = 0. Two curves which intersect at w = 0 at an angle X are transformed into two curves which intersect at an angle A/a., so that at the origin corresponding angles, though no longer equal, are proportional; in these circumstances the representation at the origin is said to be quasi-conformal. 54. Representation of a circular crescent. The area between two intersecting circular arcs or between two circles * If the wedge is first transformed into a. strip (§ 49), and this strip into a. ha.lfpla.ne (\152), the same function 153·2) is arrived at.

§§53-56]

RIEMANN SURFACES

29

having internal contact is transformed, by means of a Mobius' transformation whereby a point P common to the two circles corresponds to

Fig. 17

the point co, into a wedge or a strip respectively. Hence, by the foregoing, each of these crescent-shaped areas can be represented conformally on the interior of a circle. In the same way the exterior of a circular

Fig. 18

crescent or the exterior of two circles having external contact can also be represented on the interior of a circle. 55. Representation of Riemann surfaces. If, in equation (53'2), ex is a positive integer n, the equation effects the representation of the unit-circle Iz I < 1 on a portion of a Riemann surface of n sheets, overlying the circle Iw I < 1 and having an n-fold branch-point at w = 0. 56. It is important to determine the function which represents the unit-circle on a Riemann surface which has the point w0 as its branchpoint but is otherwise of the kind described in §55. We require, further, that w = 0 should correspond to z = 0 and that parallel directions drawn in the same sense through these two points should also correspond. · Suppose first that w 0 is a positive number k (k < 1), and displace the branch-point of the Riemann surfat:e to the origin by means of the

30

ELEMENTARY TRANSFORMATIONS

Mobius' transformation (§ 35)

11-u w=l-hu"

[CHAP. m ...... (56"1}

By §55 the equation u = t" represents this new Riemann surface on the 1

simple (scklickt)• circle Itl < 1, and in this way t = Jtfi corresponds tow= 0. Hence the desired transformation is 1

h,i-z

t=---..-, l-k"z i.e., in terms of the original variables, 1

1

(I-k"z)"- (1-k-,., z)" w=k -1 1 (1- k"z)"- k2 (1- k -..- z)" The binomial theorem shows that

...... (56"2)

...... (56"3) In the general case when w0 = ke16 the required transformation is 1

w= k

1

(1-k»e-••z)"-(1-k

e-16 z}" 1

1

e16•

• ••••• (56"4)

(1- ~t»e- 16 z)"-k2 (1- k -,.e-16 z)"

57. The case in which the Riemann surface has a logarithmic branchpoint at w 0 = k may be treated in a similar manner. We first, by means of (56"1), transform this Riemann surface into the surface already considered in § 50; then, in virtue of the same paragraph, the transformation u = e' transforms this surface into the half-plane lt (t) < 0, making u = k, i.e. w = 0, correspond tot= log k. Finally the half-plane It (t) < 0 is transformed into the circle Izl < 1 by means of the relation t= 1 +zlog/z l-z ' so that the function producing the desired transformation is seen to be ~ Ioc"

1+•1og"

to=

k-el-•

1-~logh

l-e1-•

,

~logh.

...... (57"1)

I-ke•-• l-k2e 1 -• From this equation we obtain at once

= -2klogk >O. (dw) dz •=O 1-k 2

. ..... (57•2}

* A Bieman.J surface is simple (•chlicht) if no two points of ihe surfacehaveihesame coordinate u.

§§57-60]

31

EXTERIOR OF THE ELLIPSE

58. If n is made to increase indefinitely, the Riemann surface dealt with in §56 becomes, in the limit, the Riemann surface of §57, having a logarithmic branch-point. It is therefore to be expected that, if ~. (z} denotes the right-hand member of (56"2), and 1/J (z) that of (57"1), we

..

lim ~.(z} = 1/J(z). ,._

shall have

. .•... (58"1)

The truth of (58"1) can in fact be deduced from a general theorem; but the equation can also be verified directly, as follows. In 1

(Jii. )" --=t= 1

put

k"= 1-e-,.,

I _! 6,. {log(ll.n-z)-lotrl1-ll"•l}

••••••_(~8"2)

(1-k"'z)"' so that lim e-,.=0; the exponent of e in (58"2) can

now be written as

.......

1 n{log(1-z-e-..)-log(1-z+e-,.z)} og log(1-e-,.) · ' and when e-,. tends to zero this tends to the limit 1+z -logk. 1 -z Equation (58"1) now follows at once. It can be shown in the same way that the limiting form of (56"3) is (57"2). 59. It can also be shown that, for all values of n, ~'.. (0) > ~·.+1 (0), ...... (59"1) and this inequality may perhaps rest upon some deeper, as yet unremarked 1

property of the transformations. To prove (59"1) write log li1 =-~.so that .\,.+1
~·.. (0) = (- :~1~~ k) (~~'• ;~"'!)' and the function

11'-e-A

A'

A'

2A= 1 + 3! +51+ ...

steadily decreases as A decreases. 60. Representation of the ezterior of an ellipse. We start with the problem of representing thew-plane, cut along the finite straight line -at < w 1 of the unit-circle, so that the points w = <Xl and z = <Xl correspond. The Mobius' transformation w-at 1+u W=at-u= w+at' 1-u

32

ELEMENTARY TRANSFORMATIONS

[CHAP. III

transforms the cut w-plane into the u-plane cut along the negative real axis, and the further transformation u = t 2 transforms this into the halfplane Jli (t) > 0. 'fo the point w = ~ correspond the points u = 1 and t = 1, so that the required transformation is obtained by writing 1 +t z=1-t"

Thus finally

w=~(z+D;

...... (60"1)

the relation (60"1) is very remarkable in that it represents the cut wplalie not only on the ezterior but also on the interior of the unit-circle.

61. If in (60"1) we write z=re;', where (for example) r> 1, we obtain

w=~{(r+~)cos8+i(r-Dsin0}; ...... (61"1) hence to the circle I z I = r there corresponds in the w-plane an ellipse with semi-axes

a=~ (r+ ;),

b=~ (r-;).

. ..... (61"2)

Conversely, if a, bare given, (61"2) determines ex, r: ...... (61"3)

Thus equation (60"1) transforms the exterior of the circle I z I> r> 1 (or the interior of the circle I z I <

-~ < 1) into the exterior of the ellipse

(61"1).

The representation of the interior of an ellipse on the interior of the unit-circle cannot, on the other hand, be obtained by means of the elementary transformations so far employed. But it is to be noticed that the function (60"1) represents the upper half of the ellipse (61"1) on the upper half of an annular region cut along the real axis; this last area, however, and therefore the semi-ellipse also, is easily (by a method similar to that of§ 50) transformed into a rectangle. The details of this calculation, which leads to trigonometric functions, are left to the reader.

62. Representation of an arbitrary simply-connected domain on a bounded domain. By a domain we understand an open connex (zusammenkiingend) set of points of the complex plane; thus an open set of points is a domain

§§ 61-63]

CONNEcrriVITY

33

if and only if every two points of the set can be joined by a continuous curve all of whose poil}ts belong to the set. The frontierS of a domain Tis defined as the set of limiting points of T which are not also points of T. The set 8 + T of points of a domain and of its frontier is called a closed domain and will be denoted by T. In conformity with the conventions already made (see Chapter 1, § 9), the point oo is to be treated like any other point and may, in particular, be an interior point of a domain T. If a one-one transformation is continuous at every point of a domain T, it transforms T into another domain T'. If T is not identical with the whole complex plane, its frontier contains at least one point, and hence we can, by means of a Mobius' transformation, represent T conformally on a domain T• which does uot contain the point z = oo.

63. The classification of domains according to their connectivity is important. A domain T, which may have the point oo as an interior point, is said to be m-ply connected if its frontierS consists of m distinct continua. The degree of connectivity is, of course, a topological invariant, that is to say, it is not altered by any continuous one-one transformation. Our main concern here is with simply-connected domains, whose frontier consists of a single continuum. 'rhe property expressed by the words "simply-connected" can be specified in many other ways. It can be proved by topological methods that all these specifying properties are equivalent to one another. A few examples of such properties are: (a) If the domain T* does not contain the point oo, then T• is simply-connected if and only if the interior of every polygon, whose frontier belongs to T•, consists entirely of points of T•. (b) A domain T is simply-connected if and only if every curve y, which joins two points of the frontier S and lies within T, divides T into at least two domains. (c) The same is true if every closed curve within T can be reduced to a point by continuous deformation in T. (In the course of the deformation the curve may possibly have to be taken through the point oo .) (d) A domain Tis simply-connected if it can be represented on the interior of a circle by means of a continuous one-one transformation. (e) The monodromy tkeorem may be regarded as giving a specifying property for a simply-connected domain T• which does not contain the

34

ELEMENTARY TRANSFORMATIONS

[CHAP.

ill

point oo. This characterisation is especially important in the theory of functions. The theorem states : .A aomain T• wkick does not cuntain tke point oo is simply-connected if, wlumever f (z) is an analytic functiun wkick can be continued along every curve in T"',f(z} is a singk-valuedfunction(ll). By this is meant: If an arbitrary functional element is assigned to a point ofT*, then analytic continuation of this element along paths entirely within T• must either lead to the same functional value at any point of T*, by whatever path that point is reached, (R" T* must contain a singular point of the function obtained by continuation. Every circle, for example the unit-circle I z I< 1, is, by this definition, a simply-connected domain. Koebe, in his lectures, proves this as follows: A function f(z), which can be continued along every curve within the circle, must be expressible in the neighbourhood of z = 0 by a powerseries ...... (63"1} The radius of convergence of this series is at least unity, for if it were less than unity analytic continuation of f(z) along every radius within I z I< 1 would not be possible. It follows that, for all analytic continuations within I z I< 1, the value of the function f(z) may be calculated from the relation (63"1), and the function is seen to be single-valued. From this it follows that a domain is simply-connected if it can be put into one-one correspondence with the D C unit-circle Iz I < 1 by means of a conformal transformation. Thus, for instance, the half-plane (§ 12), the semi-circle(§ 54), and hence also the quarter-circle (§ 53), are simply-connected. It can now be shown that the square is simplyconnected, for it can be regarded as the sum of two quarter-circles; every common point P of the two quarter-circles can be joined to the B middle point 0 of the square by means of a A segment which lies entirely in ABCM and enFig. 19 tirely in ANOD and the monodromy theorem applies. 64. Now let T be a simply-connected domain of the complex z-plane, such that the frontier of T contains at least two points .A" B., and let z. be a.n interior point of T, other than the point oo. Consider the circular arc (which may be a finite or infinite straight line) which joins

§§ 64, 65]

SIMPLY-CONNECTED DOMAIN

35

A, to B, and has z0 as an interior point, and denote by A the first frontier-point of T which is met in describing this arc from z0 to Ah and by B the corresponding point of the arc from z0 to B, _ Thus all interior points of the arc Az0 B are interior points of T, and its endpoints are frontier-points of T. If we now apply to the z-plane the Mobius' transformation whereby the points A, z0 , B correspond to the points 0, 1, oo in the u-plane, the domain Tis transformed into a simplyconnected domain T,, which is such that u = 0 and u = oo are frontierpoints of T, but all other points o£ the positive real axis are interior points ofT,. By means of the transformation u = v4 we now obtain four different domains in the v-plane corresponding to T, ; let T 2 denote that one ot them in which the straight line v > 0 corresponds to the straight line u>O*. We shall prove that T 2 lies in the half-plane J!{ (v)>O, and is therefore transformed by v-1 W=...••• (64'1) v+1 into a domain T' which lies inside the circle I w I < 1. For if T 2 were to contain a pointv2 for which J!{ (v) ~ 0, we coulddrawin T 2 a curve joining v 2 and v1 = 1; let P be the first point of intersection (counting from 'lh) of this curve with the imaginary v-axis, and Q the last intersection of the arc from v1 toP with the real axis, so that the arc PQ, which we will denote by Yv• lies either in the first or in the fourth quadrant of the v-plane. The relation u = v4 transforms the arc Yv into a rnrve 'Yu all of whose points belong to T,, and whose end-points lie on the real axis u > 0; by adding to y, a portion of the axis u > 0 we therefore obtain a closed curve which lies wholly in T, and surrounds the point u = 0. But nq such curve can exist, since T, is simply-connected and does not contain either of the points u = 0, u = oo. Thus: any simply-connected domain whose frontier contains at least two distinct points can, by simple tmn.iformations, be conformally represented m; a domain which lies entirely inside the unit-ci1·cle. A multiply-connected domain can be treated in precisely the same way provided that its frontier contains at least one continuum of more than one point. 65. A theorem which, though in appearance but little different from that· of§ 64, is, on account of the value of the constant involved, of the greatest importance in the general theory, is due to Koebe(l2}. * Each of the four domains corresponding to T 1 , in particular T 2 , is simple (•chlicht) (cf. §56, footnote).

36

ELEMENTARY TRANSFORMATIONS

(CHAP.

m

Let T be a simple simply-connected domain of the w-plane not containing the point w = oo ; suppose further that a.ll points of the circle I w I < 1 are points of T, but that w = 1 is a frontier-point of T. Any such domain Tis transformed by the same function 4(1 + .j2) 2 z W= •••••. (65"1) {1 + (1 + .j2)2 z} 2 into a simple simply-connected domain T' which lies inside the unit-circle I z I < 1 but always contains the fixed circle I z I < (1 + J2t 4• Consider in the w-plane the two-sheeted Riemann surface which has branch-points at w = 1 and w = oo ; the function 1-w=u}

...... (65"2)

transforms this surface into the simple u-plane, making the two points u = ± 1 correspond to w = 0, and transforms any curve drawn in T and starting from w = 0 into two curves in the u-plane. 'fhe aggregate of a.ll curves in the u-plane which correspond to curves in T and which start from u = + 1 fills a simply-connected domain which we will denote by T 1 • We shall prove that, if u0 is any complex number, the domain T 1 cannot contain both the points ± u 0 • For if it did, these two points could be joined by a curve y,. lying inside I;., and toy,. there would correspond in thew-plane a closed curve 'Yw lying inside T, some point W 0 of y"' corresponding to the end-points ± u0 of y,.. Since Tis simply-connected we can now, by continuous deformation, and keeping the curve always inside T, reduce 'Yw to an arbitrarily sma.ll curve passing through w0 • But as, in this deformation, the end-points of y,. remain fixed, we have clearly arrived at a contradiction. 66. Equation (65·2) transforms the circle I w I= 1 into a bicircular quartic curve whose equation in polar coordinates is p 2 = 2 cos 28. . ..... (66"1) This curve is symmetrical with respect to the origin; and, of the two domains bounded by its loops, one must lie entirely inside T1 , and hence, by§ 65, the other must lie entirely outside T1 • The function 1-u t=1+u

...... (66"2)

transforms the curve (66·1) into the curve C of Fig. 21, having a double point at t = 1 ; and it transforms T 1 into a domain T2 which lies entirely inside the curve C, since the exterior of C corresponds to that loop of the curve (66"1) which contains no point of T1 . H therefore follows that

§66]

KOEBE'S THEOREM

37

the inner loop of C lies entirely inside T2 • But (65'2) and (66'2) give t== 11+

'J?

-w) == {1-.j(l-wW; 1-w) w

...... ( 66 .3)

and from this we obtain at once that, if I w I== I, It I~ (1 + ../2)2• Consequently ~ lies entirely inside this last circle; also, since the curve 0 is its own inverse with respect to the unit-circle I t I == 1, we see further that T2 must contain the whole circle It I= (1 + J2t 2 in its interior.

-./2

u-p lane Fig. 20

The theorem enunciated a.t the beginning of § 65 is now obtained on using (66'3) to express was a function oft and then writing t == (1 + ../2)2 z. . ..... (66'4)

t-planc Fig. 21

Since we supposed that T does not contain the point w == oo , T 1 does not contain the point u == oo ; hence t == -1 lies outside T 2 , and, finally, z=-(1 + J2t 2 lies outside T'.

38

ELEMENTARY TRANSFORMATIONS

[CHAP. m;,

§66

Remark. Let T- be a simply-connected domain contained in T and containing the point w = 0, and denote the frontier of T- by Yw· The transformation (65·2) gives two distinct domains in the u-plane corresponding to 'J"J. One of these contains the point u = + 1 and the other the point u = -1, and their boundaries may be denoted by yu' and y,/' respectively. If ·u' is a point of Yu', the relation u" = - u' transforms it into a point u" of y,.". 'fhe relation (66"2) transforms these two domains into distinct simply-connected domains in the t-plane. One of these, say T,*, contains the point t = 0 and consists of the points lying within a certain continuum y,'; the other consists of the points lying outside a continuum y/'. Since the domains have no points in common, y/' surrounds y,'. The relation t'' = another.

f,

transforms y,' and yc" into one

CHAPTER IV

SCHWARZ'S LEMMA &7. Schwarz's Theorem. We take as starting-point the observation that if an arbitrary analytic function l(z) is regular in a closed domain T, the maximum value of 1/(z) I in Tis attained at one point at least of the frontierS of T. The familiar proof of this theorem depends upon the eleme9~ fact that every neighbourhood of a regular point Zo of a non-constant function I (z} contains points~ such that 1/(z.) I> l/(z0 ) I· Consider now a function/(z) satisfying the following three conditions: (i) I (z) is regular in the circle Iz I < 1; (ii) at all points of this circle 1/(z) I <1; (iii)I(O}=O. From conditions (i) and (iii) it follows that "'{z) =/(z)fz ...... (67"1) is also regular in Iz I < 1 ; consequently, if Zo is any point inside the unit-circle, and r denotes a positive number lying between Iz0 I and unity, there is on the perimeter of the closed circle Iz I ~ r at least one point z1 such that I"' (z.) I ~ I "' (Zo) 1. But, by hypothesis, II(~) I < 1 and I~ I = r, so that we may write

I"' (zo) I ~ I "' <~> I = Illz. I < !,. ;

hence, since r may be taken arbitrarily near to unity,

I"' (zo) I~ 1.

. ..... (67"2)

Thus for all points inside the unit-circle I"' (z) I ~ 1; but if"' (z) is not constant, this may be replaced by I"'(z) I < 1 for all interior points. For if, "' (z) not being constant, there were an interior point Zo such that I"' (z0) I = 1, there would necessarily exist other interior points for which I"' (z) I> 1, and this contradicts (67"2). On use of (67"1} there now follows: THEOREM 1. ScHWARZ's LEMMA. Ijtke analytic/unction/(z) is regular /<»" Iz~ < 1, if 1/(z) I < 1/<»" Iz I < 1, and if/urtker I (o) = o, tlten eitller

1/(z) I< lzl I<»" Iz I < 1 <»"/(z} is a linear function of tke /<»"m l(z) =e'9 z, where 8 is rea/(13). 0

...... (67"3) ._ ... (67"4)

40

SCHWARZ'S LEMMA

[CHAP. IV

By (67"1) ~ (0) =/'(0), andourformerreasoningtherefore gives at once the further theorem :

If f(z)

satisfie,g tke conditions qf &kwarz's Lemma, then either If' (0) I< 1 or/(z) is qf the form (67"4). THEOREM 2.

68. Theorem of uniqueness for the conformal representation of sJ.mply-connected domains. One of the simplest but most important applications of Scpwa.rz's Lemma is the following. Let G be a simply-connected domain in the w-plane, containing, we will suppose, the point w = 0 in its interior; and we assume that a function w = f (z) exists which representS G conformally on the circle lzl <1, and is such that/(0) =0 and/'(0) is real and positive. We shall prove that tkefunctionf (z) satisfying these conditions is necessarily unique. For if there were a second function g (t), distinct from /(z), but satisfying the same conditions, the equation ...... (68"1)

g (t) =/(z)

would transform the circle It I < 1 into the circle Iz I < 1, making the centres t = 0, z = 0 correspond to each other. Hence botk the functions

t=

~

(z),

z = yt(t),

...... (68"2)

obtained by solving (68"1), would satisfy all the conditions of Schwarz's Lemma, and hence, by §67, both the inequalities ltiSizl, lziSitl would hold for all values of zand tsatisfying (68"2). Thus Itl =I (z) I= lz I and, again by Schwarz's Lemma,¢ (z) would be of the form eifz. But it follows from the original hypotheses that ~· (0) > 0, and hence cp (z) z; that is to say f(z) =g(cp(z))=g(z),

=

so that the function/ (z) is unique.

69. Liouville's Theorem. A large number of important theorems follow readily on combining Schwarz's Lemma with earlier theorems. As an example we may mention the theorem of Liouville that a bounded integral function is necessarily a constant. For ifF (z) is an integral function such that I F(z) I< M for all values of z, and we write z = Ru, /( ) =F(z)-F(O) u

2JL[

'

§§ 68-71]

41

PICK'S THEOREM

the function /(u) satisfies the conditions of Schwarz's Lemma for any positive value of R. Hence, if Iu I < 1 , 1/(u) I < u; i.e. if Iz I < R,

2M

IF(z)-F(O) I<][ lzl. Keeping z constant and letting R tend to infinity we obtain F(z)=F(O), as was asserted.

10. Invariant enunciation of Schwarz's Lemma. In the circle I z I < 1 let the function w = f(z) be regula.,. and let lf(z) I< 1; we no longer suppose, however, that f(O) = 0. nPnoting by zoanypoint inside the unit-circle, transform the unit-circles in thez-plane and in thew-plane into themselves by means of the Mobius' transformations (cf. § 35) t= z-zo w= w-f(zo) = f(z)-f(zo) ....... (70 .1) 1-zoz' 1-f(z0)w 1-f(zo)f(z) The fun.::tion w = w (t) obtained by elimination of z from equations (70"1) satisfies all three conditions of Schwarz's Lemma, and consequently

I w (t) I ~ It I·

...... (10·2)

This inequality can be expressed by the statement that the non-Euclidean distanceD (0, w(t)) of the two points w(O)=O and w(t) is not greater than the non-Euclidean distance of the poil').ts 0, t, so that, remembering that, as was proved in Chapter II, non-Euclidean distances are invariant for the transformations (70"1), we obtain the following theorem, which includes Schwarz's Lemma as a special case and which was first stated by G. Pi~kCl4l. THEOREM 3. Let f (z) be an analytic function which is regular and such that lf(z) I< 1 in the circle Iz I< 1, and let z~> Z 2 denote any two p<Jints inside the unit-circle; then either D (f(zl), f(zs)) < D (z1, z2) ...... (70"3) for all such values of ZJ and z 2 , or ...... (70"4) D (f(z1), f(z2)) = D(z1, z2) for all such values of ZJ and z2. Further, if (70"4) holds, the function w=f(z) is necessarily a Mollius' traniformation whick transfo'rms the unit-circle into itself.

71. Comparing (70·2) with (70·1) we obtain

1/(z~- f(z

0)

It -.I (zo)f(z)

I:£ II_[ z_-~z·:"'""'I

or

1./Jz~ =~(zo)

~ ~o

I:£ I~.((z~{(z) I; 1- ~u-

(CHAP. IV

SCHWARZ'S LEMMA

and if we let z converge towards

Zo

this gives

1 1/ ,(zo)I~" 1-l/(zo) 1-lzol2 2

Thus, since

Zo

is arbitrary, we now have the theorem:

THEOREM 4. 1// (z) satisfies tke conditions of TIIRhrem 3, then either

If' (z) I< 1 ~ ~~j~ 12

000 0.. (71°1)

/or all points z inside tke unit-circle, or

1 1/' (z)I= 1-l/(z) 1-lzl2 2

•o•···(71°2)

/or all points z inside tke unit-circle; further (71°2) lwlds if and only if (70°4) koldso

From (71°1) and (71°2) it follows at once that, if Iz I ~ r < 1,

I/' (z) I~ 1 ~r;

oooooo(71"3)

hence, denoting by z1 and z2 any two points of the closed circle Iz I ~ r, and integrating along the straight line from ~ to .z,, the relation

L"'l'

(z) dz=/(z2} -/(~)

gives at once, by the Mean Va.lue theorem,

THEoREM 50 lftke/unctionf(z) satisfies tke conditions of Theorem 3, and if~, .z, are any two points of tke closed circle Iz I ~ r < 1, then

I~~l /(~)-/(z2) ~-z2 1-7

.

o.. o(71'4)

72. As a la.Rt application of equations (70'1) and (70'2) we take the folloWing: if we solve the second of equations (70'1) for I (z) and take Zo = 0, we obtain /(z)= w+!(O) ; . 0. 0(72'1) 1+/(0)w but, for Zo==O, the firr,t of equations (70'1) becomes t=z, so that, by (70'2), Iwl ~ lzl. Also {72°1) is a Mobius' transformation which transforms the .::ircle Iw I ~ Iz I into another easily constructed circle, and by consideration of the figure so obtained we arrive without difficulty at the theorem: 00

§§72-74]

43

POSITIVE REAL PARTS

1/(.z) I< 1 in tile circle Iz I <1, tAen 1/(z) I ~ Iz I + II(o) I ...... (72 .2> 1 + 1/(0) liz I

THEOREM 6. Iff (z) is regular and

at all points z inside tile unit-circle. Finally we deduce from (72"2), by an easy manipulation,

1-lf- 1- lf I >-1-I/(O) I 1-l z I ""1 + lt
< 3) ...... 72 .

73. Theorem 3 of § 70 may be regarded a.s a. special case of a. more general theorem which we will now prove. If the function w=/(z) is regular for Iz I < 1, every curve Yz in the z-pla.ne which lies entirely inside the unit-circle corresponds to a curve y., in the w-pla.ne, which will also lie entirely inside the unit-circle provided that 1/(z) I < 1 for the values of z considered. If the curve Yz is rectifiable (in the ordinary sense) it ha.s, by §43, a. non-Euclidean length L (y..) which is defined a.s the upper limit of the non-Euclidean lengths of certain inscribed curvilinear polygons; to any non-Euclidean polygon P., inscribed in y.. there corresponds a curvilinear polygon P .. inscribed in y., and, by §70, the non-Euclidean distance between two consecutive vertices of P ID is never greater than that between the two corresponding vertices of P.. From this it follows at once that the curve Yw is also rectifiable and ha.s a nonEuclidean length L (yiD) which cannot exceed L (y..). If now the two curves have the same non-Euclidean length, corresponding elementary arcs of y.. and y., must also be equal, so that (71"2) must be satisfied at all points of y •. Thus, on using Theorem 4 of §71, we obtain ~he required theorem : PICK's THEOREM. If tile function w=f(z) is regular and in the circle I z I< 1, and if L (y.. ), L (y..,) denote the nonEuclidean lengths of corresponding arcs y.. , 'Yw drawn in the unit-circle, tAen either L (y.,)
1/(z) I< 1

74:.

~ctions with positive real parts. In the circle Izl < 1 suppose that/(z) is regular, that lV(z) > 0 and also supposef(O) = 1. By the Mobius' transformation 1+w u-1 U=-to=...... (74"1) 1-to' U+l'

44

SCHWARZ'S LEMMA

[CHAP. IV

the half-plane lltu > 0 is transformed into the uni u-circle Iw I < 1, and u = 1, w = 0 are corresponding points. Schwarz's Lemma can therefore be applied to the function f(z) -1

...... (74'2)

w=f(z)+1'

Consequently, the figure in the u-pla.ne, which corresponds, by means of the relation u =f (z), to the circle Iz I < r (0 < r < 1) must lie inside the circle which corresponds, by means of the Mobius' transformation (74'1), to the circle lwl
1-r

1+r 1-r

1+r

-1 -
75. Harnack's Theorem. It will now be shown that a theorem of Harnack, the fundamental importance of which has been long recognised, is an almost obvious consequence of our last result, so that Harnack's Theorem may without loss be replaced by Schwarz's Lemma. Harnack's Theorem may be stated as follows: If U17 U2 , U3 , ... denote a monotooe increasing seq'IU3nC6 of harmonic functions in tke circle

Iz I< 1, and if U.,. tends to a finite limit at z = 0, tken U,. tends to a limit unifQI'mly in the circle I z I < r < 1. We write ...... (75'1)

f,. (z) = u,. + i?J,.,

...... (75'2)

where v,. is the harmonic function conjugate to u,. and vanishing at the origin, so that, by hypothesis, tV.. (z) = u,. ~ 0, and consequently u,. (0) =f,. (0) > 0 unless f. (z) is identically zero. The functions j,.(z)/u,.(O) therefore satisfy the conditions of §74, so that, by (74'3), 1 +t'

If. (z)l
1+r

u,.
and

Harnack'~

Theorem follows at once from this.

§§75-78]

4o

BOUNDED REAL PARTS

76. Functions with bounded real parts. Suppose the function f(z) to satisfy the following conditions: (i)/(0) = 0; (ii)/(z) is regular for Iz I< 1; (iii)there is a constant k such that ...... (76"1) I Jit/(:&) I
-

'(IJ

. ..... (76"2)

These relations determine was a function w (z) of z, and w (z) satisfies the conditions of Schwarz's Lemma. From this fact and the relation (52·3) for I z I < ,. it follows that 2/t

1 +r

If/(z) I~- log ~ 1 -~. 11" r

. ..... (76•3)

77.

Surta.ces with algebraic and logarithmic branch· points. The functions tf>,. (z) and Y, (z), by means of which, in §56 and §57, the unit-circle was represented on the circular areas having algebraic or logarithmic branch-points, satisfy all conditions of Schwarz's Lemma; therefore they must satisfy the inequalities ltf>,.(z)l,.' (o) I< 1, Iy,' (o) I< 1. ... ... (11·2) The inequalities (77"2) can also easily be verified d~rectly; for in §59 it was shown that, if n > 2, IY,' (0) I< It/>,.' (0) I< It/>2' (0) I· But, by (56·3),

4>2' (0) =

:

ft

< 1.

78. A further property of the function used in§ 57, l+ulogll

Y,(u) =

k- el-u

I+u

,

(k < 1),

...... (78"1)

1-keJ-ulogll

may be obtained as follows. Consideration of the Mobius' transformation employed shows that, if Iu I< r < 1, ) 1-r -1+r logk
-u

l+r

and hence, in particular

I e 1--_,.log Ill >8 1::-r log ll =p. l+u

l+r

. ..... (78•3)

<Jonsequently, by (78"1), the values assumed by Y,(u) for these values

46

SCHWARZ'S LEMMA

(CHAP. IV

of u must lie outside a non-Euclidean circle with non-Euclidean centre k, and from this we at once obtain, remembering that p < 1, the inequality k+p ll/t(u)-kl> 1 +kp -k>(1-k)p, I+r log h

I.e.

ll/t(u)-kl>(1-k)e 1 -r

•

...... (78"4)

79. Consider now a function w=/(z) having the three properties (i), (ii), and (iii) of§ 67, and having also the further property that there is a real number k, (0 < k < 1) such that the equation/(z) = k has no solution in the circle Iz I< 1. With these assumptions it may be shown that if from the equation ...... (79"1) 1/t (u) =/(z) (where 1/t (u) denotes the same function as in (78•1)) we obtain that solution u = .P (z) which vanishes at z = 0, the function .P (z) possesses all the properties (i), (ii), (iii) of §67. Hence lul
*

I+r log h

1/(z)-kl> (1-k)e 1 -r

.

. ..... (79"2)

80. Representation of simple domains. Let the function ...... (80·1) w=/(z) represent the simple domain T, which was discussed in § 65, on the interior of the unit-circle Iz I< 1, and suppose that/(0) = 0. If in (65·1) we replace z by - w, and if we denote (1 + J2)-2 by k, §§ 65 and 66 show that the function ...... (80"2) w=
}

w= (k- w'f =/(z), ...... (80"3) k= (1 + J2)-2, effects a conformal representati::.:. of the domain T' COD$idered at the end of § 66 which lies inside the unit-circle I "' I < 1 and does not contain the point w=k. Since .P(O)=O it follows from Schwarz's Lemma that I "'I ~ r if Iz I ~ r, and also, by the Theorem of § 79, that

f

l+rl

lw-kl>(1-k)e 1 -r

og

h

...... (80"4}

§§ 79-81]

47

SIMPLE DOMAINS

But, from the second of equations (80·3), 1-k=2 Jk, and hence, from the first equation of (80•3), using the inequalities I "' I ~,. and (8o·4), if I z I~ r, lf(z) I< re

-2 !±!log h.

...... (80"5)

1-T

Observing that 1 + J2 = 2·41 ... < 2·7 ... < e, so that - 2log k=4log(1 + J2) < 4, we see that (80·5) includes the inequality 41~ I-T.

lf(z)l
. ..... (8o·s)

Thus we have proved that, in the confonnal representation (80"1), the domain corresponding to the circle Iz I ~ r < 1 is bounded afld lies wholly 4 1+T . 1 -".

within the fixed circle Iw I~ re

On the other hand the domain T contains, by hypothesis (see § 65), the circle I w I < 1 in its interior, and hence, by Schwarz's Lemma applied to the function z=cfl(w) inverse tof(z), the figure corresponding to Iz I ~ r must contain the circle I w I ~ r in its interior. If we now apply an arbitrary magnification, we obtain the following general theorem : THEOREM. Let T be a simple domain qf the w-plane containing w = 0 in its interior but not containing w = oo ; let a be tke dista'IUJe qf tke point w=O from tke frontier qf T, and let f(z) denote a function wil,ick represents T confqrmo)Jy on tke circle Iz I < 1, malcing w = (1 correspond to z = 0. Tken, for any value qf z inside tke circle Iz I < 1. lf(z)l 41+1•1 a~ lZI
The fonnulae already established can also be used to obtain limits for the difference quotient {/(zt)-/(Zt)}/(~ -Zt) of the functionf(z). For 41~

example, by {80•7), the function f(ru)fare I-r has modulus less than unity inside the circle Iu I< 1. Hence by Theorem 5 of § 71 inside the circle 1u 1< r, i.e. inside the circle 1z 1< r,

l

4 I+r

f (zt) -f(Zt) j = jf(t"Ut)-f(rus)l s ae 1 - .. Zt-Zt r(Ux-U,) - 1-r' lzd
81. By letting Iz I tend to zero in (80•7) we see that the function f (z) of§ 80 satisfies the inequalities a~ lf{O) I~ ae'. It is known by Schwarz's

48

(CHAP. IV

SCHWARZ'S LEMMA

Lemma that a is the true lower bound for I/' (0) I; we shall now determine the true upper bound for this same number, using a very ingenious method due to Erhard &kmidt. It is published here for the first time(l5). The considerations which follow form the basis of the proof: (a) Suppose that the function} (z) is not a constant and that it is regular within and on the frontier of a simpiy-connected domain D, whose frontier is a regular curve y. Then, ifj(z) = u + iv, where u and v are real,

i

uclv>O.

. ..... (81"1)

For, if account is taken of the Cauchy-Riemann equations, Green's Theorem shows that

i

uclv= fJD (u,2 +u.,/)dxdy

(b) The inequality (81"1) can be extended to multiply-connected domains provided/(z) is regular and single-valued in the domain considered. Let y' and y" be two closed curves such that y' lies within y" and itself surrounds z = 0. Then the function log z is regular in the annular domain between y' and y" but it is not single-valued. If however z = pd-1{1 and log z = log p + itfr, then log p and dtfr are single-valued functions in the annular region, and hence

£-,logpdlft~ fr,togpdtfr.

. .. 000(81°2)

(c) If the curvesy' andy" are such that the rela.tionz" =!,transforms

z

y' into y", it is easy to see that

f logpdljl+llogpdtfr=O. J.,.. y

•ooooo(81"3)

From (81"2) and (81"3) it is seen that in this special case

f log p d.p ~ Oo

}.,·

... 00o(81°4)

(d) Let F(z) = zq, (z), where q, (z) is regular and differs from zero throughout the closed circle Iz I ~ 1. We write F(e") = p (6) ff/tt11l. . .. ooo(81"5) The hypothesis shows that the function log q, (z) is regular for I z I ~ 1, and, if 1z i =1, then log q, (e111) =log (F(el') r 111) =log p + i (tfr- 6). By (81"1) £Iogpd(tfr-6)~0, .... 0.(81"6) where " is the circle I z I= 1.

§ 82]

KOEBE'S CONSTANT

4:9

On the other hand, by the Mean Value Theorem,

r2· logpd8,

1 logicf»(O)I=I0ogcf»(0)= 27T}o

...... (81"7)

and, further, cf» (0) = F' (0 ). Comparison of ( 81 ·s) and (81"7) shows that log I F"(O)

I~ 2:.ilogpd.r.

. ..... (81"8)

82. Let/ (z) be the function considered at the end of§ 80. If r < 1 we write 1

g (z) ==-/(rz) a

...... (82"1)

and

F(z)= 1 - JI-g(z) 1 + J1-g(z) By(66'3), thefunctionF(z) transforms

_

_fliz_~-~-- ....... (82"2)

{1 + .J1-g(z)l2

lzl ~ 1 conformallyintoasimple

domain T.• which lies within the domain T9 of§ 66. Thus F(z) does not

z

vanish in the unit-circle. If therefore

F (eill) == pe'"', the inequality (81"8) holds. By (82"2) and (82·1)

F' (0) = g' (O) = .!___I' (0). 4

Thus . log

Irj~~O) I~ ;11"

4a

ilog p. "'' (8) dO= 2~ i,log p

a.r. ...... (82"3)

Reference to the last part of § 66 now shows that y' has all the properties assumed for this curve in (b) and (c) in§ 81, so that (81·4) holds. From this and (82'3) it follows that If (0) I ~ 4afr, and, since this holds for all r< 1, . ..... (82"4) I/'
50

SCHW.ARZ'S LEMMA

(CHAP. IV

83. Representation upon one another of domains containlng circular areas. Let R., and R,. be two simply-connected domains which lie entirely inside the circles Iwl <1and Iz I< 1, but contain the circles Iw I
,._1

In the first place, by Schwarz's Lemma,

l'w(z)l~! z . It at all points of I z I
at all points of R,.. But, if lzl
Iw - z I =

i; -1\zl ~ I; -1,. i

kr

.. .... (83'4)

To obtain an upper bound for this expression we now write Z==

Itt,

w (z) = F (t), F(t) , (t) z }/(0)==.,. '

...... (83·a)

so that, by (83'3), 1-k

IF(0}-11 ~ -----r• ...... (83'6)

§§ 83-85]

51

RIGIDITY OF THE TRANSFORMATION

the last inequality holding since F(O) = w' (0) was supposed real and positive. With this notation

I~ -11==·1F(t)-11 ~ IF(t)-F(O)I + IF(O)-II' and hence, by (83·6), l

1-k z- I I~ 14> (t) -II kI + T·

w

...... (83"7)

84. If a is an arbitrary complex number, represented (say) by the point P, the number I a - I I, being equal to the length of the segment UP, is not greater than the length UMP, where MP is an arc of a circle with centre 0. We therefore have the inequality Ia - J I ~ JI a I - II + Ia II IE log a i . ...... (84"I)

Fig. 22

86. By (84"1), 14> (t) -11 ~ 114> (t) I-II+ 14> (t) II lE Iog4> (t) I· ...... (85"1) But since, by (83"5}, 4> (0} = 1, the branch of the function 1/1 (t) =log 4> (t), which vanishes fort== 0, satisfies, in consequence of (83"6), the inequality lllil/l (t) I~ -2log k; and therefore, by § 76, if It I ~ r, lllog4> (t)l

~ - 4 ~ogklog ~ ~~:

...... (85"2)

Also, by (83"6}, 1 14> (t) I ~ Ji2'

114> (t) 1- II~

I- k 2

Ji2 .

. ..... (85·3)

It now follows from (85"I), (85"2) and (85"3) that l4> (t) _ II ~ I - k 2 _ 4 log It log I + r k2 7rk2 I- r' and on using (83"4) and (83"7) we obtain finally

lw-zl ~m(k, r), k ) =(Ih3 ) r _ 4rlop: k 1 .i!:_ ( m 'r k" "Irk• og 1 - r ·

l

•

0

••••

(85"4)

From this e::pression it is seen that not only is lim m (k, r) = 0 for h.-1

every fixed r, as was to be proved, but also lim m (k, h)= 0. h-1

52

SCHWARZ'S LEMMA

[CHAP. IV

Remark. By longer calculations than the above it can be shown that the function m (h, r) of (85·4) can be replaced by a much smaller function.

86. Problem. The reader may now make use of the results of§§ 82-85 to establish the following theorem. · Let Rv. and S,. be two simply-connected domains of the U-plane which contain the point u = 0 in their interiors and which are represented on the interior of the unit-circle Iz I < 1 by the functions u=f(z), (f(O)- 0, f' (0) > 0), and u = g (z), (g (0) = 0, g' (0) > 0), respectively. It is supposed that the functions u =f(z) and u = g (z) represent the circle Iz I < h < 1 on domains which lie entirely inside S. and R,. respectively, i.e. that all points of R,. or of S,. whose non-Euclidean distances from u=O are less than !log

!~! are common to both the domains.

Then it is to be

proved that, if lzl ~hr
relation7

87. Extensions of Schwarz's Lemma. Let w =/(z) again denote a function which is regular and such that all points of the circle Iz I < I. Together with any nonEuclidean circle O(z) with non-Euclidean centre z and non-Euclidean radius p (z) we consider the circle I' (z) in the w-plane whose nonEuclidean cent.e and radius are w =f(z) and p (z). If A.., denotes the set of all points belonging to an arbitrary set of such circles 0 (z) (and their interiors), and Aw that covered by the set of corresponding circles T (z) in the w-plane, it follows from Theorem 3 of§ 70 that any interior point of A. is transformed by the function w =f(z) it: to an interior point of Aw. If the set of circles 0 (z) is enumerable we can also consider the set of points B. consisting of all points which lie in all but a finite number of the circles 0 (z), and compare it with the set of points B.., obtained in the same way from the corresponding circles I' (z) in the w-plane. It follows from the theorem quoted that w =f(z) transforms any point of B. into a point of B'ID. Various applications of the above considerations can be made, and those applications are particularly interesting which throw light on the behaviour of/(::) in the neighbourhood of the frontier Iz I= I.

If (z) I < 1 at

§§ 86-89]

53

JULIA'S THEOREM

88. Suppose first that the centre z of the circle 0 (z) describes that diameter of the circle 1 z I < 1 which lies along the real axis, and that all the circles 0 (z) have the same non-Euclidean radius. In this case the domain A., consisting of all points interior to any of these circles A B O(z), is the area bounded by two circular arcs, and is symmetrical with respect to the real axis (see § i8). If then f(z) is a function which is. regular and such that lf(z) I < 1 if I z I < I, Fig. 23 and if also f(z) is real wltenever z ts real, the circles r (z) in the w-plane corresponding to these circles 0 (z) will have their nonEuclidean centres also on the real axis, and, since their non-Euclidean radii are the same as the non-Euclidean radii of the circles C (z), must lie inside precisely the same area A., but in the u·-plane. We therefore have the following theorem : THEOREM. Let f(z) be a function whick is regular and sud that If (z) I
89. Julia's Theorem. The method of§ 87 leads without difficulty to an important theorem which is due to G. Julia(!6). We note in the first place that two real points :r, k (:r
. ..... (89"1)

or, writing 1 -It= u, 1 - k = v, if 1-y v(2-u)1-:r I +y = u(2-v) I +:r"

...... (89"2)

Consider now two sequences of positive real numbers ul> u 2 , u 3 , ••• and ••.. satisfying the following conditions: . v,. IliD -=CX, lim u,.=lim v,.=O, ...... (89"3)

vh t'2o t•3 ,

n~oo

U,.

where ex is finite. Denote by K,.' the circle having the point It,.= 1 - u,. as

54

SCHW.ARZ'S LEMMA

(CHAP. IV

non-Euclidean centre and the point z on its circumference, and by r ,.' a circle of the same non-Euclidean radius and having the pointk,. = 1- v,. as non-Euclidean centre. The circle r ,.' cuts the real axis of thew-plane at a pointy,. which is obtained by replacing u, v, y in (89"2) by u,., v,., y,.. The circles K,.'

Fig. 24

converge, in consequence of (89"3), to an oricycle K of the non-Euclidean plane Iz I< 1, passing through the points z = 1 and z = z, and the circles r ,.' to an oricycle r of the non-Euclidean plane I w 1 < 1, passing through w = 1 and w = y; here y, being the limit of y,., is given by the equation (see (89"2) and (89"3)] 1-y 1-z --=ot-. 1+y 1+z

...... (89"4)

If r and p denote the Euclidean radii of K and r, then z y= 1-2p, so that, by (89"4), p=

otr 1-r(1-ot)"

-,---;-c----;-

=

1- 2r,

...... (89"5)

90. Now let/(z) be an analytic function which is regular and of modulus less than unity in the circle Iz I < 1, and suppose further that there is a sequence of numbers Zt, z 2 , z3 , ••• such that lim z,. = 1, lim/(z,.) = 1, ...... (90"1) ft.~CIO

and

I.

fi ....OO

1-l/(z,.) I

1m 1- IZn I =ot, ,._..,

...... (90"2)

where ot is finite. It follows from (72"3) that

1-l/(o) I

ot~ 1 + 1/(0)1 >O.

...... (90"3)

Write now u,. = 1 -I z,. I, v,. = 1 - If (z,.) I, and construct the circles K,.', r ,.' of § 89. Still using K and r to denote the oricycles to which K,.' and 1',.' converge, let Kn and r,. denote circles with the same non-Euclidean radii as Kn' and r,;, but having their non-Euclidean centres at z,. and f(z,.) respectively, instead of at lznl and 1/(z,.)l. Thus K,. is obtained

§§ 90-92]

JULIA'S THEOREM

55

from K,.' by a rotation about the origin; but since, by (90'1), the angle of rotation tends to zero as n tends to infinity, the circles K,. tend to the same limiting figure K as the circles K,.'. Similarly the circles 1',. tend to 1'. Thus, by §87, if the point z lies inside K, the point w=f(z) cannot lie outside 1'; we can in fact show, by an argument similar to that of § 67, thatf(z) must lie inside 1'. The above result constitutes Julia's Theorem• (16).

91. The work of § 90 may be completed by adding that if there is a point z1 of the frontier of K which is transformed into a peiatf(z1) of the frontier of 1', then f(z) must be a bilinear function. For, by §48, any two oricycles which touch the circle Iz I = 1 at z = 1 are parallel curves in the sense of non-Euclidean geometry; also equation (89'4), which represents a non-Euclidean motion, transforms the two oricycles through the points .x1 and .:v2 (.r1 < .x.) into the two oricycles through '!h andy., and.the distances between these two pairs of oricycles are the same, say, 8. If then z1 is a point on the oricycle through .x1 such that the corresponding point W1 =f(zl) lies on the oricycle through y 11 and we determine on the oricycle through .:v2 the unique point z2 for which IJ (z1 , ~) = o, then, by Julia's Theorem, the point w2 =f(~) cannot lie outside the oricycle through J/2; that is to say, IJ (w~o w2 ) ~ o. But, by Theorem 3 of§ 70, D( w~> w2) ~ o. Hence D (wit w2) = o, and this cannot be true for a single pair of points unless w =f(z) represents a nonEuclidean motion. Since also/(1) = 1,/(z) must be of the form - z-zo 1-zo /( z ) ---...... (91'1) 1-zoZ 1-z0 '

92. Julia's Theorem may be interpreted geometrically as follows: m Fig. 25 we see that AP

A.x

1-.:v

PB- .xE= 1 +.x' and similarly that

A1P1 ~ 1-y PlBI 1 +y"

Thus, if P, P 1 denote the points z,f(z) respectively, Julia's 'rheorem gives, on using (89'4), the relation A1P1~ AP ...... (92'1) PJJ1 "'(J.PB· * In ,his own proof of this theorem Julia. requires tha.t f(z) should be regula.r at z = 1. It is rema.rka.ble tha.t it should be possible to prove the theorem without

assuming regula.ri ty a.t z = 1. E

06

SCHWARZ'S LEMMA

(CHAP. IV

But by elementary geometry

AP AP AP l1-zl 2 PB = AP.PJJ= CP.PD=l-lzl 2 '

...... (92"2)

so that Julia's Theorem is equivalent to the following inequality: ...... (92"3)

Fig. 25

93. A number of interesting consequences follow from Julia's Theorem. If we continue to denote, as in (89"5), the Euclidean radii of the circles K and r by ,. and p, then, by the theorem, the real point :X=

1-2'1"

...... (93"1)

corresponds to a pointf(:x) lying inside or on the frontier of I', i.e. such that 1-lf(:x) I ~ 11-f(:x) I ~ 2p. Thus, by (89"5) and (93·1), 1-lf(:x)l
1-

.X

z-1

1-

.X

...... (93"2) (

· · · ... 93"3

)

But a: may be any number for which an equation such as (90"2) holds; thus the most favourable choice of a: will be made by choosing the sequence z,. of {90"1) and (90·2) so that a:=lim !=J/(.x)l_ -x... l 1-.x

§93]

57

JULIA'S THEOR.Elrl

When this is done, inequalities (9_?·2) and (93"3) at once establish the existence of the limits lim l-1/(x)l =lim 11 -/(x)l =Ot. . ..... (93"4) z-1

1 -X

z-1

1 -X

Since, by (90"3), at> 0, the last equation shows that .

I-1/(x}l

!~11-/(x)l

But, if we write

...... (93"5)

I.

1-f(x) = AeUI,

...... (93"6)

where A> 0 and 6 is real, we have I-1/(x)l_ 2cos6-A . 11-=/(X}T- 1 + J(1- 2A. cos 8 + A.2) ,

. ..... (93"7)

also, when x tends to unity, A tends to zero, and therefore, by (93"5) and (93"7), cos 6 (and hence also eUI) must tend to unity. From this it follows, on using (93"5) a.ud (93"6), that lim 1 -f(x)=lim 11 -f(x)leUI =at. . ..... (93"8) 1-x -• 1-x We may collect what has been proved in the form of the following theorem: .,_1

THEOREM.

If f(z) is regular and suck tkat lf(z) I< 1 in tke circk

Iz I < 1, and if there exists inside ~•

tke unit-circk a sequence qf points z1 ,

... /0'1" wkicklim z,.= I, lim/(z,.)=1, and . 1-l/(z,.)l

lrm ,._... 1- I z,. I

( ...... 93·9)

itt finite, tken tke following limit-equations kold, wherein x tends to its limit tkrougk arbitrary real values and at is a real positive constant lim 1-l.f(x)l = 11.m II-/(x)l = l•"m 1-/(x) .,_. 1-x .,_1 1-x .,_. 1-x

Ot. . ..... (93"10)

CHAPTER V

THE FUNDAMENTAL THEOREMS OF CONFORMAL REPRESENTATION

94. Continuous convergence. If A is an arbitrary set of points in the complex plane, we consider a sequence of complex functions}; (z), f 2 (z), ... which are defined and finite at all points of A ; and we introduce the fol1owing definition : if z 0 is a limiting-point of A the sequence f,. (z) will be said to be continuously. convergent at z0 if the limit lim f,.(z,.)=f(z0)

••••••

(94:1)

exists for every sequence of points z,. of A having z0 as its limit. It is readily proved that the limitf (z0 ) is independent of the particular sequence z,., and that every sub-seq uencef,.1 ,j"a, ... of the given sequence also converges continuously to f (z0) at the point Zq, i.e. that Jim f,." (zk) =f(z.),

(n1 < nz <

... ).

. ..... (94"2)

k--c:o

The following theorem also follows immediately from the definition of continuous convergence:

THEoREM. Ij (i) the sequence of functions w =f,. (z) is defined in the set of points A and converges continuously at the point z0 to the finite limit W 0 =f(z 0 ), (ii) for all points z of A and for all values of n the point w =f,. (z) belongs to a set of points B oj thew-plane, and (iii) the sequence of functions cp,. (w) is defined in B and converges continuously at. w., then the sequence F,. (z) = cp,. {f,. (z)} converges continuously at z0 • ·

95. Limiting oscillation. Letft (z),j2 (z), ... be an arbitrary sequence of complex functions which are defined in a domain R and uniformly bounded in a neighbourhood of every point Zo of R. Denoting by Zo a point of R, let C01 , Cl 2l, ••• be a sequence of circles lying in R, having their centres at Z 0 , and such that their radii decrease steadily to the limit Z')rO. Further, let o,.(k) denote the oscillation of{,. (z) in the circle Clkl, i.e., O,.fk)

=sup If. (z')- f,. (z") 1.

. ..... (95"1)

§§ 94-96] where

59

CONT~OUSOONVERGENCE

z, z" are any two points of the domain 011:1.

__..,

If we now write ..•..• (95"2)

we have, since 011:+ 11C ()1.1:1, wli:+JI ~ wiA:I; hence the limit w (z0) =

lim

U~(l:)

k-oo

exists. We see also that this limit depends only on the sequence /,. (z) and on the point z 0 , but not on the clwice of the circles 01"1• The number w (z0) is called the limiting oscillation of the sequence /,. (z) at the point z 0 •

96. We now prove the following theorems: THEOREM 1. Jf w (z0) and u (z0) denote respectiuly the limiting oscillations of a sequence offunctiuns f,. (z) and of a sub-sequence f"r (z) at the point z 0 , then CT (z0) ~ w (zo). ...... (96"1)

For, by (95·2), for every circle ()1.1:1 ulk) ~ wlk).

THEOREM 2. .({ at a point z 0 the limiting oscillation w (z0 ) > 0, the cannot cunverge contintWUSly at z 0 •

sequence/,. (z)

For, by hypothesis, for every integer lc an infinite number of functions of the sequence have, in the circle ()1~>1, an oscillation greater thanU~It) -!w(z0 ), i.e. (since wltl ~ w(z0)) greater than !w(z0). Hence there are two sequences of points z1', z2' ••• , and zt, ~", ... , and also an increasing sequence of inkgers ~ < n2 < ... such that simultaneously lim z~:' = z0 ,

k....,oo

lim z~:'' = z0 ,

]& ..... oo

I/,." (z~:') -

j "" (zA:") I > !w (zo). ...... (96"2)

Hence, of the two sequences

w,: = f,."

(z~:'),

w~:" =/,.A: (zA:"),

...... (96"3) either at least one is not convergent, or, if both converge, their limits are unequal; and each of these possibilities excludes continuous convergence, by § 94. THEoREM 3. Tlte sequence/,. (z) cunverges cuntintWUSly at Zo provided tlw.t U~ (z0 ) = 0 and tlw.t every neiglthourlwod of Zo cuntains a point ' BUCk tlw.t tAe seqtteneej,.(') is contJergent. Titus, in particular, tAe sequence converges continuously if and lim J. (Zo) exists and is finit8.

·--...

w (z0) =

0

60

THEOREMS OF CONFORMAL REPRESENTATION

[CHAP. V

Let Z 1 , Zto··· denote an arbitrary sequence of poi11ts of R having Zo as its limit. We have to show that, when the hypotheses of Theorem 3 are satisfied, the sequence w,.=f,.(z,.) ...... (96'4) is convergent. Denoting byE an arbitrary positive number, we can, since lim (1)11:1 = 0, define a circle ()1.1:1 for which k._CII

(A)(k) <}E. .. .... (96'5) In ()l.l:l there is, by hypothesis, a point ( at which the sequence f,. (z) converges; and hence there exists a number N 1 such that, if n > N 1 and m>N1, If,. fm I N 2 the oscillation 0,.11:1 < tE; finally, suppose that for n > N 3 the point z,. lies in ()1.1:1. Thus if N denotes the greatest of the numbers N1, N2, Na, we have, provided only that n >Nand m > N, in addition to (96'6), the inequalities lf,.(z,.)-f.. I< !E, lf,.(z,..)- f ... I< 1€....... (96'7) In view of (96'4) to (96'6), .. .... (96'8) I w,.-w,.l <E (n,m>N). Hence, by Cauchy's criterion, the sequence w,. converges, so that Theorem 3 is proved. THEOREM 4. Q B denotes tke set of points z of A at whick tke sequence f,. (z) converges to a function f(z), then every limiting point Z 0 of B suck tkat (A)(z0) = 0 belongs to B, and tkefunctionf(z), whick is defined in B, is continuous at z0 • From 'l'heorem 3 we have at once not only t.hat the functions f,. (z) converge at z 0 , so that z 0 is a point of B, but also that the convergence at z0 is continuous. Now consider an arbitrary sequence of points z~> Zt, ... of B, having z 0 as limit; to establish the continuity off (z) at zo we have only to prove that ...... {96'9) limf(z,.) =f(zo)·

m- m

m

m

n-co

Since the sequence/,. (z) converges at each of the points z~;, we can find .an increasing sequence of integers~. n,, ... , such that for every k 1 lf,.t (z~:)-f(z~:) I < k.

.. .... (96'10)

Equation {96'9) now follows on combining (96'10) with the fact that, on account of the continuous convergence of the sequence at zo. lim f,.t(z~;;)=f(zo)· ...... (96'11) k-co

§§97, 98]

61

NORMAL FAMILIES

97. Normal ftunillea ofbounded functions

(17) (18).

We consider a definite class {/ (z )} offunctions f(z), which are defined in a domain R and are uniformly bounded in certain neighbourhoods of every point z0 of R. For such a family of functions we can define for each point z0 of R a number w (z0), which we shall call the limiting oscillation of the family. We again consider the sequence of circles 0<" 1 defined in § 95, and we denote by w tkl the upper limit of the oscillations in ()t"l of all functions /(z) of the given family; thus w<"1 has the following two properties: if p > w<"l at most a finite number of distinct functions belonging to {f(z)} have oscillations in 0"1 which are greater than p, whereas if q < w 1"' an infinite number of distinct functions of the family have in 01"1 oscillations greater than q. Finally, we define the limiting oscillation by the equation w

(z0) = lim k._ao

w<J:I.

. ..... (97•1)

The following theorem is obvious : THEOREM 1. Tke limiting oscillation at Zo of any sequence of functions };. (z), /,. (z), ... wkich belong to {f(z)} cannot be greater than the limiting oscillation w (z0) of tkefamily {f(z)} at Zo.

98. We now introduce the following definition: a family {/(z)} of functions defined in R and uniform(y bounded in detail (im kleinen) is said to be normal on R if tke limiting oscillation w (z0 ) vaniskes at every point Z 0 of R.

We have THEoREM 2. If f 1 (z), /, (z), ... is a sequence of /unctions all belanging to a family {f(z)} which is normal on R, tken tke given sequence contains a sub-sequencef.,.1 (z)J~~s (z), ... whick is continuously convergent at every point qf R (19). Consider a countable sequence of points z,, z,, ... of R which lie everywhere dense on R. By the diagonal process ( Oantor and Hilbert), a sub-sequence /"'(z), f.,..(z), ... can be picked out from the given sequence in such a way that, at each poie -:-f the countably infinite set, the limit ...... (98•1)

t:xists. The conditions of Theorem 3, § 96, are satisfied by this sub· sequence at every point of R; this proves the present theorem.

62

THEOREMS OF CONFORMAL REPRESENTATION

[CRAP. V

99. It is easy to show that the two conceptions, (a) continuous converg61/Ce of a. sequence of continuous functions at all points of a closed set y, and (b) uniform converg61/C6 of this sequence on "f, are completely equivalent (18). The following theorem is a consequence of this equivalence, but we shall give a direct proof. THEOREM 3. If a sequene~~ J;. (z), .h (z), ... qf functions converges continuousZ11 to thefunctionf(z) at all points qf a bounded closed set y, and if f(z) =F 0 on y, then there is a positive number m and a positive integer N, such tkat, at every point of y and for eack n > N,

lf,.(z) I >m.

. ..... (99'1)

If this were not so, it would be possible to find an increasing sequence of integers, n1 < ~ < ... , and a sequence of points ~. z~, ... of "f, such that ...... (99'2) By omitting some of the points z., and ordering afresh those that remain we can ensure that lim z., = z0 • .. .... (99·3) I:_..

Since y is closed, z0 belongs to y and therefore the sequence / 1 (z), f~ (z), ... converges continuously at z0 • Hence

f(zo) =lim f,... (z.,) = 0,

...... (99'4)

k-

and a hypothesis is contradicted. This proves the theorem.

100. Eldstence of the solution in certain problems of the calculus of variations. Weierstrass' Theorem that a continuous function defined on a bounded closed set in n-dimensional space attains its maximum at some point of the set can be extended, in certain circumstances, to function-space, that is, to space in which any element is defined not by a system of n numbers but by a definite fun<;~~vn. For instance, suppose that a family {f(z)} of complex functions is given, all the functions being defined in a domain R. The family is said to be compact if every sequence / 1 (z), / 2 (z) ... , of functions of the family, contains a sub-sequence f"l (z),f-s (z), .. which converges in R to a function/0 (z), where.fo (z) also oolongs to the family (a)).

§§ 99-101]

63

EXISTENCE OF THE MAXIMUM

Secondly, we consider a functional J (f), by means of which a finite number is associated with each member f (z) of {f (z)}. The functional J (f) is said t<> be continuous if the convergence in R of the sequence ft (z),f2 (z), ... to the function/0 (z)of {f(z)}alwaysimpliesJ(f,.) ~J (fo). ·We now have: THEOREM. If J (f) is a continuqusfunctional defined in tlze.compactfamily {f(z)}, tken the problem I J(f)l =Maximum ...... (100'1)

kas a solution within thefamily {f(z)}. That is to say, there is at least one member fo (z) of the family {f(z)} such that all membersf(z) of {f(z)} satisfy I J(f)l ~I J
.. .... (100'3)

Now, since {f(z)} is compact, a sub-sequence/"~ (z).f,.. (z), ... can be selected from the sequence .It (z), / 2 (z), ... in such a way that the subsequence converges to a memberfo (z) of {f(z)}. The continuity of J (f) shows that I J(.fo)l =lim I J(f,..t)l =IX. k-.o

This not only proves the theorem but also shows that IX is finite.

101. Normal fil.mUies of regular analytic functions. We now considerfamilies {f (z)} whose members/ (z) are all defined, analytic and regular, in the interior of a domain R. For these families the following theorem holds: THEOREM 1. lf {f{z)} is a family qf analytic functions, whick are all regula!r in the interior qf a domain R, and whick are uniformly bounded on the circumference qf any circle lying strictly witMn R, then {f(z)} is normal in R, and the same is true of the family {!' (z)} of deriwd functions, obtained from the functions f (z).

Let z0 be a. point of R and Iz- z0 I ~ p a closed circle lying strictly within R. There is a. positive number M such that all members of {f(z)} satisfy the relation If (z) I < Mat all points of Iz - z0 I = p and hence also at all points of Iz- Z 0 I < p.

64

THEOREMS OF CONFORMAL REPRESENTATION

[CHAP. V

Account being taken of Theorem 5, § 71, it is seen that, if r is any number between 0 and 1, the relation

~ Mlz'-z"l < 1-r 2Mr If(z')-f(~")l ., "'p(1-r)

...... (101"1)

is satisfied by every pair (z', z") of points of the circle Iz- z0 I < pr and by every member of {f (z)}. From this it follows immediately that the limiting oscillation of {f(z)} vanishes at the point z 0 • By§ 98 the family {f(z)} is normal. To prove that the family {f' (z)}, consisting of the derived functions of the given functions, is also normal, we need only r3mark that by Theorem 4, § 71, the derived functionsf' (z) are uniformly bounded in the circle lz-zol <;,pr.

102. We shall now show that the boundary function.fo (z) of a convergent sequence/1 (z),f2 (z), ... offunctions of the family is analytic. For this, as is well known, it is sufficient to show that fo (z) is differentiable at every point of R. Take a closed circle Iz - z0 I <;, p, lying in R, and in it consider the sequence of regular analytic functions cp,. (z), which is defined by the equations ..~,. ( ) _f,. (z) -f,. (zo) ...... (102"1) ..,..,. z' (z *' Zo), z-z0

...... (102"2)

The sequence of functions cp,. (z) is uniformly bounded on the circumference Iz - Zo I = p of the circle in question, and hence, by the result of the preceding paragraph, it is normal. The sequence obviously converges at all points of the circle other than zO> and therefore it converges continuously at the point z0 itself (Theorem 3, § 96). Let the boundary function be denoted by cp (z); then (102"1) and (102"2) vield cp(z)=fo(z)-fo(Zo), Z-Z0

cp (z0)

= lim f.,.' (zo)·

(z*'zo),

...... (102"3) ....•. (102"4)

Further, by Theorem 4, § 96, cp (z) is continuous at the point z0 • The above equations therefore assert that fo' (z0 ) = cp (z0) = lim f,.' (zo). . ..... (102"5)

,.._..,

This gives THEOREM 2. If a sequence Q{ analytic functions is uniformly bounded in a domain in wkick tke sequence converges, tken tke boundar-y function is

§§ 102, 103]

REGULAR CONVERGENCE

Go

analytic and its derived function is tke limit qf tke derived functions of tke functions qf tke approximating sequence. Keeping this theorem in mind, we shall, for the sake of brevity, describe a convergent sequence of regular functions which is uniformly bounded in detail as regularly con'INJrgent. Suppose that a given sequence of analytic functions, defined in a. domain R, is uniformly bounded in detail ; the sequence is regularly convergent in R provided only that the points at which it converges possess a limiting point in the interior of R. For, if the given sequence did not converge everywhere, it would be possible to select from it two sub-sequences converging to two distinct analytic functions/ (z) and g (z). But, at each point at which thegivensequenceconverges, (f(z)- g (z)) = 0. These zeros cannot possess a limiting point within R.

103. The following theorem relating to regularly convergent sequences of analytic functions is a special case of a well-known theorem due to Hurwitz: THEOREM 3. 1./, in a domain R, the sequence offunctions J; (z),/2 (z), ... converges regularly to tkefunctionf(z), none oftkefunctionsf,.(z) vanishing at any point of R, tken either f(z) 0 or f(z) does not vanisk at any point of R. For suppose that/ (z) does not vanish identically in R. Corresponding to each point z0 of R there is at least one circle Iz- z0 I ~ p lying within R and such that on its circumference f (z) =1= 0. By Theorem 3 of § 99, there must be a number m > 0 such that If. (z) I> m ...... (103"1) for all points on the circle Iz - z0 I = p, provided that n is sufficiently large. But in this circle/,. (z) =1= 0, and therefore, if n is sufficiently large. If,. (Z0 ) I~ m. . ..... (103"2) Thus lf(zo)l = ..._ lim lf.(zo)l ~m,

=

..

and it follows t~tf(z0)=1=0. An immediate corollary of this theorem is : THEOREM 4. If in a domain R a sequence of functions .f. (z),f, (z), ... con·veryes regularly to afunctionf(z) wkick is not a constant, then any neig/WourltOOd N zo qf a point z 0 of R contains points z,. suck tkat f,. (z,.) = f (zo), if n is sufficiently large. If this were not so, the given sequence would yield an infinite subsequence,/,.1 (z),f"' (z), .~., such that, in the neighbourhood N,.. of z0 , the functions (f.~ (z)-f (z.)) all differ from zero. Since the boundary

66

THEOREMS OF CONFORMAL REPRESENTATION

[CHAP. V

function (f(z)-f(z0)) of this last sequence vanishes at the point z0 , Theorem 3 shows that (f (z)-f (z0)) vanishes identically, i.e. the functionf(z) is a constant, and a hypothesis is contradicted.

104. Application to conformal representation. The following theorem is of fundamental importance in the theory of conformal representation : THEOREM. If/ 1 (z), / 2 (z), ... is a sequence offunctions whick converges regularly in a domain R, and if thefu11ftions give conformal tram(ormatiws qf R into simple domains SlJ 8 21 ••• respectively, whick are uniformly bounded, then, either the boundary function f (z) is a constant, or it gives a cMiformal tran:iformatioo of R into a simple domain S. By hypothesis,/,. (z) =Ff,. (z0) when z andz0 are points of R, z =Fz•. The functions 4>,. (z) =f,. (z)- f,. (z0 ) do not vanish in the pricked (punlctiert) domain R- z 0 • These functions converge continuously in this domain, towards the function 4> (z) =f(z) -f(z0 ). By Theorem 3 of § 103 either

4> (z) is identically zero, andf (z) is then a constant, or 4> (z) is different from zero and sof(z)=Ff(z.). 105. The main theorem of conformal representation (21). Let R be an arbitrary bounded domain in the z-plane, containing the point z = 0 and therefore also a circular area K defined by

Iz I < f' ...... (105'1) in its interior. No assumption is made as to the connectivity of R. Consider a family {f (z)} of functions which are regular in the circle (105'1). The family is assumed to be made up of the functionf(z) 0 and also all functionsf(z) which satisfy the following conditions: (a) f(O) =0, (b) analytic continuation of f(z) is possible along every path 'Y within R and the function/ (z) is always regular, (c) if -y' and-y" are two paths joining z = 0 to the points z' and z" respectively, and if .,.F (z') and '('F (z") are the values obtained at. z' and z" by continuingf(z) along these paths, then if z' =Fz" .,.F(z')=F.,.. F(z") ..... (105"2) (in particular, .,F (z) =F 0 provided z =F 0}, (d) with the above notation I.,F(z)l< 1. ...... (105'3)

=

106. It will first be proved that the family {f(z)} is compact(§ 100). Condition (d) shows that any sequence of functions of the family conta.ins

§§ 104-107]

67

THE MAIN THEOREM

a sub-sequence .it (z), / 2 (z), ... which satisfies the relation lim /,, (z) =fo (z) ...... (106"1)

,._..,

in

We have to show that fo (z) belongs to the family {f(.:)}. It is obvious that/0 (0) = 0, so that either condition (a) is fulfilled or f(z) = 0. To verify condition (b) we must show that if y' is a path within R joining z = 0 to a point z', and if for every point { of y', other than z', the analytic continuation of fo (z) gives a function y-Fo which exists, is regular and can be obtained as the limit of the analytic continuations .,.F,. ('> of the functions f,. (z), then all these conditions are satisfied at the point z' itself and in a certain neighbourhood of that point. It is easy to show this, for the functions yFn (z) are regular in a certain neighbourhood of z' and by condition (d) they form a normal family which converges to iFo (z) at all points of a certain portion of y' (§ 102). To prove condition (c), consider two paths y' andy" with distinct endpoints z' and z", and let Nz' and Nr· be non-overlapping neighbourhoods of z' and z" respectively. By Theorem 4, § 103, there are points z,.' in Nz' and z,." in Nz'' such that the equations . K.

m

y-Fn (z,.') = yFo (z'),

y"F,. (z,.") = y-Fo (z") ...... (106"2)

hold simultaneously, n being suitably chosen. By hypothesis the terms on the left in these equations are unequal. 'fhe required result, y-Fo (z') * y"F', (z"),

follows. Finally, it is obvious that/0 (z) satisfies condition (d). 107. Conf?ider now a particular function f(z) of the family and its analytic continuations yF(z} in R. Suppose that there is a number w0 , where w0 =he;8 , (O
f, (z) = e-iBf (z),

...... (107"3}

h-f,(z) ex (z) = i -lt.f. (z)'

...... (107"4}

{3(z)=Jcx(z),

...... (107"5)

({3(0}=+Jh),

JJI-{3(z) g(z)= 1-Jk{3(z)'

and proceed to investigate their properties.

...... (107"6)

68

THEOREMS OF CONFORMAL REPRESENTATION

(cHAP. V

The function f 1 (z) clearly belongs to the family {f(z) }, and neither the function nor its analytic continuations takes the value k. The function lX {z) is connected with _A (z) by a Mobius' transformation which transforms the unit-circle into itself. The conditions (b), (c), (d) of§ 105 are all satisfied bylX{z). The same is true of f3(z}. NeitherlX{z)nor its continuations take the value zero, so that f3(z) is regular on every path y. To verify condition (c) we notice that if analytic continuation of P(z} along paths y' and y" leads to coincident values at z' and z", the same must hold for lX (z), which would lead to the conclusion that z'=z".

Finally we see, from {107"6), that g (z) satisfies the conditions (b), (c), and (d), and further, since f3 (0) = Jk, the function g (z) vanishes at z = 0, so that it is a member of the family {f(z)}.

108. If the equations (107"3) to (107"6) are solved, f3 (z), lX {z), .h (z) bein.g successively determined as functions of g(z)=t,

...... (108"1)

it is found that /(z)=ef8t2 .Jk-(I +k~t. . ..... (I08"2) (I +k)-2 Jkt The expression on the right-hand side of this equation coincides with one of the functions investigated in§ 56. It satisfies all ihe conditions of Schwarz's Lemma (as can also be verified by direct calculation) and therefore, for every value of t in the domain 0 < It I < I, 2 Jit-(I +;> tl
...... (108·3)

From (108"I) to (108·3) it follows that •..... (I08"4) lf(z)l
109. !Jet z1 be a fixed point of K, z1 =F 0, and define a continuous functional J(f} (see§ 100), by means of the equation J(f) = 1/(zi) J. ••.... (109"1)

§§ 108-111]

69

THE MAIN THEOREM

Since the family if(z)} is compact,§ 100 shows that it has at least one member .fo (z) such that · lfo(zi)I ~ lf(zi)I ...... (109·2) as f(z) varies in the family. The domain R being bounded, the functionf(z) = .\z belongs to the family, if A is a positive number taken sufficiently small, and thus, from (109'2), l.fo (~)! ;?: A Iz1l > 0, which shows that.fo(z) is not a constant. Using the theorem of§ 108 we obtain at once : THEOREM 2. The family {f(z)} contains at least onefunctionfo (z) wkick, witk its analytic continuations, takes every value in tke circular area

lwl <1. 110. We now consider functionsf(z) which satisfy conditions (b), (c) and (d) but not necessarily condition (a) of § 105, and which have in addition the following property: (e) To every point Woof the unit-circle Iwl < 1 there corresponds a path y 0 , joining z = 0 and some point z 0 , such that, with the notation used above, 'I' F(zo)=wo. .. .... (110'1) 0 The last theorem proves the existence of such functions. Further, by the property (c), the number z0 in (110'1) is a single-valued function of w0 , defined throughout the interior, Iw I < 1, of the unit-circle. This function is the inverse function of Y0 F (z) = w in some neighbourhood of each point w0 , and it is therefore regular. Thus it is seen that the function, which we denote by rp (w), is single-valued, regular and analytic throughout the circle Iw I< 1 (~ 63). The equation z = rp (w) ...... (110'2) therefore gives a conformal transformation of the circular area I w I < 1 into a Riemann surface which covers the whole domain R. <

111. A second function g (z), satisfying (b), (c), (d) and (e), yields an inverse function ...... (111'1) Z=t/t(w) which has the same properties as the function rp (w) of (110'2). By means of (110'2) a correspondence is set up between the points w0 within the unit-circle and points z 0 , with associated paths y 0 • The paths ma.y be obtained, for instance, as the paths in the z-plane resulting from the transformation of straight lines joining 0 to w 0 • If we write w 0 =Yo G (z0),

70

THEOREMS OF CONFORMAL REPRESENTATION

[CHAP. V

then wo is a single-valued regular function of w 0 • In what has just been said w and w are interchangeable, and it follows from § 68 that the equation cp (w) =.; (w) gives a non-Euclidean motion of the unit-circle on itself.

112. The monodromy theorem (§ 63) shows that if the given domain R is simply-connected the inverse of the function (110"2) must be a regular single-valtMd function of z in R. The equation z = cp (w) then gives a one-one conformal transformation of the interior of the unit-circle on the domain R. It has already been shown(§ 64) that any simply-connected domain with more than one frontier-point can be transformed conformally into a bounded domain. Combining this with the results just obtained, we have THEOREM 3. The interior of any simply-connected d<Jmain R with m(ffe than one frontier-point can be represented on the interi(ff of the unit-circle by means of a one-one conformal tran._yormation. The representation is not unique. A git·en directed line-element in R can be made to C(ffrespond to an arbitrary line-element in I w I < 1. JVhen such a C(ffresporuknce is assigned the transf(ffmation is determined uniquely.

113. On the other hand, if the inverse of the function (110·2) is a single-valued function of z in R, then by § 63 the domain R is simp\yconnected. Thus if R is multiply-connected there must be at least two points w' and 1v" in Iw I < 1 such that cp (w') = cp (w"). Hence there is at least one non-Euclidean motion of the circle Iw I < 1 for which cp (w) is invariant (§ lli). The aggregate of such non-Euclidean motions forms a group, and cp (w) is an automorphic/unction. The points w of the circle I wl
If the domain R is multiply-connected, then the function

(110·2) is automorphic, and it is invariant for certain transf(ffmations

§§ 112-115]

71

AUTOMORPHIC FUNCl'IONS

wlwse aggregate forms an infinite group of noo-Euclidean translations ancllimit-rotatimzs.

114. In the case where R is doubly-connected it is easy to form the group of transformations. Suppose first that R is a simply-connected domain from which the single point z0 has been removed. Then a one-one conformal transformation can be found, to transform R into the pricked (punktiert) circular domain ...... (114'1) O
*

Now assume that the points equivalent to a point Wo. where Iw0 I < 1, are generated by a cyclic group of limit-rotations. A Mobius transformation can be found which transforms the circle Iw I < 1 into the half-plane ltu < 0, while at the same time the limit-rotation from which the group is generated becomes a Euclidean translation determined bythevector271'i. If now t = e", the domain R is represented on the pricked circular area. 0 < It I < I by a one-one conformal transformation. Then, by Rad6's result, R coincides with a simply-connected domain from which a single point has been removed.

115. Next let the frontier of R consist of two distinct continua 0 1 and C,, each containing more than one point. Regard 02 as the frontier of a simply-connected domain containing 0 1 • This domain can be conformally transformed into the interior of a circle. In this transformation R becomes a doubly-connected domain whose frontier consists of a conF

72

THEOREHS OF CONFORMAL REPRESENTATION

(CHAP. V

tinuum 0 1' and a circle 0,'. The simply-connected domain exterior to 0 1' and containing 0 1' in its interior is now transformed into the interior of the unit-circle. In this transformation 0 2' is transformed into an analytic curve. It can therefore be assumed, without loss of generality, that the doubly-connected domain R has as its frontier the unit-circle 0 1 and an analytic curve 0,. without double-points, surrounding the point z=O. By means of the transformation z = e", s. certain periodic curvilinear strip Ru 1n the u-plane is conformally transformed into the domain R. The set of points corresponding to a given point z0 is of the form U~;=U0 +2ikr

(k=O, ±1, ±2, ... ).

Thus the points equivalent to u 0 are generated by a cyclic group. Since R,. is simply-connected, a one-one conformal transformation u = cf> ( w) can be found which represents it on Iw I < 1. The points of Iw I < 1 which are equivalent to w 0 are obtained as the images of the points u~;. 'fhe corresponding group is certainly cyclic and is generated by repetition of a certain non-Euclidean translation, the possibility of a limitrotation being excluded by the result of§ 114. 116. l\ non-Euclidean translation is a transformation with two fixed points, A and B, on the circumference of the circle. Let w=lf!(t)

I ~t I ( w) gives a one-one conformal transformation of an annular region into the doubly-connected domain R. be a function transforming the interior of the strip

THEOREM 5. A doubly-connected dmnain R wltose frontier consists of two continua eaclt containing more titan one point can be 1·epresented, by a one-one conformal transformation, on an annular region whose frontier consists of two concentric circles.

117. Let R be a simply-connected domain on which the circular area Iw I < 1 is conformally represented. Two fixed points, Z1 and z 2 of R,

§§ 116-119]

DOUBLY-CONNECTED DOMAINS

73

correspond to two points w 1 and w1 which are uniquely determined for any p088ible conformal representation of the circle on R, and whose non-Euclidean distance D(wl> w 2 ) is always the same. This number may therefore be regarded as the l.lon-Euclidean distance of the points Zt and ~. and in this way a non-Euclidean metric is set up for R. It is also possible to set up a non-Euclidean metric for a multiplyconnected domain R, but the process is somewhat different, for here an infinite number of points w correspond to a single point z. Let y. be any rectifiabld curve in R. It is transformed into an infinite number of curves y"'l, y"", ... in thew-plane. All these curves can be transformed into one another by non-Euclidean motions, and thus they all have the same non-Euclidean length. This length we associate with the curve y •. In the multiply-connected dmnain R curves have length, but the rxm.ception of a distance between two points has no meaning.

118. Consider an annular region ...... (118'1) r1 < lwl
119. Nonnal families composed of functions which transform simple domains into circles. Let {T} be a family of simple simply-connected domains in thew-plane, each domain containing the point w = 0 while none of them contains the point w = oo. A family {f (z)} is formed of analytic functions by means of which the domains Tare transformed into the circle I z I < r. We assume that for all these functions f(O) =0. Results already obtained show that the functions of the family {f (z)} are uniformly bounded in the circle Iz I < r9, where 9 is any positive number less than unity, and that in consequence the family is normal in the circle Iz I < r if one or other of the following conditions is satisfied: (ot) for each domain T the distance a of the point w = 0 from the frontier is not greater than a fixed number M(§80and § 101), (/3) all the functions/ (z) satisfy the relation If' (0) I ~ .3-I (§ 81).

74:

THEOREMS OF CONFORMAL REPRESENTATION

[CHAP. V

=

The theorem of§ 104 shows that the addition of the functionf(z) 0 makes these families compact. In case (/3) there is no need to add this function if the condition If (0) I ~ Mis replaced by the stricter condition 0< m ~If' (O)I ~.M.

120. The kernel of a sequence of domains. Take a family {T} of domains, and, to fix the ideas, suppose it satisfies condition (at) of§ !l9. In it choose a sequence T1 , T2 , ••• of domains such that the correspondingfunctionsf1 (z),f2 (z), ... converge regularly to a functionf.(z) in lzl
the limit by z 0 • Since the sequence/1 (z), } 2 (z), ... is continuously convergent fo (z0 ) = lim f.,:lr (z,.· 11 ). k-oc

w,.· 11 = w. belongs to the closed set A.,. Reference k-oo to the construction of the circle Iz I = n shows that this is impossible. Also the point lim

121. We now assume that T* is a domain which contains the point w = 0 and has the property which has just been proved to hold for T 0 , namely, any continuum A 10 which lies in T"' and contains the

point w = 0 lies in every domain T,. from some n onwards. We shall

§§ 120-122]

75

THE KERNEL

then show, first that .f. (z) is not a constant, and, secondly, that T* is contained in T 0 • Let w 0 be an arbitrary point ofT*. Consider a domain A., containing the points w0 and w = 0 and such that both A. and its frontier are contained in T*. By hypothesis, A., is covered by T,. whenever n is sufficiently large; for these values of n, the inverse functions .p,. (w) of f,. (z) are all defined in A •. These functions are all less than rinabsolute value and so form a normal family. The sequence {q,,. (w)} yields a sub-sequence {q,,..t (w)} of functions which are all defined in A. and converge continuously to a function Y, ( w). By§ 104, the function .p (w) is either a constant, and, in that case, since Y, (0) = 0, it vanishes everywhere, or the equation z = Y, (w) gives a conformal transformation of Aw into a domain lying within the domain lzl
••••••

(121"1)

it follows thatzk-z0 , and, sincez0 is a point within Izl < r,f,.,.(zk)-fo(zo) also holds. But, by (121"1), from which it follows that Wo=fo (zo).

Now Wo was an arbitrary point ofT*; thus fo (z) is not constant, and, further, the equation w =fo (z) gives a conformal transformation of Iz I< r into a domain To which must have T* as a sub-domain. The domain T 0 , obtained from Iz I <,.by means of the traniformation w =fo (z), whet·efo (z) istheboundaryfunction, hasthefollowingproperty: it is the largest domain suck that every rontinuum, rontaining the point w = 0 and contained in the domain, is coured by all the domains T,;from some value of n onwards. It has incidentally been shown that, If the sequence}; (z), f 1 (z), ... tends to the function which vanishes everywhere, there is no neighbourhood of w = 0 which is covered by all the domains T,. from some n onwards.

122. Let an arbitrary sequence 1'., T 2 , ••• of simply-connected domains each containing w = 0 be given. We may suppose, for example, that the domains satisfy condition (at) of§ 119. We associate with the sequence a certain set of points K, called the kernel of the sequence. If tke seq'U8nCe {T ,.} is suck tkat no cit·cle witk w = 0 as cent1·e is covered by all the domains T,. when n is sufficiently large, then its kernel ronsists

76

THEOREMS OF CONFORMAL REPRESENTATION

[cHAP. V

of the single point w = 0. In all other cases the kerrwi K of the sequence is the largest domain having the property that every continuum, containing w = 0 and contained inK, lies witkin every domain T,.for sufficiently large ·values of n (2'-l). The kernel of a sequence of domains is uniquely deter,uined. Suppose that w0 , a point of the w-plane, bas a neighbourhood which is covered by all the domains T,. from some n onwards. Let Kw0 be the largest circle with centre W 0 and such that all smaller concentric circles are covered by T,. for sufficiently large values of n. Now let w 0 vary within the w-plane. The sum of the interiors of all the circles Kw0 is either the null-set or an open set which may be regarded as a sum of non-overlapping domains. If one of these domains contains w = 0, then that domain is the kernel K. Otherwise the kernel is the single point w = 0.

123. A sequence {T,.} ofdomains is said to converge to its kernel K ij every sub-sequence T"'t., T,..., ... has thesamekernelas theoriginalsequence {T,.}. Since the domains T,. are assumed to be simply-connected, there are, by§ 112, functions.ft (z), / 2 (z), ... which give conformal transformations of 1;, T2, ... on the circle Iz I < 1 and satisfy

f .. {0) = 0, f ..' {0) > 0.

. ..... {123'1)

Suppose that the functions f,. (z) converge regularly to a function fo (z) in this circle. Then, if/,.1 (z)J,.. (z), ... is any sub-sequence, limf,.t (z) =fo (z).

.. .... (123'2)

k-

The function .fo (z) transforms the circle Iz I < 1 into the kernel.K' of the sequence {T,.k}. From this it follows that lC = K, that is, the sequence {T,.} converges to its kernel. The converse of this result also holds: the convergence of a sequence of domains to its kernel implies the regular convergence of the sequence {f,. (z)} in the circle Iz I < 1. For the functions/,. (z) form a normal family. If the sequence {f,. (z)} were not convergent it would be possible to find two sub-sequences, ...... (123•3) f,.l (z), f ... (z), ... ' fm 1 (z), f,... (z), ... ,

...... (123•4)

converging regularly in Iz I < 1 to two distinct functions cp (z) and 1{1 (z) respectively. But this gives a contradiction, for both functions cp (z) and 1{1 (z) transform the kernel of the sequence {T,.} into the circle Iz I < 1, and also, from (123'1),

cp (0) = 0, cp' (0) > 0;

"'{0) = 0, 1{1' (0) > 0 ....... (123•5)

§§ 123-125]

77

THE KERNEL

The uniqueness theorem for conformal representation (§ 68) shows that q, (z) 1/J (z).

=

124. Examples. (a) Let the w-plaM be cut along the negative real axis from the point -

a:;

to the point -

!n .

Denote by In (z) the function which gives

a conformal transformation of the circular area Iz I < 1 into this domain, conditions (123·1). Then 1.. (z) converges continuously to zero, for the kernel of the sequence of domains is a single point. Actually, we find

In (z) satisfying the

4z f,.(z)= n(l-z)~·

...... (12-U)

(b) Let the w-plane be cut along those parts of the imaginary axis which lie above the point ifn and below the point - ifn respectively. Consider the function fn (z) which transforms the half-plane .li (z) > 0 into this domain and satisfies the conditions

f,. ( I ) = 1,

I,.' (I ) > 0.

. ..... (124•2)

The general theory shows thatf,. (.z) converges continuously to z in the half-plane. This can be verified at once; for calculation shows that f,. (z) =

1) + ..jf+n2( 1). 21(z + z ~ z-:z

. ..... {124·3)

(c) Let the w-plane be cut along an arc of the uni~-circle joining the points e"•tn and e-i"1", and of length 271" (1- 1/n). A simply-connected domain is thus formed, with the point w = a:; as an interior point. Let the interior of the unit-circle, Iz I < 1, be transformed into this domain by means of a function/,. (z) which satisfies the conditions (123'1). The problem is analogous to (b), and it is found that _ 1 - z cos 1rj2n /,n ( Z) -Z •

cos 1rj"l.•z- z

...... (124'4)

As we might anticipate, each of these functions has a pole within However, they form a normal family. 'fhis is most readily seen if we observe that our domains can all be transformed into those of example (b) by the use of afi.red Mobius h<~.nsformation.

Iz I < 1.

125. Simultaneous conformal transformation of domains lying each within another. The solution of the following problem is important for the theorems by which arbitrary analytic functions are represented by single-valued

78

THEOREMS OF CONFORMAL REPRESENTATION

[CHAP. V

functions (cf. Chap. vm). Consider an infinite sequence of complex plane.'!, z,, .::2 , ••• , and let each plane contain two simply-connected domains, one within the other and both containing the origin.

.r,.-plane

Fig. 26

z,.+rplane

The two domains in the z,.-plane are denoted by 0~") and o~n-l) respectively. Take a w-plane and attempt to find in it a sequence 0 11,021, •.• of domains each lying within the next, all containing the point w = o, and such that the function cp<"l (z,.), with

cp(") (0) = 0,

cp'(ft) (0}

> 0,

which transforms o<:> into 0"1, also transforms o~-l) into O!R-l). The sequence {01"'} is to be such that this holds for all values of n. Suppose that the domain 0~"> is transformed conformally into a domain r~"> in the same plane, by means of a function if! (z,.) with if! (0) = 0 and 1/r' (O) > 0, and that the same function transforms 0~"- 1> into the domain r~- 1>. Then the two original domains may be replaced by their images without materially affecting the problem just stated. From this it follows that there is no loss of generality involved in the assumption that the domains o are the circles n

lz,.l
(n=1,2, ... ).

. ..... (125"1)

Let the relation which transforms the circle I z,. I < r,. into the domain o be ••+1 z,.+, = ...... (125·2) 1 (z,.),

.f.:t

.r<:!

where it is assumed that If ,.,. is fixed, 1 (0) = 0 and/~"~ 1 (0) > 0. then /'~"! 1 (0} is proportional tor,.+., andr.. +, can be determined so as t.o ensure that

f("l (0) = 0 n+l '

f,(n) (0) = n+l

1.

...... (12.5'3)

We take r 1 = 1, and require that ~onditions (125'3) shall hold for all values of n. '11hen the numbers r,. are all uniquely determined.

§§ 126-128]

79

SIMULTANEOUS TRANSFORMATIONS

126. The system of equations (n) (z) =f(n+l) (f(n) (z )) f n+2 n n+2 n+l " ' ( ) -J
...... (126"1) ...... (126"2)

defines new functions f !:'~k (z,.) which can all be calculated in succession if the functionsf!:'! 1 (z,.) are known for all values of n. From (125"3) it is seen that ...... (126"3) f ~n!k (0) = 0, f'~;!k {0) = 1. Now observe that the relation w =f~~k (z,.)

...... (126"4)

transforms the circle Iz,. I < r,. conformally into a simple domain of the w-plane. By(126"3), the functionsf~".!.k(z,.),(k==1, 2, ... , n==1, 2, ... ), satisfy condition (/3) of§ 119, so that they form a normal/amity.

127. If n is given, a sequence k~"1, k~n>, ... of natural numbers can be found such that the functions (n)

fn+k(n) p

(z,.) (p=1, 2, ... )

...... (127•1)

form a sequence which converges continuously in the circle Iz,.l < r,.. Then, by means of the diagonal process, a choice of the numbers k'"l(z,.)

(n=1,2, ... ), ...... (127"2)

P-+"'

within the domain lz,.l
128. The set of particular domains 01"1, which has just been found, forms a. figure a.bont which some further observations may be made. From (125"2)

80

THEOREMS OF CONFORMAL REPRESENTATION

[CHAP. V

using Schwarz's Lemma., it is seen that, if r,.. 1 ~ r,., then If'~"! 1 ( 0) I < I. Thus (125"3) shows that r,. < r,.+I and either lim r,. =

...... (12!:1"1)

OC),

or there is a finite number R such that lim r,. = R.

. ..... (128"2)

These two cases must be considered separately.

129. Denote by a,. the distance of the boundary of Cl"l from the point w = 0. The function ¢1"l (z,.) transforms the circle domain, and, by (82"4),

Iz,.l < r,. into a simple

a,.> r,./4. . ..... {129·1) Thus, if (128·1) holds, the sequence ~. a..z, ... tends to infinity, so that the sequence of domains Cl"l covers the whole w-plane.

130. Now assume that (128·2) holds. The construction shows that If~!~~ (z,.) I < r,..~: < R, so that, on proceeding to the limit, I¢1"1 (z,.) I < R for all values of n. 'l'hu!> the domain;; C(I), 0 2l, ... lying each within the next, are all contained in the circle Iw I < R, and the same is true of the set D formed by all points belonging to at least one of them. This set D is itself a simply-connected domain. Let t = 1/1 (w) ...... (130"1) be the relation which transforms D into the circular area It I 0. Schwarz's Lemma. shows that 1/1' (0) ~ I.

...... (130"2)

On the other hand, the functions t=>¥,.(z,.)=l/!(¢1"l(z,.))

(n=1,2, ... )

...... (130"3)

give conformal transformations of the circles Iz,. I < r,. into domains which lie within It I < R. By Schwarz's Lemma >¥,.' (0)

~ ,.,. R .

. ..... (130·4)

Since ¢''"1(0) = 1, (130·3) shows that >¥,.' (0) = "'' (0),

...... (130"5)

so that .p' (0) ~ Rfr,.. From (128"2) it follows that .p' (0) ~ 1, and in (130"2) the sign of equality holds. But this can only be the case if D coincides with the circular area I w I < R, i.e. if the domains Cl"l tend unifarmly to the domain I I < R.

w

CHAPTER VI

TRANSFORMATION OF THE FRONTIER 131. An inequality due to Lindel8f. Let R be an arbitrary domain in the z-plane, containing the point z0 • The domain R may be multiply-connected. Suppose that there is an arc of the circle Iz- z 0 I = r, subtending an angle 211"

ot>-

n at the centre z0 and lying outside the domain R. Here n is a positive integer. By means of rotations about z0 through angles

... ' the domain R gives rise to new domains R1, R2, ... , R,.-1respectively. The common part RR1R2 ... R,._ 1 of all the domains is an open set which contains the point zo but which has no point of the circle Fig. 27 Iz-zo I=r as an interior point or as a frontier-point~ Among the domains whose sum makes up this common part there is one, R 0 , which contains the point z0 • The frontier y of Ro lies in the circular area I z- z0 I < r, and evePy point of y is a frontier-point of at least one of the domains R, R 11 R,, ... , R,._1.

132. Let/(z) be a bounded analytic function defined in R, and suppose

1/(z) I <M ...... (132'1) in R. Assume fuither that if ~is an arbitrary frontier-point of R, lying within the domain I z- z0I < r, and if z1 , z2, ... is any sequence of points of R tending to ~. then ...... (132'2) lim 1/(z,.) I~ m, (m < M), ,.._"' where m is independent of the particular sequence {z,.} and of the choice of'· Consider the functions 2k1ri

/~:(z)=/(zo +e.,. (z-zo)), {k= I, 2, ... , (n-1)) ....... (132·3)

82

TRANSFORMATION OF THE FRONTIER

[CHAP. VI

The function f~< (z) is analytic in the domain RJ<. It follows that the function F(z)=/(z)/I(z) ... fn-dz) ...... (132•4) is analytic in R 0 • Thus, if {z,.} is a sequence of points of Ro tending to a frontier-point Cas limit, our hypotheses show that ...... (132•5)

and hence, since (132·5) holds for all frontier-points of R 0 ,

I F (zo) I ~ M" ; . By (132·3) and (132·4),

F(z0 )

=

(/(z0 ))n,

so that, finally,

1/(z.) I~

! j)f (';)".

. ..... (132•6)

133. The following theorem follows almost immediately from Lindelof's inequality (132·6): THEOREM. If /(z) is bounded and analytic in an arbitrary simplyconnected domain R wkose frontier contains mare than one point, and if the1·e is a frontier-point t with a neighbour/wad N~ such that I (z) is continuou..~ at tko1<e points and frontier-points of R which lie within N~, and takes the constant value ex at tkose/ro-ntier-points, then /(z) =ex. First suppose that Cis a limiting point of points which do not belong to R or to its frontier. Then, if z0 is any point of R sufficiently near C, we can construct a circle I z- z0 I ~ r lying entirely within N~ and having points which do not belong to R or its frontier on its circumference. On applying Lindelof's result to the function (/ (z)- ex) we obtain/ (z0) =ex, and the required result follows at once. If C does not fulfil the above condition, we consider the function F (t) =I (C + f-) - ex. Comparison with a similar substitution used in~ 65 shows that this function is defined in domains Rt which have the property that t = 0 is a limiting point of points exterior to Rt. The argument used above therefore serves for this case.

134. Lemma 1, on representation of the frontier. Let a bounded simply-connected domain RtD be given in the w-plane and suppose that the function w=f(z) .•.... (134:1)

§§ 133-135]

JORDAN DOMAINS

83

represents it conformally on the domain R. in the z-plane. We assume that the frontier of Rz is a closed Jordan curve c, and that the origins Ow and 0. lie within the respective domains Rw and R., and are corresponding points in the transformation. Let Yw be a cut into the interior of the domain Rw. By this is meant a Jordan curve joining an interior point of R"' to a point w of the frontier, and such that any infinite sequence of points of yfiJ either has at least one limiting point within RfiJ or else has w as its only limiting point. We shall prove that: The cut '"' is tran.iformed into a curve y. which is a cut into the interior of the dAJmain R. bounded by the Jordan cur·ve c.

135. To prove this lemma we use Jordan's Theorem, which states that c divides the plane into two and only two domains, and also the simplest

properties of the domain R •. 'rhese are, that every point of the Jordan curve c is a limiting point of points exterior toR., and that every point of ccan be joined to any interior point by means of a cut into the interior. Suppose, if possible, that sequences of points of Yz can be found, which have two distinct points J-I and N of c as limiting points. It will be shown that this leads to a contradiction. The two points .DI and N divide the Jordan curve c into two arcs, c1 and c2 • Imagine the curve c described in the positive sense and choose, in order, four points A, B, C and D, of which the first two lie on c1 and the others on c2 • Let the four points be joined to· 0, by four cuts without other intersections than that at 0.. 'l'hese cuts divide R. into four sub-domains. By hypo- C thesis, we can find on Yw a sequence of points which converges to w, but such that the corresponding points in the z-plane lie alternately in the domain U=DA and in the domain O.BC. An arc of y, joining two successive points of the transformed sequence must cross one or other of the Fig. 28 domains O.AB and o.cD. To fix the ideas suppose that there is an infinite number of these arcs crossing O.AB, and let the parts of the arcs which lie in this domain and have their end points on O.A (1) (2) • (1) (21 and O.B bfl ~, , a= , .... The correspondmg arcs aw, a,,,, ... on Yw are

84:

TRANSFORMATION OF THE FRONTIER

[CHAP. VI

!

all distinct and tend uniformly to 111. The arcs ~1 1, ~1: 1, .•• tend uniformly to the arc AB of c. Consider the domain O;AB. On AB choose a pointE whose distance from the arcs BO. and O.A may be denoted by 2"7 > 0. In the domain O.AB take a point z0 whose distance from E is less than "7, and with centre z0 draw a circle K of radius "7· The pointE

lies within K and is a limiting point of points exterior to R., so that there is an arc K 1 of K lying outside R •. Let ex be the angle subtended at z0 by this arc, and choose a positive integer n such that ex> 2rfn. Join 0. to z0 by means of an arc y lying within O.AB. If k is sufficiently large, say k> k 0 , ~~:>has no points in common with :y, so that the domain O.F~ 1 :1 G (Fig. 29) is a sub-domain of O.AB containing z 0 but not containing any point of K 1 either in its interior or on its frontier. Let ~ be an arbitrary positive number. Denote by hi the (finite) diameter of Rw and choose k greater than ko and also such that the arc ~':]lies entirely within the circle I W-1111
136. Lemma ~. Consider two cuts rw' and rw" into the interior of the domain Rw. We suppose that they join the point Ow to the points w' and w" of the

§§ 136, 137]

85

JORDAN DOMAINS

frontier and have only the point 0., in common. The result just proved shows that y.,' and y.," are the images of two cuts into the interior of R•. These cuts y,' and y." join the point o. to the points rand(;' of the Jordan curve c. We now investigate necessary and sufficU!nt conditions that ,, and r' slwuld coincide. The curve (yz' +yz") divides R. into two domains R 1 and R'!l. If(! and r' coincide, then, since cis a Jordan curve, one of the domains, say R, is such that all its frontier-points are points of "tz' or of y." or of both. This domain corresponds to a domain, R':] say, which may have points of the frontier of R., as frontier-points, but such a frontier-point (j) of R~> cannot be at a positive distance from the curve (y.,' + y.,''). For suppose that w is such a point. 'rhere is a neighbourhood N., of w such that if w,, w2 , ••• is a sequence of points converging to a point of the frontier of R::! within N.,, then the inverse of the functionf(z), say q, (w), takes values converging tor. By§ 133, cp (w) is constant; but this is impossible. On the other hand, if R':] has the property in question, so that, in particular, w' = w", then r = '"· This is readily proved if the above argument is applied to the transformation w = f (z) and the domain

!,

R'~,. z

137. Transformation of one Jordan domain into another (23). We now assume that the frontiers of both the domains R., and Rz are .Jordan curves, which may be denoted by c., and C2 • Let w be an arbitrary point of c., and let w 1 , w 2 , ••• denote an arbitrary sequence of points of R., tending to w. The corresponding points of R. are z 1 , z;., .... Since c., is a Jordan curve the points w,, w2 , ••• may be joined, each to its successor, by a sequence of arcs whose aggregate forms a cut y 111 from w into the interior of R.,. By§ 135, y 111 is the image of a cut "t• from some point' of Cz into the interior of R •. Also, y. contains the sequence of points z1 , z2o ... ,so that this sequence must converge to '· The convergence of the sequence {z,.} to 'is simply a consequence of the convergence of {w,.} to w. It follows that the point' to which {C,.} converges depends only on w, not on the choice of the sequence {w,.}. A point Ccorresponds uniquely to each point w, and, since R. and R., may be interchanged without modification of the reasoning, a point w of c., corresponds uniquely to each point Cof c•. It is seen at once that two distinct points w1 and w 2 of c.., have two distinct corresponding points C, and C2 of c.. (This also follows from

86

TRANSFORMATION OF THE FRONTIER

[cHAP.

VI

§ 136.) Further, the transformation of one frontier into the other is continuous. For let ~, ~ •... be points of Cw converging to w0 , and let ~~> '2> ... be the corresponding points of c•. If lc is a positive integer, a point wk can be found in Rw such that the relations

are satisfied simultaneously by wk and its image z,.. But the points w1 , w2 , ••• converge to w 0 • It follows that Z1o ~ •... , and therefore also the points ~I>~ •••• , tend to a point~Finally, let~. w2 and w3 be three points of cw which are passed through in this order when the curve is described in the positive sense. Suppose that threecut<>intotheinteriorof Rw join Ow to the respective points w~> "'s and w 3 , and that these cuts do not intersect one another except at Ow· We consider the corresponding figure in the z-plane and observe that the transformation is conformal at Ow· 'fhen it is dear that the points~~. ~2 and ' 3 occur on c. in this order when the curve is described in the positive sense. These results may be summarized as follows: THEOREM. If one JtYrdan domain is traniftYrmed coriftYrmally into another, then the transformation is one-one and continuo1ts in the closed domain, and the tu'o frontiers are described in the same sense by a moving point on one and the ctYrresponding point on the other.

138. A slight generalization of this theorem is easily made. Let Rw be a domain, and Cw a Jordan curve (with or without its end points), and suppose that the following conditions are satisfied: (a) every point w of c,., is a frontier-point of Rw, (b) every point w of Cw can be joined to any interior point Ow by a cut into the interior of Rw, (c) every .Jordan domain whose frontier consists of a portion w1w2 of Cw and two cuts into the interior, O,w1 and O'"w2 , lies entirely within Rw. Then Rw is said to contain a/ree JtYrdan curt•e. Suppose that w1 =!= w2 • Then § 136 shows that the cuts Ow~ and O,w 2 are the images of two cuts 0.'1 and O.Co, where ' 1=1= ( 2 • The transformation w = f (z) then transforms the interior of the Jordan domain 0.,1 { 2 0. into the interior of the Jordan domain Oww1w2 0 10 • By§ 137, any arc of the free Jordan curve c.,., and hence the whole curve, is a one-one continuous image of an arc of the frontier c. of R •. Just as in § 136 it may be shown that c" .is not the image of the whole frontier of R. except when c, is the whole frontier of Rw, i.e. when Ru- is a Jordan domain.

§§ 138, 139]

87

INVERSION

139. Inversion with respect to an anal:vtfc curve. A real analytic curve. in the :cy-plane is given either by an equation F (x, y) = 0

...... (139"1)

or in parametric form by two equations

X=
. ..... (139"2)

Here it is supposed that if the functions F (x, y), cfJ (t) and 1/1 (t) are expanded in power-series in the neighbourhood of any point of the curve the coefficients in the expansions are real. Further, we suppose that at each such point at least one of the partial derivatives F., and F, and at least one of the derivatives cfJ' (t) and .p' (t) differs from zero. Consider the function

z=f(t)=cfJ(t)+i.P(t)

.•.... (139"3)

as an analytic function of the complex variable t in the neighbourhood of a real point t 0 • By hypothesisj'(t0 ) =1= 0, so that a circle It- t 0 I < r can be chosen which is conformally represented, by means of (139"3), on a certain neighbourhood of the point z 0 =f (t0). The circle has a diameter lying on the real axis, and this diameter is transformed into part of the curve (139"2). The following definition is due to Schwarz: Two points z and z* are said to be inverse points with respect to the curve (139"2), if they are the images of two points t and t of the circle I t - t0 I < r, where t and tare conjugate complex numbers. '.rhus, to (139"3) we can add the equation z• = cfJ (t) + i.p (i). . ..... (139'4) Let the complex number conjugate to z* be denoted by w. Since cfJ (t) and -.p (t) can be expanded in the form of power-series with real coefficients, cfJ (t) and (t), 1/J (t) and if (t) form pairs of conjugate numbers. Hence ...... (139'5) w = z• =cfl (t)-i.f (t); (139·3) and (139"5) give z+w z-w . ..... (139'6) x=cfJ(t)=-2-, y=.P(t)= 2i . Substitution of these values in (139'1) gives

F(z+w ~)=o ...... (139'7) 2 ' 2i ' and from this relation w can be calculated as an analytic function of z. The relation (139'7) shows, in particular, that the operation of inversion depends only on the form of the curve (139"1), and is independent of the choice of the parameter t in (139'2) (24). G

88

TRANSFORMATION OF THE FRONTIER

(CHAP. VI

140. The equation (139"7) is especially convenient when an algebraic curve is given. If (139"1) represents a straight line or a circle, then the corresponding formulae (139"7) are exactly the inversion formulae given in Chapter 1. In the case of the ellipse ...... (140"1) (139"7) becomes (a2 -b1 ) (w2 + z2 ) -2 (a2 + b2 )zw+ 4aW= 0 ....... (140"2)

If this is solved for w a two-valued function of z is obtained. This function is regular throughout the z-plane except at the foci of the ellipse (140"1 ). The corresponding transformation (a2 -b2 )(z*2 + zt)- 2 (a 2 + b2) z*z + 4a2b2 = 0 ...... (140"3)

can only be regarded as an inversion with respect to the ellipse when the points tranRformed lie within the ellipse confocal to (140"1) and passing through the point

141. The inversion principle. Let R., be a simply-connected domain in the v-plane, containing a segment A 1 B 1 of the real axis, and symmetrical with respect to this aXIs. Suppose that the relation "'=.; (t) gives a conformal transform&-

v-pl&ne

Fig. 30

t-pla.ne

tion of R., into the interior of the unit-circle in the t-plane, the transformation being such that the origin t = 0 corresponds to some point Ov of A 1 Bh and that .;· (0) > 0. The function if! (t) i'l uniquely determined by these conditions (§ 112). Further, if the two figures are inverted with respect to A 1 B 1 and A 2 B 2 respectively; they are transformed into themselves. From this it follows that -;jr (l) =.; (t), ...... (141"1) where

~

and tare the numbers conjugate to if! and t.

§§ 140-143]

INVl!;.RBION PRINCIPLE

89

The relation (141"1) shows that .p (t) may be written in the form of a power-series with real coefficients, so that the segments A 1B 1 and A 1 B 2 correspond to one another. Hence R~' and Rc' are corresponding domains.

142. To obtain a conformal transformation of R~' into the interior of a unit-circle lzl < 1, we may first use the relation v= 1/f (t) to transform R.,' into R/ and then a relation t = q, (z) to transform R/ into Iz I < 1. The function q, (z) is already known (§54); it transforms the circular area. It I < 1 into the z-pla.ne cut along an a.rc of Iz I = 1. Two points, z1 and Zs, which are inverse points with respect to Iz I = 1, correspond to two points which are symmetrical with respect to the real axis in the t·plane. The transformation v=l/f(f/l(z)) therefore transforms lzl <1 into Rv' in such a way that the points z 1 and z 1 correspond to points v1 and v2 which are symmetrically placed with respect to A 1 B 1 • The segment A 1 B 1 is transformed into an arc of the circle Iz I = 1, and the function 1/1 (f/l (z)) is analytic on this arc. 143. Let Cw be a regular analytic curve, and suppose that the relation w = x (v) ...... (143"1) tra.nsforms it into a segment of the real axis in the v-plane. Let a domain R 11,' in thew-plane, having an arc of c.., as a free Jordan curve, be trans-

("

w

w-plane

Fig. 31

v-pla.ne

formed by (143"1) into a domain R.'. Let the domain R,/, which is obtained by inversion of R 11,' with respect to c"', be transformed conformally, by means of (143"1), into the inverse R.," of R.,'. In order to obtain a conformal transformation of Rw' into the circle Iz I < 1, it is only necessary to substitute the function v =- •/t (q, (z)) of§ 142 in equation (143"1). We obtain the transformation ...... (143"2) w=f(z),

90

TRANSFORMATION OF THE FRONTIER

(CHAP. VI

which has the following properties: the are AB of c10 , which is part of the frontier of R 10', is the image cf an are of Izl = 1 and on this arcf(z) is an analytic function; if z1 and z2 are inverse points with respect to Iz I = 1, the corresponding points w1 and W2 lie one in R 10' and the other in R 10", and w, and w. are inverse points with re~pect to c10 •

144. Let Rv:' be the domain just considered, and let Rv' be a domain lying within some circle and having an arc A 3 B 3 of the circle as part of

t!)'

its frontier. Suppose that the relation w=P(v) ----gives a conformal transformation of Rv' into R 10' /"" ", in which the circular arc A 3 B 3 corresponds to / B3 the arc AB of C10 • Then F(v) has, on A 3 B 3 , / exactly the same properties as were found to \ hold for f(z) in§ 143. For F(v) is obtained by \ R'v elimination of z between the equatiom w =f (z) ' .. and v = g (z), the latter being a relation bv which A3 Rv' is transformed into Iz I < 1. Fig. 32 Now let R~ be an arbitrary simply-connected domain in thew-plane, having an arc MN of an analytic curve c10 as a free Jordan curve. Let w=F(v) ...... (144"1) transform R~ into the circular area IvI < 1. N In general it is not possible to invert the U'hole domain R~ with respect to c10 , so that the method of§ 143 cannot be used. We know, however, that the arc MN of c10 is transformed into an arc of the ~ircle IvI = 1, and that this M is a one-one continuous transformation(§138). Fig. 33 Now let w be an arbitrary point of :J.IN. We can find a sub-domain R 10' of R~, which can be inverted with respect to c10 and which has, as part of its frontier, an arc AB of c10 , where AB has was an interior point and is itself part of :J.:IN. The relation (144"1) transforms R 10' into a domain R v' which has all the properties mentioned at the'beginning of this paragraph. Thus F(t•) is analytic at all points of the arc of Ivi= 1 which corrc · sponds to c10 • Further, every interior point of this arc has a neighbourhood such that a pair of inverse points t•1 aud v2 , both lying in the neighbourhood, are transformed into a pair of inverse points with respect to c..,. 'rhese facts constitute the famous Principle of Symmetry, or Inversion Prin('iple, associated with tltc uamc of Schwarz (25).

§§ 144-146]

91

TRANSFORMATION OF CORNERS

145. Let R be an arbitrary simply-connected domain whose frontier consists of more than one point. Suppose that the function ...... (145·1)

w =f(z)

gives a conformal transformation of the circle Iz I < 1 into the domain R. As in Chapter rr, we now regard the circle as a non-Euclidean plane. The non-Euclidean straight lines in Iz I < 1 are such that if the circular area is inverted with respect to one of them it is transformed into itself. An analytic curve in the w-plane which corresponds to one of these nonEuclidean straight lines divides R into two sub-domains R' and R", and these are interchanged when they are inverted with respect to the curve. Thus all these curves may be regarded as lines of symmetry in R.

146. Transformation of corners,. Let RIIJ be a domain whose frontier contains the free Jordan curve MN, and let A be an interior point of MN at which the portions AM and AN of the curve both possess tangents, which we denote by AP and A Q respectively. Then RVJ has a corner at A, and this is measured by the angle ex between the .two directions AP and A. Q. If RVJ is transformed into the circular area I z I < 1, the point A corresponds to a certain point A. 1 of the circumference Iz I = 1. It will now be proved that any curve in RIIJ which ends at A. and has a tangent at A. making an angle ()ex with A.Q, (0 < () < 1), is the image of a curve in the z-plane, this curve having at A 1 a tangent which ·makes an angle 61f' with the circular arc A 1 N1. It is easy to deal with the case where AM and AN a~e both straight lines. For ~ circular sector SIIJ can be found, having its straight sides on

p

M

A Fig. 34

Fig. 35

AMand AN a.nd lying entirely within R •. The function z = .P (w), which transforms R. into Iz I < 1, transforms 8 111 into a. sub-domain Bz of the unitcircle, a.nd this sub-domain has A 1 as a frontier-point. It is only necessary to show that angles a.t A. a.re transformed proportionally into angles at A.1.

92

TRANSFORMATION OF THE FRONTIER

(CHAP. VI

.. the transformation u=(w-w represents

Now if A is the point w 01 0) ; 8tD on a semi-circle 8,., and 8,. can then be transformed into 8.. But the ratio of the angles in question is clearly preserved in the transformation of 8tD into 8,., and the second transformation, 8. into 8,, is analytic (§ 143) and therefore conformaL

147. Our proof of the general theorem depends upon the following lemma: Let the domain R contain the domain 8, but suppose t.hat R and 8 have the free Jorda~ curve AOB as part of their respective frontiers. Let R and 8 sepa.T&tely be transformed into the same circular area. K in such a way that in both cases the ends A and B of the Jordan curve

B

Fig. 36

AOB are transformed into two fixed points A1 and B 1 of the circum· ference of K. Now let 1 be a circular arc within K, joining A 1 and Bu and let 1a and 1s be its images. It will be shown that 1 8 lies within the domain whose frontier consists of AOB and 'Ya· If, by a conformal transformation, the circle K is transformed into a semi-circle, the arc A 1 0 1 B1 being made to correspond to the diameter A 2 B 2 of the semi-circle, then the curve 1 is transformed into a circular arc passing through A, a~d B 2 • We may assume that A 2 B 2 lies on the real axis. From this it is seen that a transformation of Rintoasemi-circle, with correspondence Fig. 37 between the Jordan curve AOB And the diameter, transforms 1a into a circular arc 1a' passing through A 2 and Bs. The same transformation represents 8 on a. domain S' lying within the semi-circle, while 1s is transformed into a curve 1s' within the semi-circleandjoiningA, and B,. We have only to show that 1; lies between 1a' and AtBs. Now 'Ys' may be regarded as the curve obtained from 'YB' by means of the transformation which represents the semi-circle conformally on 8'

§§ 147-149]

TRANSFORMATION OF CORNERS

93

and transforms the diameter A 2 B 2 into itself. The inversion principle shows that the function which sets up this transformation is analytic throughout the circle whose diameter is A 2 B 2 • The function takes real · values on A 2 B 2 • It is now seen, as a direct consequence of the theorem of§ 88, that -y; lies between -yl and A 2 B 2 • 148. We are now in a position to prove the theorem of § 146 in the case where R has a corner A, one of whose arms, say AN, is a straight line. As before, let ex denote the measure of the corner at A. Now construct two new domains, R' and R", as follows: K is a domain containing R. · Its frontier contains AN and also a segment AM' which makes an angle (ex+~> with AN. The domain R" is contained in R. Its frontier contains AN and also a segment AM" which makes an angle (ex-~) with AN. Take three functions by means of which the three domains R, R' and K' are transformed into a fixed circle, in such a way that the segment AN corresponds to the same circular arc A 1 N 1 R'' in each case. Let 'Yo be a circular arc joining A 1 and N 1 , lying within the circle and making an angle 1r8 with the arc A 1 N 1 • Consider the images Fig. 38 -y, y' andy" of this arc in the three domains R, R', R". By § 147, -ylies between -y' and -y"; § 146 shows that the curves -y' and y" both have tangents at A, these tangents making angles 8(ex+~) and 8(ex-~) respectively with AN. Since ~ is arbitrary and-y always lies between y' and y", it is seen that y itself has a tangent at A and that this tangent makes an angle 8ex with AN; for the contrary assumption would lead to a contradiction. This proves the theorem of§ 146 for domains which have a. straight line as one arm of the corner A. 149. The general case of§ 146 can be made to depend on the result just obtained. Let R be the domain with a corner at A. Consider a domain K which contains R and has a eorner at A, one of whose arms is AM and the other a. straight line AK. First transform R' 1nto a. half-plane ; at the same time R is transformed into a. domain Rt with a. corner at Au the arm A1M1 A which corresponds to AM being now a straight line. In this transformation of R into R 1 , and a. further transformation of R 1 into a circle, the ratio of angles at A is unaltered, as is shown by § 148.

94

TRANSFORMATION OF THE FRONTIER

(cHAP. VI

150. In particular, if the frontier of the domain R has a. tangent at .A, so that A is a comer with ex=.,., then the transformation is isogonal at A. lff(z) is the function which gives the transformation, and if' is the point of the circumference corresponding to A, then isogonality is expressed analytically by the statement that the function arg

,_z

f(,)-f(z)

...... (150•1)

tends to a constant as z tends to '· An analytical proof of isogonality at A has been given by Lindelof, who used (150"1) and Poisson's integral (23). He further showed that if the curve MAN is smooth (glatt) at A, then the function arg f' (z) is continuous at '· (A curve which has a tangent at A is said to be smooth at A if every chord BG tends to the tangent as B and G tend to A simultaneously*.)

It should be noticed that the function f' (z) is not necessarily finite in the neighbourhood of ' even in the case considered by Lindelof. For instance, the function ...... (150"2) w=-z logz, where z = r~, transforms the domain lying between the imaginary axis and the curve

~=0

log r = - .p c?s .p sm¢'

(o < .p < ?:2 and - '?2: < .P < o) ,

into the domain lying between the convex

w=~r+it logr,

curv~

(O
and the real w-axis cut between the points w = derivative of (150"2) is w' =-(1 + logz),

~ and w = ~. But the ...... (150•3)

and tends to oo as z tends to zero; arg w', however, tends to zero.

151. The theorem of § 93 may be completed and can then be applied in many cases, and leads to a. more precise result than that just given. y=f(z) with /(0)=0, j(z)=z2sin!: for z*O, hasy=O as tangent z at the point z =0, but is not smooth at this point. "The

CUl'VP

§§ 150, 151]

APPROACH TO THE FRONTIER

95

Let I (z) be a function regular in Iz I < 1 and such that I/ (z) I < 1 in this domain. Consider the function/(z) at points within the triangle ABC, where A is the point z = 1, BC is perpendicular to the real axis, lies entirely within the unit-circle, A and is at a distance k from A. There is a positive number M such that, for all values of z within the triangle ABC, 11-zl <M(1-Izl), ... (151"1) and hence Fig. 40 1-1/(z)l < M ~1-/(z) ...... (151•2) 1-lzl 1-z · Thus either 1-/(z)

I

1-z

tends uniformly to infinity as z tends to A by values within the triangle, or there is a sequence of points z~> z~o ... , lying within the triangle and

I

tending to z = 1, such that j 1 ~~ ~z) and, by (151"2), 1-lf(z,.)l 1-lz,.l are bounded; the theorem of § 93 can then be used. In the latter case, let z1 , z2o ••• be an arbitrary sequence of points within ABC tending to z= ':1.. Let two sequences of numbers, r 1 , r~o ... and Pt, P2, ... , be defined by the relations 1-~z.. <Xr,. ( ) r,.=-k-' p,.=1-,·,.(l-tX)' ······ 151"3 where

has the same meaning as in (93"10). Further, let 1-z=r,.(1-t), ...... (151"4) 1-/(z) = p,. (1-
..-..

96

TRANSFORMATION OF THE FRONTIER

(CHAP. VI

Also these. functions form a normal family within I t I < 1. It follows that (151"6) holds uniformly in any closed set of points lying within

Itl < t. Now consider the original sequence of points z1 , z,, ... and define a sequence of numbers t 1 , t 2 , ••• by means of the equations 1-z,.=r,.(1-t,.); (n=l, 2, ... ); ...... (151•7) (151"3) shows that th th ... all lie on the base BC of the triangle ABC. Thus, if tj>,.(t,.)=.t,.+T,., then T,.--0, for BC is a closed set of points within the unit-circle. The relations (151"4) and (151"5) yield 1 - f (z,.) = '!!! ( 1 _ ~ ) ,

1-z,.

r,.

1-t,.

and this gives lim 1-f(z,.) =«. (151"8) ,......., 1-z,. In conclusion, we observe that since {4>,. (t)} is a normal family, so also is {4>,.' (t)}, and that lim tj>,.' (t,.) = 1. Now differentiate (151"5) with 0

0

0

•••

-

respect to t, and, taking account of (151"4), substitute t,. for t. This givesf' (z,.) r,. = p,.tj>,.' (t,.). Thus limf' (z,.) =ex. . ..... (151"9)

,._..

We have proved the following theorem : THEOREM. Q, in I z I< 1, tkefunctionf(z)isregularand lf(z) I< 1, and if z1 , z~o ... is any sequence of numbers lying uithin tke triangle ABC and tending to z = 1, tken lim 1-f(z,.) ,.._., 1-z,. e:z:ists. This limit is either + <x:> or it is a number ex > 0. In tke latter case lim f (z,.) =ex.

-

152. Conformal tranaformation on the frontier. Let R be a simply-connected domain and P a point of its frontier through which two circles K andK' can be drawn, .K'lying entirely outside and K entirely inside R. We now make a conformal transformation of R into the circular area I z I < 1. It will be shown that there is a point A on I z I = 1 such that, if z approaches A from within a triangle ABC, whose base BC lies within the circle, the corresponding pointf(z) in R approaches P, and also the derivativef' (z) tends to a unique, finite, nonzero limit. It is therefore legitimate to speak of a conformal transformation of frontier-points.

§§ Io2, Io3]

97

FRONTIER-POINTS

By means of a Mobius transformation, let the interier of the circle K' be represented on the exterior of the unit-circle I w I = 1, in such a way that P is transformed into the point w = 1. Then R is transformed into a domain R 1 lying within I w I < 1. The circle K is transformed into a circle K 1 with centre 0 1 lying within R 1 • We need ouly show that if the function giving the transformation is suitably chosen the number «. of§ 151 is finite. Take that transformation w =f(z) which makes z = 0 correspond to the centre 0 1 of~. and let p1 denote the radius of K1.

Fig. 41

Fig. 42

By Schwarz's lemma, every circle I z I = 6, (6 < 1), is transformed into a curve surrounding the circle I w - (1 - PI) I = PI6. Thus this curve passes through at least one point w' which lies on the real axis and satisfies 1 - w'
153. Attention should be called to the fact that P may be a frontierpoint at which the transformation is conformal and yet not lie on a free Jordan curve. For example, consider the unit-circle I w I< 1, and in it

n'

· la.r arcs JOIDIDg · · · -1 to+ 1 and mak"mgangles ± .,. ClrcU 2 n - 1 (n= 1,2, ··· ) with the real axis at these points. Now cut the circle along each a.rc between the point w = - 1 and the first intersection of the arc with the circle I w- ( 1 -PI) I =PI. The unit-circle, with these cuts, is a domain to which our theorem applies. The functionf(z), considered in the whole circle I z I < 1, is not even continuous at z = 1. But, within an angle at z = 1 whose arms are chords of the circle, both f(z) and f (z) are continuous and bounded (26) (27).

CHAPTER VII

TRANSFORMA'riON OF CLOSED SURFACES 1M. Blending of domains. In au-plane let three Jordan arcs a, b and c, all passing through P and Q but with no other common points, define three Jordan domains A, Band (A +B). In a v-plane let three circular arcs b', c' and d', all passing through P' and ([, define two crescents B and C'; the sum of these domains, ( B' + 0'), is a circle whose circumference is made up of the two arcs b' and d:. Further, the angle between the arcs b' and c' is ;.. , where n is a positive integer. Suppose that a known function u =if! (v) represents B' on B in such a way that b corresponds to b' and c to c'. It will be shown that two functions z =f(u) and z = g (v) can be found, such that the former transforms (A +B) into a domain (A"+ B") and the latter transforms the circle (B' + 0') into a domain (B" + 0"). Here (A"+ B" + 0") is a circular area. ;l'he transformations are to be such that A and A", 0' and 0" are corresponding areas, while B" corresponds both to B and to B'. Given any point of B" the corresponding points in the u- and v-planes are to satisfy the relation u = 1{1 (v).

a p

u-plane

v-plane Fig. 43

First take a function Ut = 4>I (u) which transforms (A +B) into the circular area I u 1 I < 1, in such a way that the centre of the circle is the image of an interior point of A. Let the new domains corresponding to A and B be A 1 and B 1 respectively. The frontier of B 1 consists of a Jordan curve b1 and a circular arc c1 • The function~= 4>I (1/J (v)) = I/J1 (v) transforms the domain B into B1 in such a way that the two circular arcs c' and c1 correspond. The in-

§§ 154-156]

REPRESENTATION OF' A SURFACE

99

version principle shows that 1{11 (v) is defined not in B' alone, but in a crescent which has angles 7r/2"-' at P and Q'. We may suppose that this crescent is bounded by arcs b' and c'. The function u = 1{11 ( v) transforms this new crescent into a domain made up of B 1 and its inverse Jj, with respect to the circle a 1 + c,. Our problem has thus been reduced to another of the same kind in which the number n has been replaced by (n- 1). By repetition of this process we obtain in succession the functions U2 =
155. 'fhe special domains B' and C' in the v-plane can now be replaced by much more general ones. We assume that the frontiers of B' and C' consist of three Jordan curves joining the points P' and Q' and having distinct tangents at these points. Let v = x (w) be a function which transforms I w I < 1 into (B' + C'). 'fhen B' is transformed into a domain whose frontier consists of a circular arc {3' and a Jordan curve which meets either end of {3' at an angle which differs from zero. This shows that the domain just described contains a crescent {3' y' whose angle is 7r/2". The function u = l{t (x (w)) transforms y' into a curve in the u-plane and, if we cut away the domain that lies between y and c, our problem is the same as that of§ 154. 156. Conformal transformation of a three-dimensional surface (28). Let a surface 8 in three-dimensional X Y Z-space be represented, in the neighbourhood of some point of 8, by means of the equations X=X(1X,{3),

Y= Y(1X,{3),

Z=Z(a.,{3), ...... (156·1)

where IX and {3 are parameters. 'fhe same surface can be represented in terms of other parameters u and ~' by substitution in (156'1) Of the functions IX= IX (u, v), {3 = {3 (u, v). We assume that the functions a. (u, v) and {3 (u, v) can be so chosen that the new equations for 8, X=~(u,v), Y=lJ(u,v), Z=,(u,v), ...... (156'2) have continuous first partial deriv11tives in some domain Rw of the uv-plane, and at the same time di' =A (u, v) (du2 + dv'), ...... (156·3) where A* 0 at all points Rw. It then appears that the domain Rw is represented on a portion R 8 of S by a one-one continuous correspondence, that any two curves in Rw which have continuously turning tangents

100

TRANSFORMATION OF CLOSED SURFACES

(CHAP. VII

and cut at an angle ot correspond to two curves which cut at an angle a on R8 , and that the scale of the !epresentation at the point of intersection is independent of direction. In other words the representation of RfiJ on R 8 is conformal. It is known that it is always possible to introduce parameters u and v such that (156"3) holds in a certain neighbourhood of a point (ot0 , Po) of (156·1) provided that in some domain of the otp-plane the functions X (ot, p), Y (ex, p) and Z (ot, P) have continuous first partial derivatives which satisfy a Lipschitz condition, and that the three Jacobians, o(X, Y) o(Y, Z) o(Z, X) a(cx, P) ' a(ot, P) ' a(ot, p) ' do not all vanish at (oto, Po). It is conceivable that the equation (156·3) may be satisfied, for a suitable choice of parameters, even on surfaces which do not satisfy alJ the conditions just mentioned.

157. Conformal representation of a closed surface on a sphere. Suppose that S is a surface which can, by a one-one continuous transformation, be made to correspond to a closed sphere ~- Further let every point P of S lie within a portion of S which can be conformally represented on a portion of a plane(§ 156). It will be shown that the whole surface S can be represented conformally on ~Consider an arbitrary one-one continuous transformation of S into ~­ The north pole N 1 of ~ corresponds to a point N of S. Stereographic projection from N 1 represents ~ on a plane T, and there is a one-one continuous correspondence between Tand the pricked surfaceS* obtained by omitting the point N from S. 158. Now consider in Tan infinite sequence of triangles 1'/, 1'2', ••• arranged spirally (like the peel of a peeled apple) and covering the whole plane T. 'rhe sum u,.' = 1'1' + 1'2' + ... + Tn' of the first n triangles always covers a simplyconnected domain. The figure gives a r-----J,___ simple example of what is meant. The triangles may have curvilinear sides; they are drawn with straight sides purely as a matter of convenience. The triangles Tn' are, however, to be so chosen that their Fig. 44 images Tn on S* satisfy the two following conditions: (a) the sides of

§§ 157-161]

REPRESENTATION ON A SPHERE

101

T,. are curves on 8- which have tangents at their end-points, and two intersecting sides have distinct tangents; (b) any two triangles with a common side lie within a portion of S that can be represented conformally on a plane domain. 159. It will be shown that for n = 1, 2, ... the sum rr,. = T 1 + T 2 + ... + T,. can be represented c~mformally on a circle I z,. I < r,. in a complex z,.-planeAssume that rr,._ 1 has already been represented on a circle I z,._ 1 I< r,._~> and that the figure ( T,._ 1 + T,.) has been represented on a domain of a v-plane. The triangle T,._1 appears in both representations, and if the image of a point of T,._ 1 in one is made to correspond to the im&ke of the same point in the other, a new conformal representation is set up. The representation of rr,. on I z I
161. Conformal transformation of polyhedral surfaces into the surface of a sphere was one of the earliest applications of the theory to be

102

TRANSFORMATION OF CLOSED SURFACES [CHAP. VII,§ 161

attempted. Schwarz showed that both the general tetrahedron and the cube can be transformed into the sphere. The transformation of the most general closed polyhedron into the closed sphere is a problem which is almost identical with the one just treated. We cannot require preservation of angles at edges and corners of the figure, and to specify our intentions we make the following conventions as to representation of neighbourhoods of these singular points. Along an edge two faces meet. We suppose them rigid, cut them from the figure and spread t:Uem out on a plane. Thus a neighbourhood of any point of the edge is represented as a plane neighbourhood. If Cis a corner we make a quasi-conformal representation of a neighbourhood of C onto a plane domain, as follows : By means of a small sphere, centre C, a small neighbourhood of Cis cut off. The pyramidlike figure so obtained is cut open along one of its edges and spread out on the u-plane, the point C corresponding to u = 0. If now 1rct is the sum 2

of the superficial angles of the polyhedron at C, then the function v = gives the required quasi-conformal representation of the corner C. The method used above now gives a solution of our problem (29).

u~

CHAPTER VIII

THE GENERAL THEOREM OF UNIFORMISATION

162. Abstract surfaces. We take a finite or infinite collection of triangles T1 , T 2 , ••• ,and suppose that with each side of each triangle Ti is associated exactly one side of some other triangle Tk. We may picture associated sides as being welded together; when this has been done, any vertex P will belong to several triangles. We assume that the triangles adjoining a common vertex form a finite cycle of adjacent triangles. The welding operation need only be done mentally; the abstract polyhedral surfaceS consists merely of the triangles T;, the pairs of associated sides (the edges of S), and the vertices.

163. We shall consider polygonal paths PP1 ..• Pn_1Q, directed from P to Q, containing a finite number of adjacent edges of S. We call S connected if any two of its vertices P and Q can be joined by such a path. A connected surface is closed if it contains only a finite number of triangles, open if it contains an infinite number. Given a path ex from P to Q, we shall denote by ex-1 the same path described from Q to P. If {3 is a path from Q toR, we shall denote by ex/3 the path PQR obtained by describing ex from P to Q followed by {3 from QtoR. We deform a path ex by making a finite number of operations of the following two kinds: (a) An edge of ex which corresponds to a side AB of a triangle ABC is replaced by AC + CB, or AC + CB is replaced by AB. (b) A pair of successive edges corresponding to AB + BA is inserted or deleted.

If ex can he deformed into {3 in this way, {3 is called homotopic to IX. Clearly this is an equivalence relation, so that the paths joining P tp Q are divided into classes {PQ} of homotopic paths. For most surfaces, there are several classes {PQ}, generally an infinite number. If there is only one class {PQ}, i.e. if all paths joining P to Q are homotopic (for any choice of P and Q), the surfaceS is called simply connected. H

104

THE GENERAL THEOREM OF UNIFORMISATION

(CHAP. VIII

164. Now consider a fixed vertex 0 of S, and two paths oc and fJ joining 0 to P. The path oc{J-1 then begins and ends at 0, and oc and fJ are homotopic if and only if oc{J-1 is homotopic to the path consisting of the single point 0. It foll?ws easily from this that we can chooRe a set of closed paths containing 0 ...... (164·1) such that, given any paths oc and fJ from 0 to P, exactly one of the paths is homotopic to fl.

oc, wllloc, w"'oc, .. .

. ••••. (164·2)

165. The universal covering surface. Consider two surfaces S and 8, formed with triangles T; and T/•' respectively. Suppose that each of the triangles T;'i', TP 1, ••• is related to the same triangle T;, which is called the projection of T/• 1• Suppose further that, if T/• 1 and Tklp.l are adjacent on 8, their projections T; and T 1e are adjacent on S. Then 8 is called a covering surface of S. A covering surface 8 of S is called unlnanched if every cycle of triangles adjoining a vertex P of 8 projects on to a similar cycle on S, both cycles having the same number of triangles.

166. We shall now show that any surface S has a simply connected, unbranched covering surface 8, called the universal covering surface of S. To construct 8, we first choose a vertex 0 of S and a set (164·1) of closed paths on S containing 0. Next, given any triangle T., of S, with vertices P, Q and R, we associate with it triangles

T"' T.,m, T.,121, ...

...... (166·1)

where corresponds to the path w<•loc (or any path joining 0 toP homotopic to w<•loc). This involves selecting a definite vertex P from each triangle T n; we restore symmetry by associating with T ..1• 1 not only a class {,8} of paths from 0 toP, but also classes {,8'} and {,8-} of paths from 0 to Q and 0 to R. The classes are related in pairs by the rule:

Tn 1• 1

A path ,8' is homotopic to ,8 + PQ, a path ,e- homotopic to ,8 + PR. Fin~lly, we arrange that the vertices p!vl, Qlvl, Rlvl of T., 1• 1 project on P, Q, R, and similarly for the sides. To complete the definition of S, we muet decide how to associate pairs of sides of triangles on S. Let T n and Tm be triangles of S with a common edge PQ, and let oc be any path from 0 to P. This path gives rise to well

§§ 164-167]

105

UNIVERSAL COVERING SURFACE

T,. 1• 1,

defined triangles T m ,,., , and we aBBociate those two sides of T,. '"' and which project on PQ. This makes the collection of triangles T,. '"' into an abstract surface S which is clearly an unbranched covering surface of S.

'I'm'"'

To prove that S is simply connected, take any two vertices P and Q of S projecting on P and Q, and let ~ and y be any two paths joining P to Q projecting on to fJ andy joining P to Q. ToP corresponds a path rx from 0 to P, and both the paths rx{J and rxy joining 0 to Q must correspond to Q. (For if~= PP1 ••• P,._1Q and fJ = PP1 ••• P,._1Q, the paths rx + PP1 .•• P; and rx + PP1 ••. Pi+ 1 belong to related classes as defined above. It follows easily by induction that rx + P ... P; corresponds to P;, and in particular that rx{J corresponds to Q.) Now since rx{J and rxy both correspond to Q, they must be homotopic, and hence so are rx-1rx{J and rx-1rxy, and therefore fJ andy are homotopic. But the operations (a) and (b) which deform fJ into y will deform ~ into y if transferred to S. Thus ~ and y are homotopic, and S is simply connected.

167. Domains and their boundaries. Given any finite collection T ,., T ns• ... , T ".t of triangles on S, their sum taken modulo 2 will be called a domain I: of S and denoted by I:= T,., t Tns t ... t T,.,. . ..... (167-1) Any finite number of domains I:1 , I: 2 , ••. ,:Em can also be added modulo 2, so that the symbol I:1 t I: 2 t ... t :Em has a meaning. The bourulary !J (T) of a triangle is the sum of its sides; the boundary of the domain (167·1) is then defined by the equation S(I:)

=

S(T,.)

t

!J(Tns)

t ... t

!J(T,.,)•...... (167·2)

This is a very convenient formalism for proving the theorems which follow:

lfrx and fJ are homotopic paths, and if[rx] and [{J] are the sums modulo 2 of their sides, then [rx] t [{J] is the boundary !J (I:) of some domain I:. For if a triangle Tis used for operation (a) of§ 163, [rx] is transformed into [rx) t S (T), whilst operation (b) leaves [rx] unchanged. Hence if fJ is homotopic to rx we can write

[{J] = [rx] and therefore

t

S

(T,.,)

[rx]

t ··· t

S

t

!J (I:).

[{J]

=

(Tn,.) = [rx]

t

S (I:), . ..... (167·3)

106

THE GENERAL THEOREM OF UNIFORMISATION

[CHAP. VIII

Every rum-empty do-main~ on an open conmcted sur,face S has a nonempty boundary 8 (~). Since S is open there exist triangles T of S which do not belong to ~. and since S is connected at least one such triangle is adjacent to a triangle of~.

Two different do-mains have the same boundary.

~

and

~·

on an open conmcted surface S cannot

For if '8 (~) = 8 (~'), then 8 (~ t ~') = 0, and this contradicts the previous theorem since ~ t ~· is not empty. It should be observed that on a closed surface there are always pairs of (complementary) domains with the same boundary. On an open simply connected surface, every simple closed polygon is the boundary of exactly one domain.

Two distinct vertices P and Q of the polygon TT divide it intv paths ex and f1 from P to Q. Since Sis simply connected, ex and f1 are homotopic. Also (since the polygon is simple) ex = [ex] and f1 = [{1], and therefore by (167·3) TT = [ex] t [/1] = 8 (~).

168. The Theorem of van der Waerden (30). Given a simply conmcted open surface S, its triangles can be arranged in a sequence ...... (168·1) in such a way th.at each domain ~ ..

= T 1 + T 2 + ... + T,

(n = 1, 2, ... )

...... (168·2)

ltas a simple closed polygon as boundary.

Suppose that T 1 , ... , T, have already been chosen. We must then find T n+I adjacent to ~ .. such that S (~, + T n+I) is a simple closed polygon. This will be so if T n+I has one or two sides in common with the polygon S (~,)but 1wt one side and the opposite vertex. We must also ensure that every triangle of S occurs in the sequence (168·1). Choose any triangle ~ of S adjacent to ~ ... The sides p, q and r of~ cannot all lie on !8 (~ .. ), because if they did S (~,) would not be a simple closed polygon. Therefore one of three things must happen: I. Two sides II. One side

of~

of~

III. One side p

lie on !8

(~,).

lies on S

(~,).

of~

the opposite vertex does not.

and the opposite vertex P both lie on 8

(~,).

§ 168]

SIMPLY CONNECTED SURFACES

107

If A satisfies I or II, we take T n+1 = A. In case III, the polygon S (~ .. ) splits into three parts p, ot and p, and ex T q, p T r are simple closed polygons, bounding domains II and II* (see Fig. 45). Thus S (II) =ex -t q, S (II*) = p -t r and S (II -t II*) = (p -t ex -t /J) -t (p -t q -t r) =

S

~

-t

(~.. )

-t S

(A).

It follows that II*=~..

+ A.

. ..... (168·3)

The triangles T 1 , ••• , T .. belong to ~n• and so each of them belongs either to II or to II*. But if T 1 belongs to II*, say, then so do T2 , ••• , T., and A, because no side of the polygonS (II*)= p + r belongs to more than one of these (n + 1) triangles. Thus II consists of triangles Olltside ~.. + A ; we now try to choose T n+1 from one of these triangles.

p

Fig. 45

Let A' be a triangle in II adjacent to ex. We repeat the argument above, with A..replaced by A', and find that: either we may choose T.,+l =A', or there is a polygon q' +-ex' bounding a domain II'. But all the vertices of q' + ex' lie in ex, and so ex' is obtained from ot by removing at least one side. We now repeat the argument, starting with a triangle A' in II' adjacent to ex'. After a finite number of steps, we must arrive at a suitable triangle T ..+l which, being adjacent to ex, belongs to II. Now replace, in the original figure, ~.. by ~..+1 = ~ .. + T n+1 and II by II - T n+l• and repeat the argument. After a finite number of steps, we obtain ~ .. + II = ~ .. + T.,+l + ··· • Tn+m = 1:n+m and are now ready to adjoin the triangle A= Tn+m+l· To show that S can be exhausted by this process, we start with any triangle T 1 ' of ~. adjoin all adjacent triangles, then adjoin all triangles adjacent to those obtained already, and so on. In this way we obtain

108

THE GENERAL THEOREM OF UNIFORMISATION

[CHAP. VIII

a sequence {T,.'} of triangles exhausting S. If we n;:,w arrange, at each stage of the process described above, that d is chosen to be the first suitable triangle T ,.', it is clear that every triangle T ,.' will appear somewhere in the sequence (168·1).

169. Riemann surfaces. So far, our arguments have not involved any individual points on S except the vertices; the edges of S may be regarded as ordered pairs of vertices, the triangles as triples of vertices. In order to obtain a Riemann surface, we must convert this framework into a two-dimensional topological space which carries the local angular metric of the ordinary plane(3l) (32).

This is done most simply by assigning to each triangle T; a Jordan curve a.; lying in the plane of a complex variable t;. On a.i we mark three points P;, Q; and R;, and we regard the interior of a.; as a conformal representation ofT;. Since any two Jordan domains can be transformed conformally into each other, the transformation being continuous if extended to their frontiers (§ 137), we see that both the shape and size of a.;, and the positions of P;, Q; and R;, are irrelevant. Nevertheless, we can now identify individual points of T;: we transform the interior of a.; conformally into a circle, so that any point of T; is mapped into a point A; of the circle and i~ characterised by the cross-ratio (P;, Q;, R;, A;)

(see§ i:i).

The angular metric which we want to set up is then well defined at all inner points ofT;, i.e. those for which (P;, Q;, R;, A;) is not real.

We must still define the angular metric on the :;ides and vertices of all our triangles. To do this, we need further assumptions which are best dealt with by considering a special case. Suppose that the triangle T 1 with sides p, l, q is to be welded along q to T2 with sides q, m, r (Fig. 46 a, b). We assume that there is at least one conformal representation of T1 such that the angular metric of the lcplane is valid on the side q of T1 , and that a strip a between q andy (Fig. 46a) is the conformal image of a part of T2 (Fig. 46b). Then the welding of T1 and T2 along q can iJe performed by the method of§ 154 (Fig. 46c). This process is uniquely defined if we know the images in the tc and t 2-planes of a particular point M of q. Suppose further that three triangles T1 , T 2 and T3 form a cycle arour.d a vertex A, am~ that accordingly T3 hfl.s sides p, r, n (Fig. 46d). We apply

§ 169]

RIEMANN SURFACES

109 the preceding method, with the difference that now the part (r + A + p) of the frontier of T3 must be welded to the corresponding part of the frontier of T1 + T 2 , and that A takes the place of M. The common vertex A lies inside the resulting circle (Fig. 46e), so that the angular metric is certainly defined there. The case when the cycle around A contains more than three triangles is tr~ted by the obvious extension of the method above.

,,

::!---...,

I

I

c

(a)

B (b)

D

, ,. /

--- ... ',s \

\

I

I

I I

B B (c)

(d)

llO

THE GENERAL THEOREM OF UXIFORMISATION

(CHAP. VIII

170. The Uniformisation Theorem. LetS be a Riemann surface. If Sis closed and simply connected, the developments of Ch. vn apply. In all other cases, the universal covering surface .-cy of S is an open, simply connected surface for which the theorem of van
~n-l·

Uniformiaation Theorem. The universal covering surface S of any open or closed Riemann su1jace S can always be represented conformally on (1) a closed sphere, (2) an Euclidean pla·ne, (3) the interior of a circle

It I <

R,

these possibilities being mtdually exclusive.

In this way we obtain a complete conformal pattern in the t-plane of the triangulation of S. Its edges are Jordan curves and do not necessarily have tangents at their end-points, so that in general the angles of the triangles are not defined. However, the given triangulation can always be replaced by an equivalent one for which the edges are analytic arcs or, even more specially, are geodesics in the particular metric involved. Remark. Care must be taken always to weld adjacent triangles of S together in the same way as their projections on S. The resulting triangulation of S in the t-plane then admits a group of displacements transforming equivalent triangles into one another.

171. The uniformisation theorem shows that we can always introduce a local metric on S which is spherical in the first case, euclidean in the second and hyperbolic (Lobatscheu·skyan) in the third. We can even define the distance between any two points of S as the distance between their images in the t-plane. 8ince two triangles of S which have the same projection on S have the same metric at equival ~nt points, we can transfer the local metric

§§ 170, 172]

REPRESENTATION OF A TORUS

Ill

from S to S. But we cannot do this for the distance between two points, since in general any point of B has an infinity of corresponding points onS.

172. Conformal repre1entatlon of a toru1. It is often important to determine which kind of metric is introduced on a given Riemarin surface by the method described above. This can sometimes be done with very little effort. Consider for instance a torus S in three-dimensional space. By combining the method of§ 156 with that of the present chapter, we hnd that ·the universal covering surface S is represented either on the infinite t-plane or on a finite circle I t I < R. The metric induced on Sis either euclidean or hyperbolic.

Fig. 47

We now consider two closed paths r:x.' and fJ' on S which are homotopic neither to zero nor to each other, and which intersect at a point 0 of S. The images of r:x.' and fJ' in the t-plane are arcs joining a point 0 1 to two different-points 0 2 and 0 3 • We replace r:x.' and fJ' by curves r:x. and fJ which are mapped on the segments (euclidean or non-euclidean as the case may be) 0 10 2 and 0 10 3 • Since r:x. and fJ are the shortest closed curves on S passing through 0 and homotopic to r:x.' and {J', they cannot intersect on S except at 0. If we cut S along r:x. and {J, we obtain a surface S' which is mapped on a quadrangle in the t-plane. This mapping is conformal at 0, so that the quadrangle has angle-sum 211. But the angle-sum of a non-euclidean triangle is less than 11, hence the angle-sum of a quadrangle is less than 211. Therefore the metric must be euclidean. Moreover, opposite sides of the quadrangle have equal length, so that the quadrangle is in fact a parallelogram. Any segment parallel to one side and joining two opposite sides is the image of a closed geodesic on S. Hence S is covered by two systems of geodesics which form a net on S.

112

THE GENERAL THEOREll OF UNIFORMISATION

(CHAP. VIII

It is well known that an infinity of such nets can be constructed on S. Among them ·is a "reduced" net, characterised by the fact that the triangle 0/)/)3 is acute-angled. The shape of the reduced parallelogram is a conformal invariant of S: two such surfaces can be transformed conformally into each other if and only if their reduced parallelograms are similar. Since any given point of the t-pla.ne can be moved to another given point by a translation, it follows that any torus S can be transformed conformally into itself in such a way that a given point P is transformed into a second given point Q.

BIBLIOGRAPHICAL NOTES I. BOOKS Encyklopii.die der mathema.tischen \Vissenschaften: L. Lichtenstein. Neuere Entwickelung der Potentialtheorie. Konforme Abbildung. n, c. 3. (Vol. II, 3, 1.) L. Bieberbach. Neuere Untersuchungen iiber Funktionen von komplexen Variablen. II, c, 4. (Vol. n, 3, 1.) H. A. Schwarz. Gesammelte Abhandlungen. II. G. Darboux. ~ODS sur la tht\orie generale des surfaces. I, Livre II (ch. IV). E. Picard. Traite d'Analyse. II. G. Fubini. Introduzione alia Teoria dei Gruppi discontinui e delle Funzioni Automorfe. (Pisa, 1908.) G. Julia. Le~ns sur les fonctions uniformes a point singulier essentiel isole. (1923.) - - Principes geometriques d'Analyse. I (1930). J. L. Coolidge. Treatise on the Circle and the Sphere. (Oxford, 1915.) Hurwitz-Courant. Funktionentheorie. (3rd ed. 1929.) H. Weyl. Die Idee der Riemannschen Flii.che. (2nd ed. 1922.) P. Montel. ~ODS sur les families normales. (Paris, 19271. C. Carathiodory. Funktionentheorie. (Basle, 1950.) II. NOTES AND PAPERS (1) p. 1. J. L. Lagrange. Sur la construction des cartes goographiques. (1779.) <Euvres, IV, pp. 637-92. (2) p. 2. C. F. Gauss. Allgemeine Auftoaung der Aufgabe die Theile einer gegebenen Flii.che so abzubilden, dass die Abbildung dem abgebildeten in den. kleinsten Theilen ii.hnlil.'h wird. (1822.) Werke, IV, pp. 189-216. (:J) p. 2. B. Riemmm. Grundlagen fiireine allgemeine Theorie der Functionen einer complexen verii.nderlichen Grosse. (Diss. Gottingen, 1851.) Werke (2nd ed.), pp. 3-41 (§ 21, p. 39). (4) p. 2. See e.g. C. Caratheodory. Variationsrechnung und pa.rtielle Differen· tialgleichungen erster Ordnung (Leipzig, 1935), pp. 305, 309, 333. · (5) p. 2. D. Hilbert. Das Dirichletsche Prinzip. (Gott. Festschrift, 1901.) Gesammelte Abhandlungen (Berlin, 1935), III, pp. 15-37. (6) p. 2. H. A. Schwarz. Gesammelte Abhandlungen (Berlin, 1~90), II, p. 145. E. Picard, Traite d'Analyse, T. II, Chap. x. (7) p. 4. A. F. MObius. Werke, 11, pp. 205-43. (8) p. 18. F. Klein. Vorlesungen tiber nichteuklidische Geometric (Berlin, 1928). (9) p. 18. H. Poincare. <Euvres, rr. (tO) p. 22. P. Firurler. tiber Kurven und Fliiehen in allgemeinen Rii.u.· en. (Diss. GOttingen, 1918.) A reprint has been published (Basle, 1951). (ll) p. 34. The monodromy theorem was originally given by Weierstrass. The proof he gave in his lectures is to be found in 0. Stolz and J. Gmeiner, Einleitung in die Funktionentheorie (Leipzig, 1910).

114

BIBLIOGRAPHICAL NOTES

(12) p. 35. Uber die Uniformisierung beliebiger a.nalytischen Kurven. GOtt. Nachr. (1907. Paper presented on 12 April.) See especially p. 13. (13) p. 39. H. A. Schwarz. Zur Theorie der Abbildung. (1869.) Gesammelte Abha.ndlungen, n, p. 108 (especially § 1). The proof given in the text is due to Erhard Schmidt. It was first published by C. CaratModory. Sur quelques generalisations du theoreme de M. Picard. C.R. 26 Dec. 1905. A similar proof had already been given by H. Poincare. Sur les groupes des equations lineaires. Acta Math. Vol. 4 (1884) especially p. 231. (14) p. 41. G. Pick. Ober eine Eigenschaft der konformen Abbildungen kreisformiger Bereiche. l\:lat.h. Annalen, 77 (1916), p. l. (15) p. 48. P. Koebe made the conjecture that If' (0) I ~ 4. The first complete proof was !{iven by G. Faber, Xeu£'r Beweis eines K(){'be-Bieberbachschen Satzes tiber konforme Abbildung. Munch. Sitzungsber., }lath. Phys. Klasse (1916), p. 39. (Compare Bieberbach, Encyklopadie, I.e. p. 511). (16) p. 53. G. Julia. Extension nouvelle d'un Iemme de Schwarz. Acta Math. 42 (1918), p. 349. The general theorem given in the text is due to J. Wolff a.nd was published in Comptes Rendus, 13 Sept. 1926. The theorelll is here presented in the form in which it was proved by the author: Uber die Winkelderivierte von beschrankten analytischen Funktionen. Berl. Sitzungsber., Phys.-Mathem. Klasse (1929), p. 39, where, in ignorance of Wolff's priority, his result was proved afresh. For the definition of continuous convergence for meromorphic functions, cf. (17) p. 61. .4. Ostrowski. Uber Folgen analytischer Funkt.ionen und einige Verscharfungen des Picardschen Satzes. Math. Zeitschrift, 24 (1926), p. 215. (18) pp. 61, 62. C. Caratheodvry. Stetige Konvergenz und normale Familien von Funktionen. Math. Annalen, 101 (1929), p. 315 and I.e. Funktionentheorie, Book IV. (19) p. 61. Cf. P. MonteZ. I.e. (20) p. 62. Cf. M. Frechet. Les espaees abstraits. (Paris, 1928.) (21) p. 66. Three distinct methods of attack have been used to prove the fundamental theorem of conformal representation: Dirichlet's Principle, the methods developed in Potential Theory for problems concerning values on the boundary, a.nd methods taken exclusively from the Theory of Functions. Proofs on this latter basis were given last of all, after the others had been disposed of. The history of the problem up to 191~ is exhaustively dealt with by Lichlensl$in in his article in the Encyklopadie. The proof presented in this book goes back to L. Fejer a.nd F. Riesz. It was published by T. Rad6. i:ber die Fundamentalabbildung schlichter Gebiete. Acta Szeged, I (1923). Rad6 draws attention t{) the important fact that simple connectivity is not made use of in the course of the proof. .For another type of proof, see C. Caratheodvry, A Proof of the first principal Theorem of Conformal Representation, Courant Anniversary Volume (New York, 1948), and Funktionentheorie, l.c. Book VI, Chap. n. Cf. aJao the interesting paper by G. Faber. Ober den Hauptsa.tz der

BIBLIOGRAPHICAL NOTES

115

Theorie der konformen Abbildungen. Miinch. Sitzungsber., Math.-Phys. Klasse (1922), p. 91. See also P. Simnnart. Surles tr&DSformations ponctuelles et leurs applications geometriques. (2" partie) la representation conforme. Ann. de la Soc. Scient. de Bruxelles, 50 (1930), Memoires, p. 81. (22) p. 76. The theory also applies to tiequences of Riemann surfaces, cf. C. Caratheodory. Untersuchungen iiber die konformen Abbildungen von festen und verii.nderlichen Gebieten. Math. Ann. 72 (1912), pp. 107-144. (23) pp. 85 and 94. The method here used is a simplification of a proof given by E. Lindewf, Sur Ia. representation conforme d'une a.ire simplement connexe sur J'a.ire d'un cercle. 4e Congres dPs ma.them. sca.ndina.ves a Stockholm (1916), pp. 59-90. (See especially pp. 75-84.) The modifications made were suggested in the course of s. conversation with T. Rad6. (24) p. 87. For extensions see: E. Lummer. Uber die konforme Abbildung bizirkularer .S:urven vierter Ordnung. Diss. Leipzig (1920). (25) p. 90. For extensions see: tr. Seidel. On the distribution of values of bounded analytic functions. Trans. Amer. Math. Soc. 36 (1934), pp. 201226. C. Caratheodory. Zum Schwarzschen Spiegelungsprinzip. Comm. Hclvetici, Vol. 19 (1947), pp. 263-278. (26) p. 97. For extensions see: Jr. Seidel. Uber die Randerzuordnung bei konformen Abbildungen. Math. Anna.len, 104 (1931), p. 182. (27) p. 97. L. Ahlfora. Untersuchungen zur Theorie der konformen Abbildungen und der ganzen Funktionen. Acta Fennicae, A, I, No.9 (1930). (28) p. 99. Cf. G. Darbvux, i.e. I, Livre 2, Chap. 3. (29) p. 102. For the application of this theorem to Plateau's problem see: T. RadO, The problem of the least area and the problem of Plateau. Math. Zeitschrift, 32 (1930), p. 763. (30) p. 106. B. L. van der Waerden. Topologie und Uniformisierung der Riemannschen Flii.chen. Leipzig. Sitzungsber. Math.-Phys. K.lasse. Vol. 93 (1941}. (31) p. 108. T. RadO. Uber den Begriff der Riemannschen Fliiche. Acta Szeged, 2 (1925), pp. IOI-121. (32) p. 108. C. Caratheodory. Bemerkung iiber die Theorie der Riemannschen Flii.chen. Math. Zeitsehrift, 52 (1950), pp. 703-8. The ~§ 131-136 contain in principle all that is necessary for the study of tre representation of the frontier in conformal transformation of the most genera; simply-connected domain. The relevant literature is mentioned by Lichtenstein in his article in the Encyklopii.die I§ 48). The results of§§ 120-123 on the kernel of a sequence of domains can be extended to the case of representation of 2n-dimensional domains on one another by systems of n analytic functions of n variables. See: C. Caratlteodory. Cber die Abbildungen die durch Systeme von analytischen Funktionen mit mehreren Verii.nderlichen erzeugt werden. Math. Zeitschrift, 37 (1932), p. 758. There is an extensive literature on the representation of multiply-connected domains and Riemann surfaces. (Cf. Lichtenstein, i.e. §§ 41, 44 and 45.)