Conformal Mapping - L . Bieberbach

CONFORMALMAPPING

BY

L. BIEBERBACHFRIEDRICH-WILHELM'S UNIVERSITY

TRANSLATED BY

F. STEINHARDTCOLUMBIA UNIVERSITY

CHELSEA PUBLISHING COMPANYNEW YORK

COPYRICHT 1953 BYCHELSEA PUBLISHING COMPANY

COPYRIGHT 1964 BYCHELSEA PUBLISHING COMPANY

PRINTED IN U.S.A.

TRANSLATOR'S PREFACE

This book is a translation of the fourth (latest)edition of Bieberbach's well-known Einfuhrungin die Konforme Abbildung, Berlin 1949. Variousminor corrections have been made (particularlyin 12 and 14) and the bibliography has beenadded to.

F. Steinhardt

...

111

TABLE OF CONTENTS

I. FOUNDATIONS. LINEAR FUNCTIONSPage

TRANSLATOR'S PREFACE .. . . iii 1. Analytic Functions and Conformal Mapping. . . . . 1 2. Integral Linear Functions . . . . . . . . . . .. 13 3. The Function w = 1/z 15Appendix to 3: Stereographic Projection. . . . . . . . . .. 21 4. Linear Functions .... . . . . . . . . . . . . . . . . . . . . . . . .. 23 5. Linear Functions (continued) ................ 33 6. Groups of Linear Functions 43

II. RATIONAL FUNCTIONS 7. w = zn. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 51 8. Rational Functions 62

III. GENERAL CONSIDERATIONS 9. The Relation Between the Conformal Mapping of

the Boundary and that of the Interior of a Region. 71 10. Schwarz' Principle of Reflection 73

IV. FURTHER STUDY OF MAPPINGS REPRESENTEDBY GIVEN FORMULAS

11. Further Study of the Geometry of w = Z2 79 12. w = Z + 1/z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84 13. The Exponential Function and the Trigonometric

Functions 92 14. The Elliptic Integral of the First Kind. . . . . . . . .. 95

v

vi TABLE OF CONTENTS

v. MAPPINGS OF GIVEN REGIONSPage

15. The Mapping of a Given Region onto the Interiorof a Circle (Illustrative Examples) 109

16. Vitali's Theorem on Double Series 119 17. A Limit Theorem for Simple Mappings 125 18. Proof of Riemann's Mapping Theorem 128 19. On the Actual Construction of the Conformal Map-

ping of a Given Region Onto a Circular Disc 131 20. Potential-Theoretic Considerations 137 21. The Correspondence Between the Boundaries under

Conformal Mapping 152 22. Distortion Theorems for Simple Mappings of the

Disc I z I < 1 ................................156 23. Distortion Theorems for Simple Mappings of

I z I > 1 168 24. On the Conformal Mapping of Non-Simple,

Simply-Connected Regions Onto a Circular Disc .. 186Remark on the Mapping of Non-Simple, Multiply-

Connected Regions Onto Simple Regions 193 25. The Problems of Uniformization 194 26. The Mapping of Multiply-Connected Plane Re-

gions Onto Canonical Regions 209BIBLIOGRAPHY ........ 230INDEX 231

CHAP'"rER ONE

Foundations. Linear Functions

I. Analytic Functions and Conformal MappingAs is well known, a function w === f (z) of a com-

plex variable z == x + iy (i == V-1) is said to beanalytic and regular throughout a region l R if it isone-valued and differentiable at every point of R.A consequence of the differentiability of f (z) ===u(x, y) + iv (x, y) are the Cauchy-Riemann dif-ferential equations for the real and imaginaryparts of f (z), viz.

au ov aU. ov(1) ox === By , oy === -ox

We further assume the reader to be familiar withthe fact that analytic functions can be developedin power series, that is, that in the neighborhood

1 A region is a point-set with the following two properties:1. If a point P belongs to the set, then so do all the points ofsome circular disc that contains P in its interior. 2. Any twopoints of the set can be connected by a continuous curve allof whose points belong to the set.-A closed region, i.e. aregion plus its boundary points, is sometimes called adomain; many writers, however, use the term "domain"to mean the same as "region."

1

2 I. FOUNDATIONS. LINEAR FUNCTIONS

of any given point a of the region R, an expansionof the form

(2) w === Co + c1(z - a) + c2(z - a)2 + holds. Now consider,in particular, functions I(z)for which I' (z) =f= 0 holds everywhere in R, andinterpret x, y and u, v as rectangular coordinates,in the usual way; it is proved in Function Theorythat if R is mapped by tv === f (z) on a point-set R',then R' is itself a region (Theorem on the Preser-vation of Neighborhoods).

That is to say, 1. if Co denotes a point of R' whichis such that the a in Co === I (a) is an interior pointof R, then all the points within a sufficiently smallcircle with center at Co also belong to the point-setR'; 2. the point-set R' is connected.

The first part of this theorem is merely thegeometric expression of the fact that powerseries (with Cl =f= 0) have an inverse. For, fromw === Co + C l (z - a) + . .. it follows that z ===a + (llc l ) (w-co) + .... Let a be an interiorpoint of R; then the last power series convergeswithin some circle with center at w === co. By con-fining ourselves to a sufficiently small such circle,we can make sure that its points correspond underw === I(z) to z-values from a neighborhood ofz === a that belongs to R. But then, the circle wehave chosen must have all its points in R'.

The second part of our theorem states that any

1. ANALYTIC FUNCTIONS, CONFORMAL MAPPING 3two interior points of R' can be connected by acontinuous curve consisting entirely of interiorpoints of R' ; but this follows immediately from thepossibility of doing the same for the correspondingpoints of R and from the fact that u(x, Y)and v(x, y) are continuous functions.

Remarks. 1. If the mapping function is regularon the boundary of the region as well, then theboundary points of R are mapped onto boundarypoints of R', since we would otherwise be led to acontradiction with the neighborhood-preservingcharacter of the inverse mapping (of R' on R).

2. We have proved our theorem only undercertain restrictions. We shall soon see that it holdsfor all functions that are regular except for poles,and that it also holds for infinite regions providedonly that we extend our definition of region a little;see 7.

3. It may happen that one and the same pointof the w-plane occurs both as an interior point andas a boundary point of R'; this has to do with thepossible many-valuedness of the inverse of themapping function f(z). It is not at all a foregoneconclusion that f(z) will assume everyone of itsvalues only once in R, and thus it may also happenthat it assumes one and the same value in theinterior and on the boundary of R. The point ofthe w-plane corresponding to such a value is thenan interior point as well as a boundary point of R'.

FIG. 1

4 I. FOUNDATIO!\S. LINEAR FUNCTIONS

This may at first make some trouble for one'svisualization of the geometric situation. But eversince Riemann's time, this stumbling block to anintuitive geometric grasp has been overcome satis-factorily. What is needed here can be made clearas follows. In Fig. 1, imagine a long "tongue"attached to the rectangle along AB, and let thattongue overlap the rectangle in the shaded part.This constitutes an example of a region R' of thekind we wish to consider. The point C, for in-stance, is a boundary point as well as an interior

point of R'; as a pointof the rectangle, it is aboundary point, and asa point of the tongue,it is an interior point.The reader will easilybe able to locate pointsthat occur twice as in-terior points of R'-for

example, D. To get a clear picture of things ofthat kind, it is best to make a paper model of theregion. For the time being, the simple examplewe have just given must suffice in the way of in-tuitive clarification. In the sequel, we shall call aregion simple (schlicht) if it covers no point morethan once; otherwise we shall call it non-simple.

An application of the Theorem on Preservationof Neighborhoods: If f (z) is regular in the in-

1. ANALYTIC FUNCTIONS, CONFORMAL MAPPING 5

terior of a region R, then I f (z) I can not assume amaximum in the interior of the region (Maximum-Modulus Principle).

This fact (an easy consequence, as is well known,of Cauchy's Integral Formula) can also be de-duced immediately from the Preservation-of-Neighborhoods Theorem. We need only observethat I f(z) I gives the distance of the image pointof z from the origin of the w-plane, and that anyimage point of an interior point of R is the centerof a circular disc made up entirely of image pointswhose pre-images fill up some neighborhood of theabove interior point of R in the z-plane; in par-ticular, any point of R' which might claim to bethe farthest away from the origin of the 'lv-planewould also be the center of such a disc.

The property of preserving neighborhoods isone that the mappings given by analytic functionsshare with all mappings that are continuous atevery point and one-to-one, orfinitely-many-valued,throughout the region. The additional character-istic which singles out the mappings effected byanalytic functions, and which is decisive for allour subsequent investigations, is contained in theTheorem of Isogonality, or Preservation of Angles,which we now proceed to state and prove.

An analytic mapping w == f (z) is angle-preserving (or isogonal) ; that is, if ~1 and


entiable at a and intersect there at an angle 1}, thentheir image curves ~t' and

1. ANALYTIC FUNCTIONS, CONFORMAL MAPPING 7Cauchy-Riemann differential equations.

Let the two curves be given by Z = Zl (t) andZ = Z2 ( t), and let the point Z = a correspond tothe value t = 0 of the parameter, on both curves.Also, let the assigned sense of traversal, for eachcurve, correspond to increasing t. Assume thatthe derivatives zt' (t) and Z2' (t) exist, and thatzt'(O) =F 0 and z/(O) =F O-an assumption which,as is well known, merely serves to exclude singularpoints of the curves or a poor choice of parameter.Then

z~(O){} = arg zi(O)represents the angle through which the directionof ~l at z = a must be rotated, in the positivesense, to be made coincident with that of (f2 atz = a. For if z' = reitP, with fP real and r > 0,then cp = arg z' is called the amplitude (or argu-ment) of z'. Hence if ~(O) = r1gffJ1 and ~ = r2ef,fJJa ,then

z~(O){} = arg z~() = ({J2 - ({Jlis the angle through which (;l must be rotated inthe positive sense to make its direction coincidewith that of (t2.

The equations of the image curves


and for their angle {)' we findI w;(O) f' (a,) . z~(O) z~(O){} = arg w~(O) = arg f'(a). z~(O)= arg z~(O) = {},

which proves isogonality, given that f' (a) =f= O.We note the following consequence of the Iso-

gonality Theorem. Assume a given region R tohave a differentiable boundary curve, and let Rbe mapped on a region R' of the 'lv-plane by meansof a function f(z) which is regular in the interiorand on the boundary of R. Assign a sense of tra-versal to the boundary curve

1. ANALYTIC FUNCTIONS, CONFORMAL MAPPING 9~' points into the interior of R', the region R' alsolies to the left of its (sensed) boundary ~'.

Next let us verify the fact that our function!(z), regular at Z =f= a and with non-vanishingderivative f' (a), gives a conformal mapping of aneighborhood of z === a. By this is meant that themapping gives an image which is similar in thesmall to its pre-image, i.e. not only are angles atthe point z === a preserved, but so are the ratiosof lengths of small segments near z === a; to putit precisely, at z == a \ve also have d8 1 /ds 1 ==d82 /ds 2 , where S1 and 82 denote arc-length alongQ: 1 and (I2 respectively, and 8 1 and 8 2 , arc-lengthalong crt' and ~2' respectively. To prove this, notethat ds 1 /dt === Idzt/dt I, ds2 /dt === I dz2 /dt I, etc.,whence it follows that d8 1 /ds t === dS2 /ds 2 ===I dw /dz I, Q.E.D.

Note also that d8/ds measures the ratio ofmagnification of "small" segments at a under themapping. This "scale factor" of the mappingdepends only on the location of a, and not on thedirection of the segments; it is the same for SI asfor S2.

The theorems on isogonality and isometry whichwe have just proved have converses in a certainsense; for it can be shown that all isogonal map-pings, and likewise all isometric mappings, aregiven by analytic functions or by functions closelyallied to analytic ones.


We shall go into the proof of the first of thesestatements. Using the notation of p. 1, let usassume that a given mapping u = u(x, y),v = v (x, y) preserves the angle of any pair ofcurves emanating from z = a. Here we assume,as we shall always do in the sequel whenever wespeak of mappings, that u(x, y) and v (x, y) havecontinuous first partial derivatives (are "of classCO)"). We assume here, furthermore, that thefunctional determinant ("Jacobian")

oud(u, V) oX

-d(x, y) ovox

ouoyovoy

is not equal to zero.[With u and v denoting, as usual, the real and

imaginary parts of an analytic function f (z), wehave, on account of (1) on p. 1:

au ov

d(u, v) _ ox - ox === (aU) 2 (OV) 2 === I f'(z) 12d(x, y) OV ou ox + ax '

ox oxso that the non-vanishing of the Jacobian isequivalent with the non-vanishing of f' (z).J

Now a curve x=x(t), y=y(t) is mappedontou=u(x(t), y(t,v=v(x(t), y(t.Thecomponents of the tangent vector to the imagecurve are then given by

(3)

1. ANALYTIC FUNCTIONS, CONFORMAL MAPPING 11

ou ouu'(t) = OX x'(t) + oy y'(t)

V'(t) = ~; X'(t) + ~ y'(t).At a fixed point a of the z-plane, the partial

derivatives of u and v have fixed values. Thus (3)represents a linear transformation to which thegiven mapping w = f(z) subjects the tangentvectors to curves in the z-plane at the point a,transforming these tangent vectors into those tothe image curves in the w-plane. But as is wellknown, such a linear transformation preservesangles if and only if it is a similarity transforma-tion. As we know from Analytic GeometrY,l anecessary and sufficient condition for this is repre-sented, in our case, precisely by the Cauchy-Riemann differential equations (1) on p. 1. Butas is shown in Function Theory, these imply thatu + i v is an analytic function of z. (Cf. Bieber-bach, Lehrbuch der Funktionentheorie, Vol. I,p. 39.) We thus have proved the following result:

Every isogonal mapping of a region is repre-sented by an analytic function.

1 In Analytic Geometry, where integral linear transform-ations of the cartesian coordinates are called affine trans-formations, it is shown that the only angle-preserving affinetransformations are the similarity transformations (simili-tudes) .


Let us now go back to our definition of con-formal mapping; its requirement of "similitude inthe small" includes isogonality, as one part. Letus relax this requirement to the extent that onlythe magnitude, but not necessarily the sense ofrotation, of every angle is to be preserved. A con-formal mapping which preserves the sense ofrotation will be called strictly conformal, and onewhich reverses the sense of every angle will becalled anti-conformal. We may then ask whetheror not every conformal mapping is given by ananalytic function, or-which by our last resultamounts to the same-whether or not every con-formal mapping is strictly conformal.

A simple example shows that the answer is no.If z denotes the complex conjugate of z, thenw = z represents a mapping of the z-plane ontothe w-plane. In geometric terms, this mapping isa reflection in the real axis; that is, if we accom-modate z and w both in the same plane in such away that equal values of z and w correspond tothe same point, then the mapping sends everypoint of the z-plane into its mirror image withrespect to the real axis. This mapping is evidentlyconformal; the length of any curve is the same asthat of its image curve, and the magnitude of everyangle is preserved. However, the sense of rotationof every angle is reversed under the mapping.Thus the mapping is anti-conformal. We can

2. INTEGRAL LINEAR FUNCTIONS 13obtain additional examples of this kind by com-bining any given strictly conformal mapping withthe mapping just considered.

We may still ask whether every conformal map-ping is either a strictly conformal mapping or elsea combination of a strictly conformal mappingwith the above reflection (and therefore anti-conformal). The answer here is yes, as can beshown by an argument very similar to the oneused in proving our last isogonality result, andwhich we omit here for that reason. Hence everyconformal mapping is represented either byw = f(z) (strictly conformal mapping) or byW= f(z) (anti-conformal mapping), where f(z)is an analytic function. Every conformal mappingpreserves the magnitude, and either preserves orreverses the sense of rotation, of every angle.

In what follows we shall be interested only inconformal mappings that are strictly conformal,and that are therefore represented by analyticfunctions.

2. Integral Linear FunctionsThe simplest example to illustrate the general

discussion of 1 is furnished by the integral linearfunctions w = az + b . We can distinguish severaltypes among these, as follows:

1. w = z + b. If we interpret wand z as


points in the same plane, then this mapping, geo-metrically interpreted, is a translation. For, asis well known, to every complex number therecorresponds a vector, and the addition of complexnumbers then corresponds to vector addition.Therefore the translation must be one in thedirection of the vector b, and the magnitude ofthe translation is the length of b. Thus any givenregion R is mapped onto a congruent region whichis obtained from R by a translation.

2. w == e;rp z represents a rotation of the planethrough the angle C{J, about the fixed center ofrotation z == O.

3. w == rz, with r > 0, represents a similaritytransformation (magnification in the ratio r: 1) .

4. The most general integral linear transform-ation w == az + b can be built up step by stepfrom the three types just considered. We set

This shows how the given transformation isbuilt up from the above three types. Anothermethod, just as simple as the one just used, willgive us an even better insight into the geometricnature of the mapping w == az + b: Observethat it can always be brought into the formW -

3. THE FUNCTION W == liz 15tegral linear transformation represents either atranslation (a === 1), or a "rotation with may,tti-jication" (a =f= 1), which reduces to a pure rota-tion if r === 1 and to a pure magnification if cp === o.

3. The Function w === lizThe discussion of this function offers no par-

ticular difficulty, at least at all those points atwhich neither z nor w become infinite, that is tosay, at all finite points of the z-plane other thanz = o. The point z === 0 itself is not covered bythe general investigations of 1 ; thus if we nowinclude that point in the discussion of our func-tion, we shall at the same time be supplementingthe material of 1 in a special case.

We shall find it useful to introduce polar co-ordinates, by setting z == reirp , W = eeifJ Then ourmapping is expressed by e === 11r, {) === - cp. Thiswill give us a clear geometric picture of the map-ping, as follows. Let us once more locate z and win one and the same plane. The points for whichr === 1 obviously playa special role in the mapping.These points make up a circle! whose radius isunity and whose center is at z === 0, the unit circleas we shall henceforth call it for short; and thiscircle is mapped onto itself under our mapping

1 By circle we shall always mean the periphery of a cir-cular disc.


FIG. 2

1V === 1/z. The point r === 1, cp === cpo is mapped onthe point e === 1, {) === - cpo, Le. on the point whichis obtained from the first one by "reflecting" theunit circle in the real axis. By the "real axis"we mean the line y === 0 (z === x + iy), and by

LJ reflection in this axis we mean thepassing from any given point to

Are---r--YC its symmetric image,2 or in other~ words, the passing from x + iy

A I1 to x - i y . As has already beenmentioned in 1, such a mappingpreserves only the magnitude ofangles but not 3 their sense of ro-tation, as can be seen from Fig. 2.

Let us go on to the consideration of arbitrary z.Clearly, the mapping IW === l/z is one-to-one (ex-cepting the cases z === 0 and w === 0, which will bediscussed later); to every z there correspondsexactly one w, and vice versa. To obtain a clearover-all picture of the mapping, it is useful todecompose it into the following two mappingswhich are to be applied consecutively:

1I r 1 = 1, {)1 = - cp; II e= -, {} = {)1 .11

The first of these is simply the reflection in the

2 It is clear from this what will be meant by reflection inany arbitrary straight line.

aWe speak of a "reversal" of angles in such cases.

3. THE FUNCTION UJ == liz 17

real axis, discussed above, and is therefore ananti-conformal mapping (Le., conformal mappingwhich reverses angles). Hence the second trans-formation, e == 1/r1 , {) === {)t, called an inversionin the unit circle, must also be an anti-conformalmapping, since the combination of the two map-pings is a strictly conformal mapping.

Let us investigate the inversion in the unit circlein more detail. We see first that it maps any pointr < 1, Le. any point of the interior of the unitcircle, onto a point with (J > 1, Le. onto a pointof the exterior of the circle; and vice versa, itmaps the exterior of the unit circle into its in-terior. The points of the unit circle itself are eachleft fixed. Thus inversion in a circle interchangesthe interior and exterior just as reflection in astraight line interchanges the two sides (half-planes) of the line each of whose points remainsfixed. For this reason, the mapping e== l/r,{) == cp is also called a reflection in the unit circle.Another, and deeper, reason for this terminologywill be brought out later in this book. How to findthe image under inversion of any given point ismade immediately apparent by recalling a fami-liar theorem on right triangles. Draw the half-line from 0 through P; the image point PI of Pmust lie on the same line, since {) == cp. Nowif P lies, say, in the interior of the unit circle,then we draw a perpendicular at P on the radius

FIG. 3


through P, intersecting the circle at T and T', andat these two points we draw the tangents to thecircle. PI is then the point of intersection of thesetwo tangent lines. If, on the other hand, PI is thegiven point, then its image P is found by simplycarrying ('ut the construction we have just

described in the reverse direction(see Fig. 3). The validity of the

, construction rests on the theoremreferred to above, according towhich 1 === OT2 === Op OP I

The above also makes obviouswhat is to be meant by a reflection in a circle ofradius R about z === 0, namely, the mapping re ===R2, {) = cp.

The geometric meaning of w === liz is nowclear: The given point is to be reflected in the unitcircle as well as in the real axis in order to arriveat its image point (Inversion plus Reflection).

The mapping is one-to-one, except for the pointz === 0 to which there does not correspond anyimage point-at the moment, for we shall pres-ently remove this exception-and except for w === 0,which is not-at the moment-to be found amongthe image points. Now we observe, however, thatthe exterior of any circle about z === 0 as centeris mapped into the interior of a circle about w === 0,and that the latter shrinks down to w === 0 as theradius of the former is made to increase indefin-

3. THE FUNCTION W == liz 19itely. It is just as though there were a point inthe z-plane which is outside every circle aboutz = 0 and is mapped onto w = 0, and as thoughthere were a point in the 1v-plane which is outsideevery circle about w = 0 and onto which z = 0 ismapped. The reader may be familiar with asimilar state of affairs in Projective Geometry,where one introduces an "improper" or "ideal"straight line, also called the "line at infinity." Inour present case, of inversion plus reflection, weintroduce a single improper point which we denoteby z = 00 (or w = 00, respectively). We shallalso speak of it as the point at infinity. Thenw = 00 is the image of z = 0 under our mapping,and z = 00 is the pre-image of w = o. With thisagreement, we have made w = liz a one-to-onemapping, without any exceptions.

Our mapping by reciprocals (Le., inversion plusreflection) is of great fundamental importance.For, just as one uses collineations in ProjectiveGeometry in order to study the behavior of curvesat infinity, so one uses the mapping by reciprocalsin Function Theory in order to study the behaviorof a function at infinity. We call a function f(z)regular at (the point at) infinity if f (l I w ) isregular at w = 0, so that it can be expanded inpowers of w in a neighborhood of w = o. Thusa function regular at z = 00 can be expanded inpowers of liz, and such an expansion will be valid


in some neighborhood of z~ 00, Le. in a region ofthe z-plane-such as the exterior of a circle aboutz~ O-which is the pre-image under w ~ liz ofa neighborhood of w ~ o. The mapping I(z) iscalled isogonal at z~ 00 if I(l/w) is isogonal at\V~ O. The angle formed by two curves at z~ 00is defined to be the angle at which their imagecurves under w == liz intersect in the w-planeat w ~ O.

The function w ~ liz is also a useful tool whenit comes to investigate the points at which a givenfunction w ~ I(z) becomes infinite. If I(z) doesnot remain bounded in the neighborhood of, say,z ~ a, then we consider instead of I (z) the func-tion 111 (z) ; if the latter is one-valued and boundedin a neighborhood of z == a, we can write down anexpansion of the form

1I I (z) ~ (z - a) 11 ( ao+ a1 (z - a) +...),with ao =f= o. Hence we obtain

f(z) = (z~ a)n (~+ b1(z - a) + .JWe call z == a, in this case, a non-essential singu-larity, or a pole, of I (z). If n ~ 1, then the map-ping represented by I (z) is isogonal at z ~ a, inaccordance with our agreements above.

ApPENDIX TO 3: STEREOGRAPHIC PROJECTION 21

Appendix to 3: Stereographic ProjectionIt is often useful to help logical considerations

along by illustrating them, if possible, by meansof intuitive or pictorial devices. Thus we will gainby illustrating the introduction of the point atinfinity by means of a model which is entirelyin the finite domain. The addition of the point atinfinity entails the possibility of mapping the planeone-to-one and isogonally onto (the surface of) asphere. This is done by a mapping called stereo-graphic projection. We take a sphere of diameterunity and lay it on the plane in such a way that itslowest point coincides with z == o. This lowestpoint we call the south pole, and the diametricallyopposite point we call the north pole. With thenorth pole as center of projection, we now projectthe plane onto the sphere. The points of the sphereand the points of the plane are thereby put in aone-to-one correspondence, under which the southpole, for instance, corresponds to z == 0, while thenorth pole corresponds to the point at infinity ofthe plane. A short argument will now show thatthis mapping is isogonal.

By the angle between two plane curves is meantthe angle between their tangents at their point ofintersection; by the angle between two curves onthe sphere is meant the angle between the tangentsto the sphere that are also tangent to the curves


at their point of intersection. Now let [1 and ~2be two curves in the plane that intersect at P, andlet ~t' and (;/ be their spherical images (understereographic projection), intersecting at theimage P' of P. Let us pass two planes through theprojecting ray P P', containing the tangents atP to ~1 and to ~2 respectively; these planes clearlycontain also the tangents to the sphere that aretangent at P' to [t' and (f/ respectively. The twolast-mentioned tangents, in turn, lie in the tangentplane to the sphere at P'. Now pass a meridian

t plane of the sphere (Le., aplane containing both thenorth and the south poles)throu.'5h P P'- this is theplane in which Fig. 4 is

e drawn. In Fig. 4, t is theFIG. 4 trace of the tangent plane,

e is the trace of the z-plane, s is the projecting ray,M is the center of the sphere, and N is the northpole. If we consider the dotted lines drawn inFig. 4 and recall certain familiar theorems ofelementary geometry, we see that t and e formthe same angle a with s . Now the two planes whichwe passed through s are seen to be intersected bytwo planes, through e and t respectively (viz., thez-plane and the tangent plane at P'), that can beobtained from each other by a reflection in theplane of the perpendicular bisectors of P P'. There-

4. LINEAR FUNCTIONS 23

fore the two pairs of lines of intersection-viz.,the two pairs of tangents (at P and at P')-formequal angles, and we have proved that stereo-graphic projection is isogonal at any finite point P;finally, the isogonality of the mapping at the pointat infinity, whose image is the north pole, followsfrom our eonvention on how to measure angles atinfinity.

4. Linear FunctionsIt would seem natural to begin the investigation

of the linear (more properly: fractional linear)

function w = ::t ~ by dividing the denominatorinto the numerator, which in the case c =F 0 yields

a bc- adw = c+ (cz+ d)c

And then it would be easy to represent our func-tion as built up from four simple types of func-tions such as were discussed in 2 and 3. Butsuch a procedure would make the further studyof the linear function somewhat laborious, and forthis reason we shall prefer a different approach.Let us, however, note the following corollary toour initial calculation:


THEOREM I. The linear function

w=== (az + b)/(cz + d)is non-constant if and only if the determinantad - bc does not vanish.

We shall always assume this condition to besatisfied in the sequel. The linear function then

has an inverse, which we calculate as z = dw ~b .-cw a

We deduce from this the following:THEOREM II. Every (non-constant) linear

function represents a one-to-one mapping of theplane onto itself, and this mapping is isogonal atevery point (including z === 00).

That this holds at z === 00 follows from firstsubstituting z === 1/3 and then noting that

(~t ~) =w is regular at 3 = 0, except ifc === 0, and that (ddW ) = be~ ad does not vanish.

3 3=0 cThis being so, we are justified in saying that w isisogonal at z === 00, in accordance with our agree-ment of 3. But if c === 0, we consider d/ (az + b)at z === 00, in accordance with 3. Finally, theisogonality at z === - d/c follows immediatelyfrom the fact that : t ~ is isogonal at this point.

Let us introduce a few abbreviations. We shall


use 8 to stand for any linear function, whose in-verse-that we have just seen how to calculateabove-we shall then denote by 8-I, as is usualin algebra. The following further result is nowalmost immediate:

THEOREM III: The composition of any numberof non-constant linear functions always leads tofurther non-constant linear functions.

In proof, let 8 1 === II (z) and 8 2 = l2 (z) ; then8 1 8 2 stands for l1 (l2 (z) ). The inverse of this lastis 821 8 t 1 The determinant of 8 18 2 is the productof the determinants of 8 1 and 8 2 , and can notvanish since neither of the factors vanishes.

THEOREM IV: If the z-plane is mapped ontothe w-plane by means of a non-constant linearfunction, then the totality of straight lines andcircles of the z-plane is mapped onto the totalityof straight lines and circles of the w-plane.

In proof, note first that the equations of linesand circles can always be written as follows inrectangular coordinates: a z z + fJ z + f3 z + y === 0,where a and yare real, fJ and f3 are complex con-jugates, z === x + iy, and z === x - iy. Theorem IVcan then be easily verified by simply going throughthe actual calculations. Under such a mapping,as the calculations would show, a straight linemay very well be mapped onto a circle, but neveronto any other conic section nor, say, onto a curve


of the third order. For example, w = liz mapsany circle through z = 0 onto a circle throughz = 00, Le. onto a straight line; in particular, itmaps any straight line through z = 00 and z = 0onto a straight line of the w-plane.

THEOREM V: Given any three distinct pointsa, f3, y of the z-plane, and any three distinct pointsa', fJ', y' of the w-plane, there al1.vays exists a suit-able linear function 10hich 1r~ap8 a, (3, y onto a', fJ', y'respectively, i.e. which maps the first triple ofpoints onto the second triple in a given order.Furthermore, the function which accomplishes thismapping is thereby uniquely determined.

COROLLARY. Since three points determine acircle and since by Theorem IV, circles (includingstraight lines) are mapped on circles (or lines),Theorem V may also be given the following geo-metric interpretation: Any given circle can bemapped conformally onto any other circle in sucha way that any three given points of the first circleare mapped onto any three given points of thesecond.

Proof of Theorem V. A function such as thetheorem requires can obviously be obtained byelimination of 3 from

Z-(X fJ-y&= z-y ~-()(, and

I f3' IW'-lX -y~ == ( R' I w-y p-C(.


For, the above maps z === u, (J, y on 3 === 0, 1, 00,and maps 1V === a', fJ', y' on "3 === 0, 1, 00. It is alsoeasy to show that the function thus constructed isthe only one which satisfies the conditions of thetheorem, as follows: If there were two differentsuch functions, say 8 1 and 8 2 , then 8'2181 wouldleave fixed the three points a, ~, r. But then thelast mapping \vould have to leave all points fixed,and 8 1 and 8 2 could not be distinct. To prove thislast statement, let us assume that

-1 _ _ az+ b82 81 = W - -I dcz -

leaves fixed the three points a, f3, y; then thequadratic equation

az+ bz==--

ez+ dor

z2c+ z(d-a)-b=Omust have the three solutions a, f3, y. But thenall the coefficients of the quadratic equation mustvanish, by an elementary theorem of algebra.This gives b === 0, c = 0, a === d. Hence w = z isthe only linear function which leaves more thanthree points fixed. We have thus obtained thefollowing further result:

THEOREM VI. Every linear function other thanw === z leaves at most two points fixed.

28 I. FOUND.4TIONS. LINEAR FUNCTIONS

To find the coordinates of these fixed points,as we shall call them, we go back to the abovequadratic equation, from which we find, in caseC=f=O,

a-dV(a-d)2-r 4bc(1) z = 2cThe two fixed points coincide if

(a-d)2+ 4bc=O.If C === 0, we are dealing with an integral linearfunction, which leaves z === 00 fixed, and whosefinite fixed point is

z===b/(d-a).Now to begin a more detailed study, let us first

investigate those linear functions that have tUJOdistinct finite fixed points. Let Zl and zz, then, be

. az+ bthe two fixed pOInts of w = cz + d - S, where Zl

corresponds to, say, the upper sign in (1). Inorder to be better able to visualize what is goingon, \ve shall again interpret z and w as points inone and the same plane. We shall also use anauxiliary plane in which we accommodate thevariables 10 and 3, defined by

w-z""_ 1..., - ,

w -z"..

The linear function LSL-t, which expresses m in


terms of 3, has 0 and 00 as its fixed points, andmust therefore be of the form w === a 0. This im-plies that S itself may be written in the form

w-z] ==eX Z-Zl (Normal Form in the case ofW-Z2 Z-Z2

two distinct finite fixed points). In order to deter-mine the value of a in terms of the original co-

efficients of S, note that

whence a short calculation yields

a+ d+ V(a-d)2+ 4bceX=-----:.,:===============_

a+ d-V(a-d)2+ 4bc

The relations we have just discussed make itpossible for us to restrict ourselves, at least tobegin with, to the function tv === a 3, since \ve canalways pass from this to the general linear func-tion under discussion by making the substitutions

Three distinct cases now present themselves:1. If a is a positive real number, our linear func-tion is said to be hyperbolic; 2. if eX == eiw (and anot positive), the function is said to be elliptic;3. all other linear functions with two finite fixedpoints, and eX === eeiw , are said to be loxodromic.


The geometric meaning of these mappings iseasily understood in terms of l11 and 3, recallingthat l11 ~ a3. The hyperbolic mappings are mag-nifications, the elliptic ones are rotations, and theloxodromic ones are a combination-referred toon p. 15 above as "rotation plus magnification"-of the first two types. Let us scrutinize the firsttwo types a little more closely. For these, a specialrole is played on the one hand by the system ofstraight lines through 3~ 0, and on the other handby the system of circles about 3~ 0 as center, asthese two families of curves are mapped onto them-selves by the two types of mappings. In particular,any hyperbolic function maps each of the abovestraight lines onto itself while permuting theabove circles among themselves; whereas anyelliptic function maps each of the circles ontoitself while permuting the straight lines amongthemselves. It only remains to locate the familiesof circles which take the place of the above twofamilies when we return to the general case oftwo arbitrary fixed points instead of the spe-cial fixed points 3~ 0 and UJ === 0 that we havejust considered, Le. when we return from theauxiliary variables 0 and l11 to the original vari-ables z and w. The desired families are clearlythose that are obtained from the above two by the

. w-z z-zmappIng tt>===w_z1 ,5==z_Zl in the z, w-plane.

2 2

~ 4. LINEAR FUNCTIONS 31

They are the system of circles through the twofixed points and the system of orthogonal trajec-tories of these circles (see Fig. 5).

FIG. 5

In the case of loxodromic mappings, systems ofcircles playing roles as described above do notoccur unless w == 1t.

The circles that remain fixed individually aresometimes called path curves or trajectories, andthose that are permuted among themselves, levelcurves of the linear function. We shall explainthe reason for this terminology, which is obviouslyborrowed from kinematics, in the case of tu == a 13.If tu == a3 is any similarity transformation witha =F 1, then by using a real parameter t we canwrite all hyperbolic substitutions in the form


y' === aty. Similarly, \\Te can obtain all rotationsfrom one given rotation. If we interpret t as time,we can see that every point moves along its tra-jectory as time passes, and that the level curvesare changed into each other.

Similarly, we can generate a whole family ofmappings out of a single loxodromic mapping;ho\vever, not all members of such a family willbe loxodromic. If we folIo,v the path of a pointas time goes on in this case, we obtain spirals bothfor the trajectories and for the level curves.

Next, let us study the linear mappings that haveonly one fixed point Zl ; for these, as they constitutea limiting case, we shall use the name paraholicmappings. Once more \\Te interpret z and 1V aspoints in one and the same plane, and pass to anauxiliary plane by means of the substitution

1 1to == tv _ Z ' 5== Z _ Z This yields \v === 0 + f3,

1 1

whence we obtain 1 = 1 + fJ as theW-Z1 Z-Zl

Normal Form for parabolic mappings. We alsosee that the special case \u === 3 + (3 just consideredbelongs to the translations, discussed earlier.Under this translation, every straight line that isparallel to the direction 0 (3 is mapped onto itself,while their orthogonal trajectories form a secondsystem of (parallel) lines, and these are permutedamong themselves by the translation. In the

FIG. 6

5. LINEAR FUNCTIONS (continued) 33Z, w-plane, the role of the two systems of linesjust discussed is taken overby two systems of circlespassing through Zl. Thecircles of each system, be-ing conformal images of asystem of parallel lines,have a common direction(Le. a common tangent) atZt. The fixed point Zl is theintersection of two mutu-ally perpendicular straight lines, each of which istangent at Zt to all the circles of one of the twosystems of circles (see Fig. 6).

5. Linear Fllnctions (continued)We shall investigate next all those linear func-

tions that correspond, under stereographic projec-tion, to rotations of the sphere. Any rotation ofthe sphere maps the (surface of the) sphere iso-gonally onto itself. Since, furthermore, stereo-graphic projection maps the sphere isogonallyonto the plane, it follows that the rotations of thesphere must correspond to one-to-one isogonalmappings of the plane onto itself. In this con-nection, the following theorem holds:

Every one-to-one isogonal mapping of the planeonJo itself is a linear 1napping.


For, according to the results in 1, every suchmapping is represented by an analytic function.Let w == f (z) be this function; either it has z == 00as a fixed point, or it maps z == 00 onto a finitepoint w == a. In the latter case, we form the func-

tion 3= fI 1 , which represents a one-to-one(z)-aconformal mapping (from z to 3) that leaves 00fixed. Since it assumes each value only once, itcan not come arbitrarily close to every value inthe neighborhood of z == 00, and must thereforehave a pole at infinity. But since it is regular inthe entire finite z-plane, it must be an integralrational function, and its degree must be unity,since it assumes no value more than once.

We are now ready to determine all the linearmappings that correspond to rotations of thesphere. Every such mapping must have two fixed

jJT points, namely the twopoints in the plane thatcorrespond to the twointersections of thesphere with the axis ofrotation. Where arethese two fixed points?

Let PI and P 2 be the two intersections just referredto, and let N be the north pole of the sphere. Thesethree points determine a meridian on the sphere,in whose plane Fig. 7 is drawn. The segments R I

5. LINEAR FUNCTIONS (continued) 35and Rz are located on the intersection of thismeridian plane with the z-plane. Since PIP2 is adiameter (of length unity), the angle at N is aright angle. Therefore R 1R 2 === 1, by a familiartheorem on right triangles. From this it followsthat if a is one of the fixed points we are seeking,then the other one must be - l/a. If we observefurther that the trajectories of the rotation of thesphere (or more precisely: of the linear mappingthat corresponds to that rotation) are the stereo-graphic images of the circles on the sphere thatare cut out by planes perpendicular to the axis ofrotation, we see that rotations of the sphere giverise to elliptic mappings and that, vice versa, allelliptic mappings with fixed points as describedabove correspond to rotations of the sphere. Usingthese facts and solving for w in the normal formfor elliptic mappings (cf. the preceding section),we find the following general form for the linearfunctions corresponding to rotations of the sphere:

az+ bW=

-bz+ a

We note that this formula could also be derivedimmediately from the fact that a rotation of thesphere sends any pair of diametrically oppositepoints of the sphere into another such pair.


As the second example to be worked out, wenow choose the linear mappings of a circular disconto itself.

1. As we know from the Corollary to Theorem Von p. 26, any given circle can be mapped onto anyother circle by a linear function, and what is more,it is also possible in doing this to prescribe themapping of any three given points of the givencircle. In particular, we can map any given circleonto the real axis (Le., onto the "circle" through00, 0, 1). Under such a mapping, the interior ofthe circle must be mapped either into the upperhalf-plane y > 0 or into the lower half-planey < 0 (z === x + iy), since conformal mappingspreserve neighborhoods. We can always arrangefor the upper half-plane, say, to be the image ofthe interior of the circle, since if necessary wecan always use w === liz to interchange the twohalf-planes.

2. Let us no\v consider, in particular, thoselinear functions which map the upper half-planeonto itself. To obtain the most general linearfunction that maps the real axis onto itself, wewrite down that function which maps the threearbitrary real points a, p, y onto the three points0, 1, 00, using the appropriate formula on p. 26above.

This yields(1)

5. LINEAR FlTNCTIONS (continued) 37

z-(X fJ-yw- 0--

- z-y p -(X This is evidently 1 the most general linear functionW = az + b with real coefficients. All of these

cz+ dfunctions map the real axis onto itself, but not allof them map the upper half-plane onto itself; theymay interchange the two half-planes, as does, forinstance, w = liz. How can we distinguish be-tween the two cases? We shall show that the upperhalf-plane is mapped onto itself if and only ifad - b c > 0; indeed, this follows from the repre-sentation (1) of our functions. For, the deter-minant of (1) equals (fJ-y) (3-a) (a-y).Now for the upper half-plane to be mapped ontoitself, it is necessary and sufficient that the orderof a,;3, y agree with that of 0,1, 00; but this im-plies that the value of the determinant is positive,and vice versa. We thus have the following theo-rem: All linear conformal mappings of the 'upper

half-plane onto itself are given by w~ az + bcz+ d

with real coefficients and with ad - be> 0.(This is of course not the most general way of

1 By p. 26 above, every linear mapping is determineduniquely by the specification of the images of any threegiven points. Therefore every real linear mapping can bewritten in the form (1), with real (, {J, y


writing such mappings; we may, for instance,multiply numerator and denominator by commonfactors, and these may be non-real complexnumbers.)

3. Since we have now learned how to find alllinear mappings of a given circular disc onto ahalf-plane, we can also solve the problem of find-ing all linear mappings of any given circular disconto itself. We shall only note down the result forthe circular disc of radius unity with center atz === 0, for whose mappings onto itself we obtain

az -+- b -w = ,aa - bb > 0 .

bz+ aThe fixed points of a linear function that maps

the upper half-plane onto itself are either real orcomplex conjugates, as can be seen from (1) onp. 28. If the fixed points are real, the mapping iseither hyperbolic or parabolic or loxodromic withnegative multiplier a; if they are complex con-jugates, the nlapping is elliptic (cf. the calcula-tion of a and of the fixed points on pp. 28 and 29) .Hence a loxodromic mapping whose multiplier isnon-real can never map a circle onto itself.

In order to obtain similar information concern-ing the location of the fixed points in the case offunctions that map the interior of a circle intoitself, we need only find out what happens to apair of points symmetric with respect to the realaxis when the upper half-plane is mapped onto a

5. LINEAR FUNCTIONS (continued) 39circular disc. The following general theorem con-tains the answer:

If a linear function maps one circular disc ontoanother, it maps any pa.ir of points related byinversion in the first circle onto a pair similarlyrelated with respect to the second circle.

This follows from the following remark: If acircle K' is passed through two points P and Qthat are mutually inverse with respect to a circleK, then K' and K intersect at right angles. Forif K is a straight line, then the center of K' mustlie on K, while if K' has radius R, we may firstdraw the tangents to K' through the center M ofK ; the square of their length is = IMP I IM Q I,by a well-known theorem of elementary geometry.This last expression, however, has the value R2,since P and Q are inverses with respect to K.Thus the tangents to K' from M are of length R,and therefore their points of tangency to K' arethe points of intersection of K and K', whence K'and K intersect at right angles. Vice versa, thesame theorem of elementary geometry that wasjust used implies that every circle K' perpendicularto K consists entirely of pairs of points mutuallyinverse with respect to K. Because of the isogon-ality of linear mappings, they map any circle per-pendicular to K onto a circle perpendicular to theimage of K, and since any pair of points mutually


inverse with respect to K lies on a circle perpen-dicular to K, the points of the image pair mustbe mutually inverse with respect to the image ofK, which is what we wished to prove.

4. The solution, indicated above under 3., ofthe problem of determining all linear mappingsof a circular disc onto itself, acquires an evengreater importance through the fact, to be provedpresently, that there are no other one-to-one con-formal mappings of a circular disc onto itself.

To prove this, it will suffice to prove that allone-to-one conformal mappings of the circulardisc I z I < 1 that leave its center z~ 0 fixed,are linear. For, any other given point of this disccan be mapped onto z~ 0 by means of a suitablelinear mapping of I z I < 1 onto itself, e.g. bymeans of a suitable hyperbolic function whose twofixed points are the end-points of the diameter onwhich the given point lies. For the class of func-tions that leave z~ 0 fixed, we shall base the proofon the following lemma.

Schwarz' Lemma. Let f(z) ~ a1z + a 2 z2 + ...be convergent for I z I < 1, and let I f (z) I < 1 forI z I < 1. T hen I f (z) I < I z I for all I z I < 1,with the equality sign not holding for any I z I < 1unless f (z) === exz (a real).

To prove Schwarz' Lemma, note first thatf(z)- === al + a2z+ likewise converges for Iz 1

5. LINEAR FUNCTIONS (continued) 41Hence in accordance with the maximum-modulus

principle mentioned on p. 5, the function fez)z

can not have a maximum of its modulus occurringin the interior of the circular disc I z I < e < 1,

whence f (z) < 1 for I z I < (). This holds forz -e - ~

every fixed z and any e < 1 that satisfies I z I < e.

But this implies that f~) I < 1 for every I z I < 1,i.e. that I f(z) I

42 1. FOUNDATIONS. LINEAR FUNCTIONS

everywhere, and therefore f(z) == rf-rxz everywhere,\vhich completes the proof of Schwarz' Lemma.

From this we now derive the following in shortorder:

All one-to-one conformal mappings of the in-terior of the unit circle onto itself are linear.

For if u; == f(z) is such a mapping that inaddition leaves z == 0 fixed, then according toSchwarz' Lemma, the mapping sends every pointinto an image point that is at least as close to theorigin z == 0 as is the original point, and the sameholds of course for the inverse mapping. These twofacts are compatible only if the mapping does notchange the distance from z == 0 of any point inI z I < 1. But then, again by Schwarz' Lemma,we must have w == f(z) == eitXz , which is a linearfunction, and the proof is through in case z === 0\vas fixed under the mapping. In case f I z) doesnot leave z === 0 fixed, then a suitable linear func-tion of f (z) will do so (cf. the remark precedingSchwarz' Lemma), and our proof is finished.

Remark. The hypothesis of "one-to-one"-nessis essential to the validity of the theorem justproved, as the example w === Z2 (which maps theunit circle onto itself) shows.

Exercises. 1. Given two circular annuli, thefirst formed by two eccentric circles and thesecond by two concentric ones; find a hyperbolic

6. GROUPS OF LINEAR FUNCTIONS 43

linear function that maps the first annulus ontothe second.

2. Find the most general "triangle" formed bycircular arcs that is mapped onto an ordinary(straight-line) triangle by 1V === 1/ (z - a).

3. Find a function that maps a crescent, formedby two mutually tangent circles, onto an infinitestrip bounded by two parallel straight lines.

6. Groups of Linear FunctionsBy a group of linear functions is meant a set of

linear functions such that the composition 8 1 8 2of any two functions 8 1 and 8 2 of the set is itselfan element 8 3 of the set, and such that the set alsocontains the inverse function 8- 1 of any function8 that belongs to the set. (Cf. the notation intro-duced on p. 25.)

We shall determine a fundamental region ofsuch a group. By a fundamental region is meanta region of the following sort: If all the mappingscontained in the group are applied, one after theother, to such a region, then the totality of imageregions thus obtained should constitute a simplecovering either of the whole plane or of a partthereof, and the region should not be a proper sub-region of a larger one that also has the coveringproperty just described.

A few examples will serve to illustrate these

2(h1 +ha)inz" == e n Z


definitions. Consider, for instance, a group ofrotations about the point z === o. Let the mappingscontained in this group be the following ones:

2hin-

z' === e n Z (h == 1, 2, ... n), where n is an integer.Thus the group consists of the rotations aboutz === 0 through the angle 2rr/n and the repetitionsof this rotation. We see immediately that the

2h1incomposition of two rotations z' == e n Z and

2h.in

z" == e n z' of the group yields the rotation

of the group. Furthermore, the2hin 2(n - h)in

rotation inverse to z' == e n Z is z == e n z',which is itself contained in the group. As a funda-mental region of this group we may take, say,the sector bounded by two rays emanating fromz === 0 that form the angle 2Jt/n at z === 0, one ofthe two bounding rays being included in theregion. For if all rotations of our group areapplied to this region, we obtain a complete cover-ing of the whole plane by n sectors. Or in otherwords: Every point of the plane can be mapped,by a suitable rotation of the group, onto a pointof the fundamental region, so that for everygiven point of the plane the fundamental regioncontains exactly one corresponding point (pro-vided only that one of the bounding half-lines

6. GROUPS OF LINEAR FUNCTIONS 45

is counted as belonging to the region while theother one is not, as was specified above). If werestricted ourselves to that part of the sector thatlies inside the unit circle, we would no longer havea fundamental region, even though the rotationsof the group applied to this part would lead to acovering of a portion of the plane (viz., of theunit circle) by congruent circular sectors; and thereason is simply that a region properly containingthe finite sector (viz., the whole infinite sector)also gives rise to a simple covering. Nor can weuse a sector with double the above angle at z === 0as a fundamental region, since the rotations ofthe group applied to such a region would yield acovering of the plane which, to be sure, is com-plete, but which is a double instead of a simplecovering. This much must suffice here in the wayof an explanation of our definition.

We note further that the fundamental regionof a group is by no means uniquely determined bythe group. We can find quite diverse fundamentalregions belonging to one and the same group.Above, for instance, we can replace the angularsector bounded by two straight half-lines with asector bounded by any two curves that lead fromzero to infinity without self-intersections and suchthat one of them is obtained from the other by the

rotation z' = e n Z


We also note that not every group of linearmappings need have a fundamental region; forexample, the group of all rotations about the pointz === 0 does not have a fundamental region, nordoes the group of all those linear functions thathave a non-vanishing determinant, as we shalldeduce from the following remark: A funda-mental region, by its very definition, can not con-tain two points one of which is obtainable fromthe other by a mapping belonging to the group,for this would contradict the requirement ofobtaining a simple covering. Now in the aboveexamples, any given point can be moved to adifferent one as close to the given one as we please,by a suitable mapping in the group. Thus thefundamental region could not contain any interiorpoints, since an interior point would have to bethe center of some circular disc that contains noimages of P under any mappings in the group.

The above considerations contain a necessarycondition for a group to have a fundamentalregion, namely that there should be regions con-taining no pair of points one of which is the imageof the other under some mapping in the group.This condition also turns out to be sufficient, ascan be seen by enlarging as much as possible someinitial region which is free of pairs of points ofthe kind just described. We shall not carrythrough the details of such a construction, as this

6. GROUPS OF LINEAR FUNCTIONS 47would lead us too far afield here. We shall, how-ever, give a few more examples of groups andtheir fundamental regions:

1. The group of mappings 1V === Z + h, whereh is an integer, has as a fundamental region astrip of width unity, bounded, say, by two parallelsto the imaginary axis.

2. z' == Z + 2h1 + 2h2 W (where h 1 and h'2 areintegers and w is a non-real complex number).As a fundamental region we may take a parallelo-gram two of whose sides are the vectors joiningthe origin to 1 and w.

1 2in3. z' == -, z' == e It z and their composite map-z

pings, with n an integer. This is a so-calleddihedral group. A fundamental region is the cir-

cular sector with its vertices at the origin, at eft

and at en; the boundary is made up of the arcof the unit circle through z === 1 that connects thelast two vertices, and of the two radii from z === 0

in into e1& and to e n

4. The groups of rotations of the other regularsolids. Note that under stereographic projection(cf. pp. 21-22), the groups in 3. above correspondto groups of rotations of the sphere that mapdihedra onto themselves, these dihedra beingdouble pyramids whose "points" are at the north

48 1. FOUNDATIONS. LINEAR FUNCTIONS

and south poles of the sphere. The octahedron,one of the five "regular solids," is among thesedihedra. Now the remaining regular solids sim-ilarly give rise to groups of rotations, each suchgroup consisting of all rotations that bring thecorresponding regular solid into self-coincidence.To find the fundamental regions of these groups,it is best to locate first the corresponding regionson the sphere and then pass to the plane by stereo-graphic projection. To locate the regions on thesphere, however, one proceeds as follows: Thetriangular faces of the given regular solid areprojected onto the surface of the sphere, with thecenter of the sphere as the center of projection;in each spherical triangle thus obtained, the alti-tudes are drawn from each vertex to the commonpoint of intersection of the altitudes. The newspherical triangles thus constructed are thenstereographic images of fundamental regions ofthe group associated with the given regular solid.(The cube and the dodecahedron may be omitted,since their groups are identical with those of theoctahedron and icosahedron, respectively.)

5. The covering obtained from a fundamentalregion by the application of the mappings in thegroup need not be a covering of the whole plane,as it was in the above examples. It may be acovering of some part of the plane only, such asthe interior of a circle, or the upper half-plane.

6. GROUPS OF LINEAR FUNCTIONS 49The latter, for instance, is mapped onto itself bythe elliptic modular group, consisting of the sub-stitutions z' = az: b , where a, b, c are rational

cz a

integers satisfying ad - be === 1. A fundamental

I 'I \

FIG. 8

~I I

I I

region for this group, shadedin Fig. 8, is the part outside theunit circle of the strip betweenthe two lines x === -1/2 andx= + 1/2 (z=x + iy).

For the proof, we refer thereader to more detailed exposi-tions (such as Vol. II of theauthor's Lehrbuch der Funktio1tentheorie, ChelseaPub!. Co., New York 1945). Here we merely addthat all the mappings of the group can be gener-ated by composition from two of them, namelyfrom the parabolic mapping z' = z + 1 and theelliptic mapping w = -l/z. The former mapsone of the two boundary lines of the strip ontothe other, while the latter has i and - i as itsfixed points and maps the two arcs of the unit

.

circle from + i to - t + ~V3 and from + i to.

+ t +; V3 onto each other.An important branch of modern Function

Theory is the theory of automorphic functions.


It is concerned with functions that remain un-changed under groups of linear functions, in otherwords, with functions f(z) that satisfy all func-tional equations f(z) == f(li(Z, where the li(Z)represent all the mappings of the given group oflinear functions. In simple cases it is easy to findsuch functions. For example, W == zn remains

2hinunchanged by the rotations ~' == e 11 z. Similarly,

]w == zn +- is an automorphic function of the

ZR

dihedral group of example 3. above. Automorphicfunctions of the group of Example 2. are givenby the elliptic functions; of the group of Example1., by the function w = e2inz ; of the group ofExample 5., by the elliptic modular function; ofthe group of Example 4., by functions of the form

n

W = ~ r(Li(z)) , where r(z) is a suitable func-1

tion, the l.(z) are the mappings in the group, andn is the number of mappings, Le. the order ofthe group.

CHAPTER TWO

Rational Functions

7. w == znIn 1 we found it necessary to exclude from

our discussion, temporarily at least, the singulari-ties of the functions we studied, as well as thoseof their inverses. In 3, where we studied thefunction w = liz, we took the first step towardclosing that gap, and we were able to extend ourresults in that connection to any function havingsimple poles only. We shall now take up the func-tion w == zit, whose study will require us to mastera new situation.

At z == 0, the derivative of the function w = znvanishes. The inverse function is not regular atthis point; its singularity at z = 0 is a so-calledbranch-point of order n. To get a picture of howthe mapping w == zn behaves at z = 0, we intro-duce polar coordinates by setting z == rei'P, W == eei/) Then e= rn, {) == ncp. Thus every circle r = const.is mapped under w === zn onto a circle e== const.,and every straight line qJ === const. is mapped ontoa straight line {) === const. Now it will be con-venient to do what the relation {) === nqJ suggestsdoing, namely to consider at first only a part of

51

52 II. RATIONAL FUNCTIONS

the w-plane, viz., the sector r > 0, 0 < fP < 2rc/n.Its vertices are z = 0 and z = 00, and it isbounded by the lines ffJ = 0 and ffJ = 2rc/n. Thissector, it now turns out, is mapped onto the fullw-plane; for, with rand cp ranging over the sector,eand {) can independently take on any values what-soever. The lines ffJ. = 0 and fP === 2rc/n are bothmapped onto the real axis of the w-plane. Thisdiscussion gives us an insight into the specialnature of the point z === 0; the mapping is notisogonal at this point; rather it changes everyangle at z = 0 into its n-fold in the w-plane, atw = o. For if two curves of the z-plane passthrough z = 0 with their tangents there inter-secting at an angle a, then the tangents to theirimage curves at w = 0 intersect at the angle na.The same holds, as we can see by referring to ourabove sector, at z === 00 ; at this point, too, everyangle is mapped onto its n-fold. In particular, itfollows that the image of our sector covers thewhole w-plane. But if the image of only the n-thpart of the z-plane covers the whole w-plane,where can the image of all the rest of the z-planebe accommodated? We have no choice but to coverthe w-plane a second time, then a third, etc., asoften (viz., n times) as necessary. And indeed,the neighboring sector in the z-plane, bounded bycp = 2rc/n and ffJ === 2-271/n, is also mapped ontoa whole w-plane by w = zn. In this way we obtain,

53

corresponding to the n sectors in the z-plane, n fullcoverings of the w-plane; every point of thew-plane thus appears as the image of n distinctpoints of the z-plane, and these n pre-images arefurnished by the n values of the inverse function

n

Z = VW. As the above discussion shows, these nvalues all lie on a circle about z = 0 as center inthe z-plane, and they constitute the vertices of aregular n-gon. Only the points w = 0 and w = 00are exceptional, in that each of these has only asingle point of the z-plane as its pre-image, viz.,z = 0 and z = 00, respectively. These two points,then, may be said to be part of all n coverings ofthe w-plane. We shall interpret each separatecovering of the w-plane as filling out a separate"sheet" of the w-plane, a device that goes back toRiemann. We think of the n sheets correspondingto the n sectors as lying one on top of the other,so that the n points of the n sheets associated withany given value of w lie vertically above eachother. The sheets being arranged in the sameorder (vertically) as are the corresponding sec-tors (cyclically), we shall fasten each sheet to thenext in a manner we shall now describe in detail.To facilitate this description, we shall distinguishtwo "banks" of the positive real axis of thew-plane, viz., a right one and a left one. The rightbank is the image, under the mapping of the sector


bounded by cp === 0 and qJ === 2rc/n, of the linecp = 2rc/n, and the left bank is the image of theline cp === o. When the next sector is subjected tothe mapping (the one that borders on the firstsector along cp=== 2rc/n, Le. No.2 in Fig. 9), theline cp === 271/n is mapped onto the left bankand the line qJ === 2 2rr/n onto the right bank.Any given point of the line cp === 2rt/n goes intotwo opposite points of the two banks under themapping of the two sectors bordering on this line,equal values of w being associated with the twoimage points. Now we shall think of the two banksof the positive real axis that correspond to thecommon boundary of the two sectors as beingjoined together, point by point, in the same wayas the two sectors hang together along cp === 2rc/n.We thus obtain a region which gives a doublecovering of the w-plane. We proceed with theremaining sectors in the same way in which wejust treated the mapping of the second sector; wethink of the corresponding sheets overlying thew-plane as being joined together along edges cor-responding to boundaries common to adjacentsectors. If there are only two sectors altogether,as is the case with w = Z2, we must think of thetwo banks that still remain free in Fig. 10 asbeing joined together. The fact that this can notbe done without introducing self-intersections ofthe resulting surface may be a practical difficulty

7. w == zn 55in the construction of a model, but it should notbe a stumbling-block to our intuitive visualizationof that construction. As is usual also in othercontexts in the Theory of Surfaces, we shall herecount the curve of self-intersection as two differentcurves of the surface, having nothing to do witheach other except for their incidental coincidencein a drawing, or on a paper model, whose con-struction the reader is urged to undertake. In thegeneral case, we thus obtain a surface of n sheets,a so-called Riemann surface. Two points, called

FIG. 9 FIG. 10

the branch points of our Riemann surface - viz.,w = 0 and w = 00 - are common to all n sheets.The fact that the sheets were joined togetheralong the real axis is only incidental; all kinds ofdifferent systems of cuts could be used to give a


decomposition of the Riemann surface into nsheets each of which covers the w-plane. Suchdifferent systems would correspond to decomposi-tions of the z-plane into sectors different from theones used above.

A few pages back we saw how the content ofthe isogonality theorem must be modified to fitthe function under discussion (viz., w = zn) atw = 0 and w = 00. Let us now ask, what aboutthe Preservation-of-Neighborhoods Theorem inconnection with our function? It obviously applieswithout any modification to the neighborhood ofany point other than zero and infinity. Peculiari-ties are encountered, however, when we deal withthe mapping of neighborhoods of z = 0 andz = 00, or with the mapping of the whole z-plane.To be sure, the Riemann surface is a closed, con-nected point set; but it is not a region in the sensein which this term was defined in 1. It is notpossible to describe a circle about w = 0 as centerand whose interior, covered simply, is a sub-regionof the surface. However, if we take an n-tuplycovered circle with its center and branch-point atw = 0, then we are dealing with a sub-region ofthe surface. This leads us to an extension of theconcept of region: Henceforth we shall includeamong the interior points of a region, points nearwhich the region behaves as does our surface nearw = o. In other words, any given interior point

7. w=zn 57will have, among its neighborhoods, a simply ormultiply covered circle whose center, and onlybranch-point (if any), is the given point. Thusit will always be possible to map a neighborhood(Le. all points within some small enough distance)of any interior point a of a region on the surfaceonto a simple and full neighborhood of z === 0 by

n

means of a function s = V"-w-:'--a , where n is asuitable integer, and we adopt this as a necessaryand sufficient condition for calling the given pointa an interior point of the surface. In the neigh-borhood of w === 00, the new definition just givenmust of course be modified, in accordance withour conventions in 3, in so far as the function

n

Z= v'w-=.-ti is to be replaced by the functionn

z=VT t()All this being established, we can now say

that the Preservation-of-Neighborhoods Theoremremains fully valid for the function w === zn.

The specific Riemann surface considered aboveis called the Riemann surface of the function

n

Z = Vw because it is suitable for giving us a geo-metric picture in the large, so to speak, of the

n

mapping represented by z = V;. For if we label


every point of the Riemann surface with thez-value of its pre-image under the conformal map-ping w = zn, then we have defined a single-valued

n

function VW on the Riemann surface, whereasthat same labelling applied to the w-plane as such(instead of to the Riemann surface) defines ann-valued (instead of a single-valued) function.If we observe the values assumed by z when wecontinue along a curve on the surface by meansof the well-known process of analytic continua-tion, we see that they correspond to our abovelabelling of points on the Riemann surface. If thefunction is continued around a closed curve on theRiemann surface, its value will return, upon onefull traversal of the closed curve, to its initialvalue-a statement which will not always be validif we replace the surface by the simply-covered

n

w-plane. To each of the n values that z = VWassumes for one and the same value of w, therecorresponds exactly one point on one of the nsheets of the Riemann surface.

We shall see just below how to constructRiemann surfaces for more general functions, toserve the purpose of furnishing a compact pictureof how the various branches of a given functionare connected. Right now, let us give yet an-other description of how the construction of the

7. w == zn 59

Riemann surface' in the above example can becarried out. We take n duplicates (sheets) of thew-plane and slit each of them open by a cut alongthe positive real axis from zero to infinity. Oneach one of the sheets we then accommodate thevalues of one of the branches of our root function,different sheets being used for different branches;this we can do, say, as follows: We first labeln interior points on the n sheets, say w = i oneach of them, with the n values that the functionassumes there, so that each value is associatedwith a separate sheet. We then imagine the indi-vidual branches of the function to be continuedanalytically, from the interior points chosen, overthe individual sheets, as far as this is possiblewithout crossing the cut on each sheet. In thisway we distribute the n branches of the functionover the n sheets. We then think of the sheets asbeing joined along their cuts whenever the func-tion assumes equal values along two edges repre-senting the positive real axis. This process ofjoining completes our construction of the Riemann

n

surface of the function z = y'W .The detailed study of the function w = zn which

we have just made, enables us in the case of otherfunctions also, to see exactly what happens to theconformal mapping in the neighborhood of a zeroof the derivative. Let, for example, w - b =


an(z - a)n + ... , where an =F 0, be such a func-tion. To see how the mapping behaves at z = a,w = b, we introduce an auxiliary variable t by set-ting w-b=tn We findtn=an(z-a)n+ ... ,whence

n

Here, PI = Van =F o. Solving for z - a, we findz - a = (Xlt + (X2t2 + ..

Thus both z and ware one-valued functions of tin the neighborhood of t = 0, where both areregular. The mapping w - b = an (z -- a-) n + ...has thereby been carried out in two steps, namelya first mapping t = Pl (z - a) + ... that maps asimple neighborhood of z = a onto a simpleneighborhood of t = 0, and a second, mappingw - b = tn that maps our simple neighborhoodof t = 0 onto a surface of n sheets winding aboutw = b as a branch point. Altogether, a simpleregion of the z-plane is being mapped onto ann-sheeted region, with a branch point, of (or"over") the w-plane. Taking into account ourextension on p. 56 of the definition of a region,we may now state that the Preservation-of-Neighborhoods Theorem applies to all function.sthat are regular to within poles.

The facts just discussed may also ,be inter-preted in a slightly different way. Because of the

7. w= zn 61role it has just been shown to play, the auxiliaryvariable t is called a local uniformizing variablefor the functional relation represented by w-b=an (z - a) n + ... , since both z and ware one-valued functions of t in the neighborhood ofz === a, w === b. This uniformizing variable playsa role in function-theoretic problems similar tothat played by time in problems in mechanics,where the quantities involved in even the mostcomplicated processes of motion may be regardedas single-valued functions of time; the onlydifference is that in the mechanics problem, thecoordinates are single-valued functions of timeat all times, whereas in our function-theoreticproblem, z and ware in general single-valuedfunctions of t in a sufficiently small neighborhoodof t === 0 only. Later on we shall touch on theimportant problem of representing a functionalrelation f (z, w) === 0 in its entirety by means ofparametric equations z === z (t), w === w (t), wherez(t) and w(t) are single-valued functions of t.What was done above by means of the local uni-formizing variable t, was to map onto a simpleneighborhood of t === 0 a neighborhood of theplace w === b on the Riemann surface of the func-tion z (w) defined by w - b === a.", (z - a) n + ....By this means we were able to express wand z assingle-valued functions of t in the neighborhoodof t == O. The solution of the general problem of


complete unilormization in the large of a givenfunction will depend on the possibility of map-ping the entire Riemann surface onto a simpleregion of a t-plane. If this can be done, then allfunctions that are single-valued on the Riemannsurface-in particular, z and w-can be expressedeverywhere as single-valued functions of t in theabove region of the t-plane.

8. Rational FunctionsTo supplement our investigations, we shall here

study some rational functions from the point ofview of how the mappings represented by thembehave "in the large." Consider first an integralrational function w === I (z) of degree m; anygiven value will be assumed by w at m points ofthe z-plane. Thus in order to obtain the completeimage region of the z-plane, we need m sheets ofthe w-plane, and we must then join these m sheetsin the proper way to obtain a Riemann surface.The best way to approach this problem, from thepoint of view of developing a systematic theory,is to first determine all those values of w thatcould be associated with branch points of theRiemann surface. These are the points at whichthe mapping fails to be isogonal, and to find themwe must set I' (z) === 0 . We must further check,by the familiar methods, on the multiple poles

8. RATIONAL FUNCTIONS 63(Le. those of an order greater than unity) and onz = 00. In the w-plane we mark the correspond-ing values of w. At each of these, the inversefunction z = qJ (w) may have a branch-point ofsome sort or other. Now we number the actualbranch-points in some definite way and then jointhem in order, starting with the first one, by acontinuous, differentiable curve (' that does notintersect itself. We then think of the w-plane asbeing cut along this curve. (In the examplew = ZIt, we had cut the w-plane from 0 to 00 alongthe real axis.) In the w-plane as cut in this man-ner, the function qJ ( w) is single-valued, and it isregular, except for poles, at every interior pointof the region bounded by our curve. (In general,Z = cp (w) will of course assume different valuesat opposite points of the two banks of our cut.)In order to find out how the n sheets of theRiemann surface must be connected, we firstdetermine the pre-image G of ~' in the z-plane.

~ will consist of several curves. (In the examplew = zn, they were the straight lines


the Riemann surface of the function z~ fP (w) ,which is single-valued on this surface and maps itonto the z-plane one-to-one, and, except at thebrancll-points, isogonally.

One fact that should be stressed here is that notevery point that was marked above is of necessitya branch-point on every sheet.

The procedure just outlined may often lead totedious lengths if all the details are carriedthrough, but it is usually quite adequate to givea schematic idea, so to speak, of what the mappinglooks like. Let us take up an example. Supposewe are given that

w = Z3 + 3z 2 + 6z + 1.We see that w = 00 is a branch-point of order 3.To locate the points where w' ~ 0, we must solvethe equation

3Z2 + 6z + 6~ O.This yields z = -1 i. We find the followingbranch-points in the w-plane: a = - 3 - 2i,{j=-3+2i.

As the curve (;' connecting the three branch-points, we choose the straight line leading froma to 00 and from there to j3 We must then deter-mine the corresponding curve ~ in the z-plane; itconsists of the straight line through -1 + i,- 1 - i, and 00, and the hyperbola having thisline as its transverse axis and having its vertices

8. RATIONAL FUNCTIONS 65at - 1 + i and - 1 - i. This can be verified byseparating real and imaginary parts.

As a parametric representation for the straightline ~' in the w-plane, we take w = - 3 + it,where the real parameter t must be > 2 in abso-lute value. For the corresponding curve of thez-plane, we find

- 3 = x 3 - 3 Xy2 + 3 x! - 3 y2 + 6x + 1,t = 3x2 y - yS + 6xy + 6y. We see that thestraight line x = - 1 is part of this curve. Thecurve being of third order, there remains a conicsection which is seen to be the hyperbola describedabove. Its equation is 3y2_(X + 1)2=3.

II~

FIG. 11

In Fig. 11, we have numbered with Romannumerals the regions that correspond to the threesheets of the Riemann surface.


The dotted segment of the line x = - 1, be-tween fl = - 1 + 2i and v = - 1 - 2i, corres-ponds to the finite segment between a and f3 ofour straight line in the 'lv-plane, as can easily beseen from t = y (3 - y2), and therefore does notcorrespond to any part of the system of cuts.Only the two branches of the hyperbola play anessential role in the construction of the Riemannsurface, since the two half-lines each lie entirelywithin one of the three regions and therefore arenot part of the common boundary of t\VO differentsheets; in joining the Riemann surface together,one merely closes up a cut within a sheet if thatcut corresponds to one of the two half-lines justmentioned. Sheets I and II of the Riemann sur-face are joined along the half-line from - 3 + 2ito 00 of the w-plane; sheets II and III, along thehalf-line from - 3 - 2i to 00. All the remainingcuts are to be closed up within the individualsheets where they occur. This completes the con-struction of the Riemann surface.

We shall see presently that the example we havejust treated is typical of the mappings representedby integral rational functions of degree three.For, all Riemann surfaces associated with suchfunctions must have a third-order branch-point atinfinity, and the finite branch-points are deter-mined from a quadratic equation in z. Hence thethree values of w which are possible branch-points

8. RATIONAL FUNCTIONS 67always lie on a straight line in the w-plane. Underthe inverse mapping, back into the z-plane, thisstraight line (drawn on the three sheets of theRiemann surface) goes into a curve of the thirdorder, as can easily be seen by separating real andimaginary parts. The straight line, however,passes twice through each finite branch-point, andany such branch-point must go into a double point(point of self-intersection) of the third-ordercurve. This curve must therefore have two finitedouble points. The line joining these two doublepoints thus intersects the curve in at least fourpoints and must tllerefore itself belong entirely tothe curve. The curve then is made up of the con-necting' line plus a conic section which turns outto be a hyperbola except in the case where the twofinite branch-points happen to coincide, in whichcase the hyperbola degenerates into two straightlines. This case is actually realized for the integralrational function w = zS; in all other cases, oneobtains a Riemann surface of the same structureas the one we have discussed in this section, exceptthat of course the finite branch-points may belocated at different points.

With certain modifications, the method justdiscussed can be extended to arbitrary algebraicfunctions. These are functions that are obtainedby solving (theoretically, at least) an algebraicequation f (z, w) === 0 between z and w. The new


feature in this situation is that we now have twoRiemann surfaces to consider, one over the z-planefor the function w = w (z) and one over thew-plane for the function z = z (w). The first ofthese surfaces was identical with the z-plane itselfin all the simple special cases of rational functionsthat we have discussed so far; for, w (z) is asingle-valued function of z if it is rational in z.In the general algebraic case (as also in our spe-cial cases), the two Riemann surfaces are mappedonto each other one-to-one and conformally bythe two functions w (z) and z (w) . Thus, forexample, w n = zm (with relatively prime integersn, m) maps a surface of m sheets over thew-plane, with branch-points at 0 and 00, onto asurface of n sheets over the z-plane, with branchpoints at 0 and 00.

We shall conclude this section with an example.The Riemann surface of the function

w=V(z--a) (z-b) (z-c) (z-d)has its branch-points at z = a, b, c, d; for atz = a, for instance, w can not be developed accord-ing to integral powers of z = a. The surface overthe z-plane has two sheets. It is obtained by join-ing the two sheets along two arcs each of whichconnects two of the branch-points (one of them,say, a and b, the other one, C and d) without meet-ing the other branch-points. This surface is map-

8. RATIONAL FUNCTIONS 69ped by w = w (z) onto a Riemann surface of foursheets over the w-plane. It is not possible to mapthe first Riemann surface one-to-one onto a simpleplane by any function whatsoever if a, b, C, and dare four distinct points; for if we draw in one ofthe sheets a curve whose projection into thez-plane loops around a and b while leaving C and din its exterior, then this curve does not decomposethe surface into two separate regions, since it ispossible to connect the two /'dsides of the given curve by /means of a suitably chosen ()

Iother curve that does not ... _~_. __/cross the given curve (cf. FIG. 12Fig. 12). But if our Riemann surface could bemapped one-to-one onto the z-plane, then therewould have to be closed curves in the z-plane,according to the above, that do not decompose theplane. There are no such curves; we shall notprove this fact here but merely appeal to thereader's geometric intuition. We only wish, inpassing, to call the reader's attention to certainfacts that are of fundamental importance forvarious deeper problems of Function Theory, e.g.for the problem of uniformization upon which Wshall touch a few more times. This problem, as wesaw on pp. 61-62, depends on the conformal map-ping of the Riemann surface onto a simple regionof the plane. For the Riemann surfaces to which


we were led by considering rational functionsw=!(z) and their inverses z=qJ(w), such amapping was always possible; but we can easilyguess here that such a one-to-one mapping willnot always be possible in more general cases. TheRiemann surfaces of the rational functions thatwe considered were mapped one-to-one onto thesimple z-plane by their very make-up in terms ofthe function z === fP (w ). In this case, every func-tion of w that was single-valued on the Riemannsurface could be regarded as a single-valuedfunction of z.

CHAPTER THREE

General Considerations

9. The Relation Between the ConformalMapping of the Boundary and that of the

Interior of a Region

The following theorem has already been usedimplicitly several times, although it has not so farbeen necessary, in view of the simplicity of ourexamples, to state it explicitly:

THEOREM. Let there be given a simple andsimply connected region R lying entirely in thefinite part of the z-plane and having one singleboundary curve ~. Let ~ consist of a finite num-ber of arcs of curves that are continuous anddifferentiable. Let the function w === f(z) beregular and let I f (z) I be less than some fixedfinite bound within and on the boundary of R.Let the boundary curve ~ of R be mapped one-to-one by w = f (z) onto a closed curve ~' that doesnot intersect itself. Then the function w = f (z)maps the region R one-to-one onto a finite, simpleregion R' whose boundary is ~'.

Proof. To make things easier, we assume fami-liarity with the fact that the curve ~' divides the

71

72 III. GENERAL CONSIDERATIONS

w-plane into exactly two regions, a finite one calledthe interior of ~', and an infinite one, the exterior.It is then obvious (by the Preservation-of-Neighborhoods Theorem) that the image regionR' must have a region in common with one of thetwo regions just mentioned, and that the entireboundary of R' is given by the curve (I'. Butthen R' can not intersect the exterior of ~', sinceotherwise R', being a finite region, would needsome boundary curve outside of ~' to separate itfro

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Conformal Mapping - L . Bieberbach