Meromorphic Functions and Analytic Curves. (AM-12) 9781400882281

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ANNALS OF MATHEMATICS STUDIES NUMBER 12

MEROMORPHIC FUNCTIONS AND ANALYTIC CURVES BY

HERMANN WEYL In collaboration with F. JOACHIM W EYL

PR INCETO N PR INCETO N

U N I V E R S I T Y PRES S

LONDON: HUM PH REY MILFORD OXFORD UNIVERSITY PRESS

1943

Copyright 1948 PRINCETON UNIVERSITY PRESS

Li t h o p r i n t e d in U . S . A .

EDWARDS A N N

BROTHERS,

A R B O R .

*943

INC.

M I C H I G A N

PREFACE Five years ago my son Joachim and I discovered and brought home from the primeval forest of mathematics, a sapling which we called Meromorphic Curves (Armais of Mathematics, 1938). It looked healthy and attractive, but we did not know much about it. Soon after, a gar­ dener from the North came along, - a skillfulKind man of great experience, L. Ahlfors was his name, he knew; and under his care the plant, almost overnight, grew into a beautiful tree (Acta Soc. Sci. Fenn., 1 9 J+1 ). Having learned the lesson, we set ourselves to carry out an idea dimly conceived before (Annals of Math. 1 9 ^1 , Proc. Nat. Acad. 1 9 ^2 ), namely to transplant the tree Meromorph from the z-plane into the mountainous terrain of an arbi­ trary Riemann surface (a landscape of which I have been fond since the early days of my youth). The experiment seems to have succeeded. The leaves are out, a few buds are visible, but only the future can teach what fruits the tree will bear. In the meantime the howling storm of war has cut us off from our wise gardener.

The material of this "Study", including many details of design, shaped itself in conversations and correspon­ dence with my son; in this sense it is a joint enterprise. But the bulk of the manuscript was actually written down by the undersigned as hour by hour notea of a course given in the Institute for Advanced Study during the first term of 1 9 ^2 - 1 9 4 3 , and responsibility for the final arrangement and text rests with him. I wish to acknowledge valuable assistance rendered by

vi Dr. Audrey Wishard McMillan and Mr. Fumio Yagi in pre­ paring the manuscript, and helpful criticisms and sug­ gestions offered during the course by Professor Claude Chevalley. Hermann Weyl April 19^3 Institute for Advanced Study Princeton, New Jersey

CONTENTS INTRODUCTION.

EARLY HISTORY AND BIBLIOGRAPHY........... 1

CHAPTER I. GEOMETRIC AND FUNCTION-THEORETIC FOUNDATIONS............................ A.

B.

Projective apace and unitary metric.......... §1 . Plucker coordinates and thecalculus of forms............................... §2 . Duality................................ §3. Unitary geometry........................ §4. Projection.............................. §5 . Distortion.............................. Generalities concerning analytic curves...... §6 . Analytic branch. Associated curves...... §7. Stationarity indices.................... §8 . Dual curve............................. §9. Rational curves......................... §1 0 . Analytic and algebraiccurves............. §1 1 . Meromorphic curves......................

CHAPTER II. §1 . §2 . §3 . §^. §5 . §6. §7 . §8. §9 .

10 10 10

15 19 26 30

35 35 ^1 kj 51 56 65

FIRST MAIN THEOREM FOR MEROMORPHIC CURVES................................ 69 The condenser formula....................... 69 The first main theorem...................... 7b Meromorphic functions. The exponential function................................... 82 A fundamental lemma......................... 87 Exponential curves.......................... 9k Kronecker multiplication. Intersections with an algebraic surface.................... 1 06 Projection.................................. 109 Poincare !s theorem for meromorphicfunctions.. 11 2 Hadamardfs second theorem for meromorphic functions................................... 1 1 7

CONTENTS

viii CHAPTER III. §1. §2. §3 . §1+. §5 . §6. §7 .

THE SECOND MAIN THEOREM FOR MEROMORPHIC CURVES................................123 The formula ofthe second main theorem........ 1 23 Average of thedefect........................ 1 25 On integrationin general.................... 1 3 0 Average of theN-term........................ 1 3 8 The order of rank p ..........................1^3 The compensating term for the stationary points..................................... 1 50 Inequalities for orders and stationarity valences of all ranks....................... 1 57

§8. A peculiar relation..........................160 CHAPTER IV. §1. §2. §3 . §^. §5 . §6. §7 . §8. §9 .

FIRST AND SECOND MAIN THEOREMS FOR ANALYTIC CURVES........................ 163 Green’s formula............................. 163 The condenser formula....... ................ 1 68 The first main theorem and the valence integral for T .............................. 1 72 Existence of the condenser potential and relaxation of conditions for boundary........ 1 7 7 Second main theorem.......................... 1 8 ^ Positive and zero capacity. The "little" terms m° and ............................ 189 The fundamental inequality................... 1 95 Special cases of rotational symmetry.......... 203 Appendix. The banks of a Jordan contour...... 205

CHAPTER V. THE DEFECT RELATIONS..................... 211 §1. Picardfs theorem............................. 211 §2 . Weighted averages. The basic general formula.....................................2 1 3 §3 . Defect relation for meromorphic functions on a Riemann surface........................ 2 1 6 §^. Convergence of an integral................... 223 §5 . Starting the investigation for arbitrary n. ...22^ §6. Consequences for arbitrary rank............... 231 §7 . Sum into product.......................... . . 2 3 7

____________________ CONTENTS_____________________ lx CHAPTER V . (Cont iniied) §8. The II-m-relation for points................ 241 §9. Discussion of the defect relations for points.................................... 247 §10. Incidences of higher order. Formulation of the general defect relation.............. 253 §11. Proof of the general defect relation........ 259 §12. Computation of coefficients. The fundamental relation for the actual defects..264

(4.12) refers to the formula labeled (4.12) in §4 of the same Chapter. (II, 4.12) refers to formula (4.12) of Chapter II.

INTRODUCTION EARLY HISTORY AND BIBLIOGRAPHY The meromorphic functions bear the same relation to the entire functions as the rational functions do to the polynomials. The most important characteristic of a polynomial f(z) is its degree. It has a two-fold signi­ ficance: it counts the number of zeros and it describes the growth of |f(z)| with increasing |z|. The only polynomials without zeros are the constants, and a poly­ nomial with the n zeros a 1 ,...,an (each counted with its proper multiplicity is of the form C - r ^ i(z-ai )

or

C^TliO

- § 7 )*

The second form presupposes that all roots a^ 4s 0 . h of them equal zero, one must write instead

If

with the product now extending only over the n - h roots a^ which are different from zero. The type of growth of a polynomial of degree n is either described by an in­ equality If (z )I 0, convergence of the integral ,0 0

f J

log .M(r) d _A+1

-A implies convergence of the series ^~lan l « o(r^+1) (k integral) then

If log M(p)

According to this definition, genus k and order A are by no means uniquely determined by the function. In the literature it is more usual to modify (and complicate) the definition so as to bring about uniqueness.

EARLY HISTORY AND BIBLIOGRAPHY f ( z ) = eP ( z ) • lim

T T

1

E(Jc;~—)

R-*co |an I^R

n

where P(z) is a polynomial of formal degree k. The driv­ ing force for Hadamard*s investigations was the wish to obtain sufficient information about the zeros of the f function to establish the asymptotic law for the distri­ bution of prime*numbers. That law states that the number Tt(n) of primes less than n becomes infinite with n oo exactly as strongly as n/log n: (8)

g(n,l^„log n

,

for n - c o .

Riemann had shown how this prime number problem depends on the zeros of the C-function, and in 1896 both Hada­ mard himself and de la Vallee Poussin were able to draw the conclusion (8 ) from Hadamard1 s results concerning en­ tire functions. Besides the zeros, i. e. the solutions z of the equa­ tion f(z) = 0 , one shpuld study, for any preassigned val­ ue c, the c- places of f, i. e. the points z for which f(z) assumes the value c. We have seen that the 0-places might be spread much more thinly than the growth of the function leads one to expect (there might be even no zeros at all). But if this is so, then 0 is an excep­ tional value of f inasmuch as for almost all values c + 0 we can expect the distribution of c-places to conform with the growth. The first and most important step in this direction is Picard*s famous theorem (1 8 8 0 ) stating that an entire function assumes all values with the pos­ sible exception of only one. If one counts also w = 00 as a value, one may say instead that a meromorphic func­ tion w = f(z) omits at most two values. For if it omits the value c, then l/(f(z)-c) is an entire function. Picard*s proof was simple, but made use of a highly transcendental tool, the modular function. By more ele­ mentary, yet much more complicated methods, E. Borel sue-

8

INTRODUCTION

ceeded in bringing Picard1a theorem into closer contact with the main body of the Poincare-Hadamard theory of en­ tire functions. Two of the most penetrating investigators in the field of entire functions, along the lines opened by these great initiators, are Georges Valiron and Anders Wiman. A long time passed by and the first World War shook our civilization before the basic paper on the theory of meromorphic functions was written by R. Nevanlinna in 1925- It put, at the same time, the more special theory of entire functions on a new foundation, greatly improv­ ing the older results. The appearance of this paper has been one of the few great mathematical events in our cen­ tury. Rolf Nevanlinna^ approach was greatly simplified by his brother Prithiof, and by L. Ahlfors; Ahlfors is also the author of a new method in this field, of half topological character. The preface tells how the theory of meromorphic curves came into being. A good part of the theory of meromorphic functions is better understood when looked upon as a special case of the theory of mero­ morphic curves. One knows that an algebraic function y(x) (or an alge­ braic plane curve) is defined by an algebraic equation y® +

+ ... + fm (x) = 0

whose coefficients are rational functions of x. After in­ troduction of the corresponding Riemann surface !R, both the independent and the dependent variables x and y ap­ pear as meromorphic functions on R. There is no reason why one should not simultaneously study three or four, or even n meromorphic functions on !R instead of two, thus passing from plane to n-dimensional algebraic curves. This will be our viewpoint. We shall spend a good deal of time on meromorphic curves, but afterwards we shall carry all the main results over to the general case of analytic curves where the z-plane is replaced by an arbitrary

EARLY HISTORY AND BIBLIOGRAPHY

3l

Riemann surface. I conclude this introduction by quoting the papers and books which form the chief landmarks of the historic de­ velopment of our subject. (The numbers in front are used for reference; books are marked by an asterisk.) [1]

1 876

IZ]

1 880

11]

1883

[it]

1893

Weierstrass, Zur Theorie der eindeutigen analytischen Funktionen, Abhdlg. Ak. Wiss. Berlin, 1 8 7 6 , 1 1 - 6 0 = Math. Werke II, 77-124. E. Picard, Memoire sur les fonctions en­ tires, Ann. Ec. Norm. (2 ) _2 , 1 4 7 - 1 6 6 . H. Poincar^, Sur les fonctions entieres, Bull. Soc. Math, de France J_1_, 1 3 6 . J. Badamard, ]£tude sur les propri6 t£s des fonctions enti&res ..., Jour, de Math.

LSI

1 900 *E.

[6 ]

1 903 *E.

til

1925

R.

[8]

1935

L.

L2]

1935

L.

[12]

1936 *R.

[JJ]

1938

H.

llSl

1 9 V1

L.

(4) _2 , 1 7 1 -2 1 5 . Borel, Logons sur les fonctions enti&res, Paris, 1900 (2e ed. Kind 1 9 2 1 ). Borel, Logons sur les fonctions m£romorphes, Paris, 1 9 0 3 . Nevanlinna, Zur Theorie der meromorphen Funktionen, Act. Math. _46, 1 9 2 5 , 1. Ahlfors, Ueber eine Methode in der The­ orie der meromorphen Funktionen, Soc. Sci. Fenn. Comm, phys.-math. _8, Nr. 10. Ahlfors, Zur Theorie der Ueberlagerungsflflchen, Act. Math. 6 5 , 157-194. Nevanlinna, Eindeutige analytische Funk­ tionen, Berlin, 1 9 3 6 . and J. Weyl, Meromorphic Curves, Ann. of Math. 22 , 1 9 3 8 , 516-538. Ahlfors, The Theory of Meromorphic Curves, Acta Soc. Sci. Fenn., Ser. A, vol. 3, No. 4.

CHAPTER I GEOMETRIC AND FUNCTION-THEORETIC FOUNDATIONS A. PROJECTIVE SPACE AND UNITARY METRIC Pliicker coordinates and the calculus of forms In an n-dimensional vector space p (^n) linearly in­ dependent vectors a^,. ,,a span a p-dimensional linear subspace or, as we shall briefly say, a p-spread Ja1,...,ap[. Pliicker taught us how to characterize such p-spreads by coordinates. First we choose n independent vectors e^.^.^e as a basis or coordinate system in terms of which any vector x is represented by its co­ ordinates (x1,...>xn ),

11

x = x 1e1 +

n n• If various vectors are distinguished by a subscript, the subscript characterizing the coordinate is written sec­ ond. From the coordinates of p given vectors a^...^^, 11

+

‘in pn

we form the determinants of degree p, 1i (1.1)

A(ir ..i ) = p

These are the Pliicker coordinates. If and only if they all vanish, the p vectors are linearly dependent. Each of the p indices l.j,...,i in A(ir .ip) runs from 1 to n, and A is a skew-symmetric function of them. Spanning the same p-spread by another set b.j,...,b we obtain the same coordinates except for a common non-vanishing factor.

§1 . PLUCKER COORDINATES AND THE CALCULUS OF FORMS

11

Indeed if bk = Z h Akhah

(k,h=l,... ,p)

then B(i^...Ip) = p*A(i^...ip),

p = det (A^)*

In this sense the Pliicker coordinates are homogeneous coorinates of the spreads. Any skew-symmetric function A( i1••.i ) of p indices i ranging over the values 1,...,n shall be called a p-ad. If desirable, its "rank" p is indicated by a superscript, A = Ap . For p > n there is no other p-ad than zero. For p n it suffices to know the values of the (p) or­ dered components A(i1...ip) for which i1 < ig < ... < 1^; they may be independently chosen. It is clear that one can add two p-ads and multiply a p-ad by; a number. Hence the p-ads are the "vectors" of a (p)-dimensional vector space. We express the definition (1.1) of the Pliicker coordinates by the formula A = [a1,..., ap ]. The p-ads of this form are said to be special. The p-ad E of which all ordered components vanish except E(l2...p) = 1 is special, namely = [e.j,...,e ]. We shall not investigate here what relations between the components of A characterize the special A. Let us mention only the simplest non-trlvial case n = 4, p = 2 where this condi­ tion reads A(12) A(34) + A(31 ) A(24) + A(23) A(14) = 0. If a p-spread is spanned by p vectors a ^ . - ^ a and a q-spread by q vectors b 1,...,b^, we can form the spread spanned by a1,...,ap , b 1,...,b^. In case the two spreads have no vector in common except zero, the resulting spread Is of dimensionality r = p + q and commonly called their direct sum. But we propose to use a multiplication

12

I.

GEOMETRIC AND FUNCTION-THEORETIC FOUNDATIONS

sign » for the representing polyads and hence write (1.2) [a^,...,ap ] * [b^,...

] = [a.j,...

This multiplication is evidently associative. commutativity holds in the modified form

* •••

^•

The law of

B » A = (-1)pq(A«B) for the product of a p-ad A and a q-ad B. One can write the definition (1.2) in terms of the ordered components as follows: (A*B) (i1,...,

+ A( j^ •jp) * B(k1,... >k^)•

Here i1 < ... < i^, and the sum at the rigiht extendsaltematingly over all (^) permutations j-j>•••>jp > ^ .j,..., kg

(j*|^ •••O p J ^ ^

)

of i^...,! . Thus oneconvinces oneself that the com­ ponents of the product are uniquely determined by the components of the two factors. The definition extends at once to arbitrary polyads. Then our multiplication of course is distributive with respect to addition. The re­ sulting calculus of general polyads has recently proved of considerable importance in several applications; it Is the algebraic part of E. Cartan’s calculus of differential forms. Here our interest is concentrated on the special polyads representing spreads; by (1.2) the product of two special polyads is again special. After defining multiplication, the p-ad [a1,a2,...,ap ] may now be written as the product a.^ * a2 * ... * ap. Thus there is a redundance of notations and we should adopt either the cross or the square bracket. Following the classical works on Ausdehnungslehre by Grassmann and on electromagnetism by H. A. Lorentz, I choose the latter, and hence will now write [A,B], with or without comma, instead of A * B. If a ^ . - . ^ p are linearly independent

§1 . PLUCKER COORDINATES AND THE CALCULUS OF FORMS vectors, x belongs to the spread

13

if and only

if [x,a1,...,ap ] = 0, i. e. if [x,A] = 0 . This shows that the p-ad A = [a^...^^] determines uniquely the spread {a.j,...,a | = fA(. The relation [AB] = 0 for two special polyads A = [a1#..a ], B = [^...b^] indicates that the vectors a1,... ,ap , b 1,... ,bq are linearly depen­ dent. If A + 0, B + 0 and p, q ^ 1 this means that the two spreads fAi, fBi have a non-vanishing vector in com­ mon (intersection). Up to now we have used a fixed coordinate system el>...,en . On replacing it by another e£ = (e^ ,... ,6^), X = J 'L j X j e j ,

the coordinates x^ undergo the non-singular transforma­ tion (1.3) while the components X(i1...ip ) of the product [x1,... ,Xp] of p vectors x 1,... ,xp undergo the transform­ ation (1 A )

Xtj^.jp) =

(±1 ••‘ip)0^ ji ••"eipjp*

We denote the latter by 5 Tp

Xp ] =

[1 (1.*«) X( J,... Jp) - X i x,(1T - - 1p) eipJr " ,eV p

1b I.

GEOMETRIC AND FUNCTION-THEORETIC FOUNDATIONS

A sum over the indices I,,..., i of a p-ad will be indi1 P Kind i i, If each index runs independently from 1 to ^ if the restriction I1 < ... < i is imposed. Kind In affine vector geometry all coordinate systems are treated as having equal rights. A linear transformation cr (even if singular) may also be interpreted as a linear mapping; the role of x and x 1 is then exchanged. A linear mapping carrying e^ into e| will carry x = Into the vector x 1 = / x^ej! = tfx with the coordinates xl = r^ lG capping

X J ' ( i 1. . . i p)x(i 1. . . t p) = rx)

(XT')

for T = [^1.. .^p] and X = [x1 .. .Xp]. The formula proves that the contravariant p-ads are transformed contragrediently to covariant p-ads. If

rP - *cn-p,

XP = vrn-p^

then clearly

r pxp) = (cn- % n- p). The unit n-ad E?1 = [e1 ...en ] with the components +1 if (i^...i^) is an even permutation, eV

...i ) =

-1 if (i.j...in ) is an odd permutation,

18

I.

GEOMETRIC AND FUNCTION-THEORETIC FOUNDATIONS

is Invariant with respect to unimodular transformations of the basis. Our definition of the star operation, rp = *cn'p, may now be written In the form (2 .2 )

[XpCn_p] = (X^P) • E11,

Xp = [x1 ...Xp] denoting a variable covariant p-ad. If we drop the limitation to unimodular transforma­ tions, the * operation is no longer invariant, but the equation V •P p = *Cn~p establishes an invariant relation between a covariant (n-p)-ad Cn"p , a contravariant p-ad r p and a volume fac­ tor V. Any linear combination of cp+i>**->cn Kind a vector which "lies in" the(n-p)-spread fCp+1,•••,cn i = fC|. On the other hand, let us say that a vector £ of 6 Kind "goes Straightforward or down-to-earth through" fCi if satisfying the linear equations (2 .3 )

^p+1 ^ =

’** 9 (cn ^) =

According to the theory of linear equations, these 5 form a p-spread fr| in 6 . Hence there is a one-to-one corres­ pondence between the (n-p)-spreads in S and the p-spreads in 6 *, (Cl^fPi. This is the essence of the duality phe­ nomenon in projective geometry. Its algebraic expression is the star operation. Indeed, assume cp+i>**'>cn to be Independent and extend them by p further vectors c1,...,Cp so as to obtain a basis c1,...,c for the full space S. Introduce the corresponding basis ,... , 7 In 6 : (Ci>j) From the multiplication theorem of determinants one gets at once |x .j,...,Xp, Cp+.j,•••,c^l • l7 *|*,##>7 nJ — I

•

§3. The x are from 1 to

UNITARY GEOMETRY

19

variable vectors in 6 and the indices k, h run p. In particular, for p = 0, Ic1 >•••>cn I * 17 -j> •••>7n I =

1•

Hence multiplication by the volume factor V =|c1,.. yields the desired relation V * ^ xk7h ^ = |X1' —

'V

•>cn l

°P+1'**#,Cn^

or [7*1 ^

•>7p 3 = * [CP+1'*•#'cn^*

This construction proves that the * operation carries a special (n-p)-ad into a special p-ad. §5 . Unitary geometry From affine, we pass to Euclidean metric geometry by introduction of the positive-definite quadratic form 2 2 x 1 + ... + xn . However, this form loses its positive character as soon as the coordinates are no longer lim­ ited to real values but become capable of arbitrary comp­ lex values, which will be the case in our applications. Then it must be replaced by the Hermitian unit form x 1x 1 + ... + xnxn . Starting from a definite coordinate system, we are thus led to introduce the scalar product (x|y) of two vectors x,y by (3.1)

(x|y) = x|y1

At thesame time we define ads X, Y by

+ ... + xnyn . the scalar product of two p-

(3.2)(X| Y) = J ^ X ( i 1...ip )Y(i1...ip ). The same expressions are used in the dual space. Two vectors x, y are said to be perpendicular if (x|y) = 0 . We ascribe to the vector x a length or magni­ tude |x| ^>0 given by |x|2 = (x|x), and in the same way

20

I.

GEOMETRIC AND FUNCTION-THEORETIC FOUNDATIONS

set |XI 2 = (X|X). Geometrically I[x«j,... ,x ]| maybe in­ terpreted as the magnitude of the p-dimensional parallelotope spanned by x 1 ,Xp. Notice that |*X| * |X| . "Distances" in the projective space which depend on the ratios of coordinates only can be formed as follows. Distance of a point x and a plane %: (3.3)

IIx S|| =

The Cauchy-Schwarz inequality | u i v

1 + - - * + u a v m | 2

i (u1u 1+...+iy^)(v1v1+...+vmvm )

shows at once that this distance ||x g|| ^ o and 1 . It vanishes in case of incidence, (x5 ) = 0 . In the same manner for p-elements:

But all cases may be subsumed under the distance (3.5)

IX:YI

-

of a p-element X and a q-element Y. unity if p or q = o. For p ^ 1 , q ^ dicates intersection of (X) and (Y). arises from it by substitution of relation (3 .6 )

|XsY|

£ 1,

This distance equals 1 its vanishing in­ The distance (3.4) for Y (q^n-p). The

I[XY] | £ IXM Y I

does not seem to hold for general polyads X and Y, but we shall presently prove it in case X and Y are special. In order to free these algebraic formulas from the ab­ solute coordinate system on which they are based, we re­ place the explicit definition (3 .1 ) of the scalar product by an axiomatic description which makes no reference to

21

§3 * UNITARY GEOMETRY

any special coordinates. Let us therefore say that our vector space is endowed with a metric if any two vectors x, y determine a scalar product (x|y) which has the fol­ lowing four properties. It is linear in x, (i)

o except for x = o.

Because of the last property (x|y) is non-degenerate, i. e. there is no vector a except 0 for which (a|y) = 0 iden­ tically in y. In any coordinate system e^, x =» x 0 except for x = 0. Hence we may simply say that a metric is the affine vector space by designating a Hermitian form as metric ground form. We establish the existence of a basis terms of which (x|y) is expressed by the form

introduced into positive definite e1,...,en in Hermitian unit

22

I.

GEOMETRIC AND FUNCTION-THEORETIC FOUNDATIONS

In other words, we are going to construct n vectors e^ such that (3-7)

( e 11®j ) =

( o V * ' a n ) *

If X a iP i(t) does not vanish identically, it vanishes for t = 0 with a certain order v ^ 0. It is essential that we use a reduced representation x^ -= p^(t) for the deter-

38

I.

GEOMETRIC AND FUNCTION-THEORETIC FOUNDATIONS

mination of the multiplicity v . Then it is evident that V is affected neither by the gauge factor p(t) (because of p0+o) nor by the transformation (6 .3 ) of the parameter t (because of b^O). in passing to another coordinate system, but keeping the plane (a) fixed, one has of course to submit the coefficients to the contragredi­ ent transformation; in this sense the multiplicity v of intersection is also independent of the coordinate systan. In case i-

The first summand equals the stationarity index vp - 1 . Of the two operations of replacing p^(t) by R^ = t^p^ and d/dt by d/dz, the first has no effect on our quotient, whereas the second adds a factor dz/dt because {0 +1 +•.•+(p-2 )| + !0 +...+p| - 2 f0 +...(p- 1 )| = K Hence the equation ( a P - ' r P - ' ) ( a P * ,bP*')

______£_________ z (A^)2

ea

„

, __ _ yp dt **

The first factor at the left is a rational function of z (which is not affected by the gauge factor Q(z)) and thus its total order equals zero. The second factor dz/dt equals 1 for the finite spots and - 1 /t2 for the infinite spot (z = 1 /t); hence its total order is -2 . Therefore, by summation over all spots, (9 .5 )

crp + (vp+ 1 - 2 Vp+Vp-. 1 ) = - 2

(p=1 ,...,n- 1 )

where and m further functions fk (€;t), also regular in 6 but de­ pending, besides on t, on a positive parameter € in such a way that fk (€;t) — * Let 1.J, . positive.

with € — ► 0 uniformly for t € (& .

, 1^ be non-negative numbers, 11 actually

LEMMA 4. C. Under the hypotheses just enumerated there exist numbers €Q > 0 and B such that (**4) h

lo8 -C

for all £ for which

+ ... + S mfm (€jt)| dt ^ -B

§4.

A FUNDAMENTAL LEMMA

1^1 I - 1-j y °>

91

l?2 ^ “ ^2’

“ \i

(torus) and all positive € £ €Q. Proceed exactly as before: Lemma A with f 0 (t) = Z a kfk (t),

For any

a on the torus,

f(t) = Z s kfk (€;t)

yields two numbers €a > 0 and Ba and a neighborhood of a such that theleft member of (4.4) hasthe lower Caring bound -Ba for 0< € o

(i^k),

if they intersect at all, have a certain open angle ^ in common. Let us call k of the first or second kind according to whether this angle of intersec­ tion does or does not exist. For directions & in the open 0^ the kth of the quantities P^(*) = a^ cos & + b^ sin is actually larger, in the closed angle ^ £ & £ not smaller, than the others. Hence the full rose is divided into a number of angles in each of which a different one of the P^(W rides on top. We label them

Fio. 1

§5.

EXPONENTIAL CURVES

97

* * aa 0 ^ (\x = 1 ,... ,v) with the dividing directions as they follow one another around the rose. (The index p ranges over the integers modulo a certain n in the cyclic order in which ja is followed by ft + 1 ; the angle * * * * 0 ,, is bounded by the directions # and & ., or & separM J* . P H + 1’ H ates 0 . and © .) Let k = k indicate the largest of H-1 m * H & the P^(^) in 0^., 0^ = 0^. The points P^ = P^ are the consecutive vertices of a^polygon Q which deserves to be called the convex polygon surrounding the points P1, ... , P of our diagram, for the following reason: the line * * joining the two points P^ _ 1 and P^ of the diagram, 5 cos

+ t) sin

= h(i>*),

leaves all points of the diagram on one side, 5 COS /

+ I) sin /

£

fo r

= (ak ,bk^*

The contribution of the angle 0 ^ to the integral (5 .2) amounts to

j (a^cos $ + b^sin &) di9-= k 8^( 3 in ^.-sin &^) - b^cos ^-cos i>k ),

therefore L= /

Ja*(sin ,-sin £*) M H+ M = 2_ "(a ”a 1 ) sin * • z — n v n jj- 1 7 p + * If & passes through if in positive r expression (5.k)

- b*(cos .-cos tf*)| K H +1 H (b -b ) cos i 9 . v fi ja-1 7 h’ sense, the sign of the

* * * * (a cos £ + b sin *>) - (a -cos £ + b sin & ) fj fJ. r ' r ' changes from - to +; therefore

< - V i = - V in V

< - bM-i - V

03


one that is Independent of the radius r, and thus an upper bound of the form Const./r for the integral (5 .8 ). The delicate point is the lower bound. Because of the semi-definite character of G the form

GU) - g n lglx 1 + ...+ g nxn l 2

(g, = i ,

g± = g n / S n )>

which no longer involves the variable x 1, is also semidefinite; hence G(x) ^ g n I g ^ ^ . - . + ^ ^ l 2 ,

and our problem is reduced to finding a sufficiently shaip lower bound for H/r

(5 .1 0 )

£

-H/r

lQg lg 1x i +* .* +gn3Sl l ' d*.

Preparing to apply Lemma 4. C we now‘let t assume complex values but limit It to a bounded region © surrounding the Segment 9 . In that region we find for the modulus of z - r « r(ei*'/r-i) - 1$ eU /r -dT 0

§5.

EXPONENTIAL CURVES

103

an upper bound H 1 which is independent of t and r, and thus after the substitution (5*9) , av r |Ak |H' ak r Ix^l^e.e = Const. e

(5.11)

For k - 1, ... , m the explicit expressions xk = e

^

= e

^

• expf-iblcr(eit/r-i )|,

in particular x 1 = 1, will be used. f,(r;t) = 1 + ( % f1xIW1 + ..*+gnxn)»

Set fl(t) = 1;

fk (r;t) = expj-ibk;r(eit/r-l )|, fk (t) = exp (t^t) (k=2,... ,m). fk(r;t) tends to (for k=2,...,m) with r — ► oo uni­ formly for t € ® . In the same manner f^r^’t) tends to f1(t) because of (5.11) and a^. < o for k = m + 1, ... , n. Our integral (5.10) assumes the form (5.12)

1

-

H

X lc)g l?1(r)-f1(r;t)+...+5m (r)-fm (r;t)| dt -H

where the coefficients are "rotating" functions of the parameter v, (5*13)

-mr ^k ^ ^ == ^k * ^

^k’””

(in particular, ^ ( r ) = 1 ). We apply Lemma k. C to the integral H (5.14) J log |$1f 1(r;t)+...+?mfm (r;t)| dt —H in which the parameters 2^, ... , torus (5-15)

lSk l -

lgjj.1 = lk

vary freely over the

(k>»1,...,m).

104

II.

FIRST MAIN THEOREM FOR MEROMORPHIC CURVES

Notice that 1 1 = 1 and that the in functiona ^(t) = exp (b^/t) are linearly independent. The lemma yields a lower bound -B for (5.14) valid for sufficiently large r and all (^ ,... ,?m ) on the torus, and since the point (5 .1 3 ) never leaves the torus, -B/r is a lower bound for (5 .1 2 ). This Is the decisive step in our argument: let the point (5 ^ roam freely over the torus without binding it to the complicated Lissajous figure described by the point (5 .1 3 ) on the torus. From the order T we pass to the order T of rank p. .(p-1 )] are The components of = [x,x!, 1, A j , "‘‘I

• • .

9A p_1 f\ j

1 exp f(A1 +. •.+A, )z\ ■*■1

Xp (i.,,...,ip ) = 1, A i ,...,AP " 1 P P

(±i
) = q 1 (.*) + ... + qp (£) •and we find zrc

(5 .1 8 )

Lp =

$

|qi (*) + ...+qp (*) | d*.

Since the Wronskian of the exponential curve vanishes nowhere, the curve is without stationary points of any rank. Instead of (5.1 ) we could discuss a more general ex­ ponential curve & with coordinates y 1, ... , yn , defined by equations (5.19)

V

7 ± = aj_ie

V

+ a1Ne

We naturally assume that the exponents A^, ... , A^ are distinct, and that no single column of the matrix 11 nl • vanishes. We may consider this curve & as a projection of the curve in N-space, A,z x1 = e upon an n-dimensional subspace.

a Nz

By the substitution

106

II.

FIRST MAIN THEOREM FOR MEROMORPHIC CURVES ” ailx l + #** + aiNxN

the square sum l y j 2 + ..- + lyn !2 changes into a semidefinite Hermitian form G of the variables x 1, ... , x^ the principal coefficients of which are actually positive. Applying our lemma to this G and CL = |x 1 I2 + ... + Ix^l2 we realize at once that CD and ' are of the same order, and thus the order function T(r) of C is equiva­ lent to where L is the length of the convex polygon surrounding the points A 1, ... , A"^. For any plane p1y 1 + ... + pnyn = o we find N(r;P) ~ ~

* r,

L' denoting the length of the convex polygon spanned around those among the points A^ for which Pla 1K + ... +

^ 0,

and hence the corresponding defect m(r;f3) ~ IckL .r< In a number of ways the example of exponential curves, even of the general exponential curves of type (5 .1 9 ), proves to be more easily manageable and more informative than that of the rational curves. §6.

Kronecker multiplication. Intersections with an algebraic surface Suppose we have a meromorphic curve with the homogeneous coordinates x$1 ... , ^ in n-space and a ^ (2 ) (?) curve CD2 with the coordinates x!j ', ... , in mspace. Let the branches associated with the point z = zQ be given in reduced representations,

§6.

(6.1)

x[1) =

)(z-z0),x£2) = ip£2)(z-z0).

We form the n*m coordinatea x^ ^ = x^1 responding expansions xlik.

(6.2)

Caring

KRONECKER MULTIPLICATION

with the cor-

? < , > p ( li>(Z-Z o ).

They constitute a reduced representation, and multiplica­ tion of the two representations (6,1 ) by arbitrary gauge factors p ^ has the effect of multiplying (6.2) by the gauge factor p^1 ^p^2 ^. Hence (6.2) defines a mero­ morphic curve in n-m dimensions which is called the Kronecker product £^ * &2. THEOREM. The order Tj£j of the Kronecker product £ = (D1 x (D2 equals T f€»1 } + T f&2 i. In other words, the order shows a logarithmic behavior under multiplication of curves. The proof Is very simple. The order of £ could be computed by an arbitrary non-vanIshing linear form

(1L(2)

k ” ^ — aikxi ^xk i,k aikxi 1Kri^K i,k *

But we specialize the coefficients a., as follows: cl., = ■ . I ')**2 *; then

(6.3) Thus the order In which the left member vanishes for z = zQ is the sum of the orders of the two factors at the right-hand side, and moreover by combining (6 .3 ) with Z i ^ i 2 “ Z i 4 1)|2£ |ak 2)|2' ^ |xi,k|2 = & ! 1)|2& k 2)|2 we find

108

II.

FIRST MAIN THEOREM FOR MEROMORPHIC CURVES

lOg — — = lOg -- r A V -.’V + lOg -- 7-pY 7-p-t . ||ax|| ||a' 'x '|| IIac x II Therefore N(Rja) = N (1)(R;a(1)) + N (2 ^(R;a(2 )), m(r;a) = n/1^(r;a^1^) + m ^ 2 ^(r;a^2 ^). Multiplication by itself of the curve CD in n-apace with the coordinates x. gives rise to a curve with the 2 n coordinates x^x^. However, in this special case it is not reasonable to carry x^x^. and x^x^ (for i ^ k) as two different coordinates. Hence for the fth power of we use the monomials

x ( f i ,.. ., f n ) -

• x 11

£

(6.h)

(f 1 ^ 0,..., f^

f'-] +•••+ f^ — f )

as coordinates. The numerical factors are added to.bring about the relation F x (f.j,..., f^ ) * x(f 1,...,f^ ) = (x-jX«|+... +X£|,X£L) • We then have the simple statement: f The order of (D is f-times the order of ••->fn ) 'x(f1 ,...,fn )

(6.5)

is a homogeneous form P(x1,...,xn ) of degree f of the variables x 1,...,x . The specialization by which we prove our theorem is indicated by the formulas a(f1 ,

.,fn ) - \/(f ^

fi

f.

Caring

Ot(f«j,..., f^ ) * X(f^,...,f^) = (d^X^+...

) «

§7.

Caring

PROJECTION

f The intersections of with the plane defined by the vanishing of any linear form (6 .5 ) are nothing else but the intersections of G with the algebraic surface of or­ der f, F(x1 ,...,xn ) = 0 • Hence our result is a generali­ zation of the fact that an algebraic curve of order ^ intersects an algebraic surface of order f In f • ✓ points. §7 . Projection In general the central projection of an algebraic curve from a point has the same order V as the original curve. However, if the center of projection lies on the curve the order is reduced by the multiplicity of the in­ tersection between point and curve. This theorem may be generalized in two ways, - by substituting any h-element as center of projection for a point or 1 -element, and by studying the order or rank p instead of v*. It is in this generality that we are going to formulate the cor­ responding fact for meromorphic curves CD . Let e.j,...,e be a normal coordinate system. Projec­ tion from the unit h-spread [E^i = fe1 ,...,e^i changes a vector (x1,... ,xn ) into (x^+1 ,... ,xn ) and any (special) pad Xp into Xp where X^(i1 ,...,ip ) = X(i1 ,...,ip ) with the range h + 1 , ... , n for the indices i. Set ^ “ [e ^,...,e^] so that IE11! = 1 . We compute the order T^(R) of rank p of the given meromorphic curve CD by means of a linear form CAX^) in which the contravariant A = [a^.-cx^] is spanned by vectors of the special form ol = (0 ,...,0 , a h+1>•••>an )> 30 that A(i 1 ,...,ip) vanishes if one of the figures l,...,h appears among the indices i^...,!^. Normalizing by |A| = 1 we have n /r ,

14^11 |Xp |

1

I

1

~ r log -}■ — d* 2lT 0 I(A xp )!

R 0

pv

The corresponding order T^(R) = Tp (R;El1) of the projected curve CD is the same expression, except that |X^| is to

110

II.

FIRST MAIN THEOREM FOR MEROMORPHIC CURVES

be replaced by IX?I . Observing that IX^ = ItE^x15]! we thus find Tp (R) - Tp (R ; # ) . N^R, II^ H -

)

2ir

hS.

l0®

d* [aXP ]l

The auxiliary contravariant p-ad A has disappeared. In this form the equation holds for any h-element (E?1) with the normalization (E^l = 1 because proceeding from a given unitary metric we can always construct a normal coordinate system e ^ . , . , 6 such that its first h vectors el9 -#",eh sPan the given • The first term in the right member, which we denote by Np (R;Eh ), may be written as [7.1 )

Np (R;E^)

j*(R;z)

I,

if (z„) + ^ ( b q :^) is the order in which the components It is thus ] vanish simultaneously at z = zf of [ the valence of the intersections of the center of projec­ tion fE*1! with £pin the circle of radius R, whereas the the non-negative (7.2)

a u r j ^ 1) = j - f * log — 3-1

p

o

-d*

urstf'n

plays the role of the corresponding compensating term. The latter depends on the choice of the unitary metric. However, by passing to another unitary metric it is changed by an additive function of r varying between the fixed limits + log ^min (p h)' as con]Puteci 1x1 Chapter I, §5 . Thus we have arrived at the following fomula des­ cribing the relation of the orders of rank p of a given curve and its projection from fE?1!: (7.3)

Tp (R) - Tp (R;E^) = ^ ( R ; ^ ) + [ f i p d * ; ^ ) .

§7.

PROJECTION

3030

We were somewhat arbitrary to normalize an additive constant in the order function by the condition T(rQ ) = a If we deviate from this for the projected curve and intro­ duce TpCR;^) - mp(r0 ;E1:l) as its order of rank p, then it is true that the order of the projected curve does not exceed that of the original curve, and the "loss by pro­ jection" Tp (R ) -

fTp ( R ; ^ ) - m p d ^ -E 11)! = ^ ( R j E * 1)

is given by /^(RiE11) = NpCRjE*1) + SpCRjE11). The quantities (7.1), (7 -2 ) coincide with (2 .6 ), (2 .6 1) ift the highest case h = n - p if fA| is the dual of (En_p|: Np (R;Ap ) = Np (R;An_p), mp(r;Ap ) = mp(r;An"p ) for Ap = *An-p. Suppose (E*1 - 1 | C |E^|. The multiplicity ^(ZqIE*1) with which [E^X^e^] vanishes at zQ caxmot be less than the vanishing order v (zQ iE^1-1 ) of [E*1 1 Hence y

v

35*1) ^ y

zo:Eh

>>

NpCRiE11) ^ Np (R ;E h_1 ).

The relation (I, 4.1) or

1 0 8

IIE5 ^

*

1 0 8

yields S p C r ^ ) ^ S p ^ E * 1-1 ). By addition A p t R ^ ) ^ A p ( R ; ^ M ).

112

II.

FIRST MAIN THEOREM FOR MEROMORPHIC CURVES

More surprising than these facts are the consequences of the inequality (7.4)

IIE?1:Xp || ^ IIE^xP’1 II

which follows by the same lemma I, 4.A from [X^” 1 j C fXp !. It implies that the order (zQ: ) with which ||E^:X^|| vanishescannot be less than ^ - 1 (ZqIE*1). The obvious remark that [E^X13] vanishes at least In the same order as [E^X-P” 1 ] yields only the inequality !dp(z0 ) - dp-i(zo)! + * V zo:Eh) _ V i (zo:Eh)l ^ °* Our stronger result leads to (7.5)

SptR;^) ^ Sip^RjE11),

and moreover, by taking the logarithm and integrating, (7.4) gives rise to the complementary inequality (7.6)

SptrjE11) ^ ffip_ 1 (r;Eh ).

In consequence of the last two relations, the loss A p (R;Eh ) by projection along fE11! of rank p is not smaller than the same loss ^p_i (R;Ej of rank p - 1 . These are quite remarkable facts. §8 . Poincare !s theorem for meromorphic functions We do not pretend that the theory of meromorphic curves as yet comprises all important known facts about meromorphic functions. Indeed, of the Poincar^-Hadamard theory it includes only Hadamard!s first theorem. For this reason the two next sections dealing with Poincares and with Hadamard*s second theorem form a sort of enclave In our pattern. We generalize these two pro­ positions from entire to meromorphic functions and prove them in this generality, closely following R. Nevanlinna's approach in his fundamental paper 1 9 2 5 .

§8.

POINCARE1S THEOREM FOR MEROMORPHIC FUNCTIONS

115

THEOREM. Suppose we are given a non-negative integer k and a number A in the interval k < A £ k + 1; moreover two sequences a . b of complex Straightforward or down-to-earth lam l > converge. With an arbitrary polynomial P(z) of formal degree k form the function (8.1)

f(z) =

Pfz)

O Sdc^z/a. ) -rif--- ---lnIE(k;z/bn ) JLX

,

J.JL

composed of Weierstrass !s primary factors of typ.e k, (8.2)

(y + . . . + ^-). E(k;u) = (1-u) • exp (j

Then the order T(R) of the meromorphic function f(z) satisfies the limit equation (8.5)

T(R) = o(RA)

(and (8.U) T(r) • r"(A+1 5dr is convergent provided A is not integral). knowthat thatTfl/fi Tfl/fi= =T Tfffff.f. Moreover Moreover Proof. WeWeknow Tjfgi £ T|f[ + Tfg|

+ const.

for any meromorphic functions f , g. This follows from a combination of Kronecker multiplication with projection. One forms the theproduct productofofthe thetwo twocurves curves in in 2-space 2-space with with the coordinates (x^,x2 ), respectively; f = x^/x^, (x^x2 (y-|>y2 ) respectively; f = x 1/x2, g “ y-j/y2 * Kind Pproduct r o d u c t is i s a ccurve u r v e in i n k-space k-s p a c e with w i t h the the coordinates x ^ , x^72 , x 2y 1, X 2J 2’ but we use only coordinates x ^ , x^J2 , x 2y l' X 2J 2’but we U3e only its lts projection in 2-space with the two coordinates x-y., Pf 77 )) P( X 2y 2 - ForKind :fac'tor e> P(z) > P(z) = polynomial = polynomial of formal of formal degree k, we have log M(r) £ c C * and thus

114

II.

FIRST MAIN THEOREM FOR MEROMORPHIC CURVES

For these reasons it is sufficient to prove the estimate log M(r) = o(rA) for the canonical product (8.5)

f (z) =

TTE(k;z/a ).

n First we must try to estimate fairly accurately the primary factor E(k;u). For small u, log |E(k;u)| is in first approximation |u|^+1, for large u it behaves like jj; |u|k . Hence it is reasonable to expect something like the following LEMMA 8. A. •

ik+1

log |E(k;u) | £ B

if k > 0.

The constant B may be chosen as (8.6)

B = 2(3+ log k).

[For k = 0 one has log IE(0;u)| £ log (1 + Iu| ).] Take this lemma for granted and assume k > 0. the inequality v— (8.7)

Then

kc+1

l o g M ( r ) ^ B - 2 _ ---- § a j (r+la^l)

holds for the entire function (8.5). We suppose the la^l to be arranged in ascending order. With the notation Ia |/r = v,

a » A - k £ 1

we may write the general summand in (8 .7 ) as

§8.

POINCARE!S THEOREM FOR MEROMORPHIC FUNCTIONS

115

Observe that the factor v a/(1+v) £ 1. Indeed va £ 1, 1 + v y 1 for 0 £ v £ 1, whereas v a £ v, 1 + v ^ v for v ^ 1. Consequently (8.8)

log M(r) £ BrA - ^ J a n r A = 0(rA).

The improvement to o(rA) is obtained by splitting the sum in (8.7) into ^ and y ,]. For a fixed N the first part is obviously O(r^) = o(rA ) while for the second part we find by the same procedure followed above the upper bound

For an arbitrarily given € ) 0 we may fix N so that this becomes < € • rA. After having done so the first part will also, from a certain r on, be less than € • rA. The case k = 0 is no exception. Indeed, if 0 < A £ 1 the inequality ’ or 7+x ^ xA_1

0

implies by integration log (1+x) £ J x A. Hence

( 8 . 9 ) log M(r) ^ X log ( 1 + J i f ) ^ XrA ,^ Z |an l" A in complete analogy to (8.8). In carrying out the integration required by the supple­ mentary statement (8.4) of our theorem we make use of the values of the following Integrals: oo k-A ^ 1 +u du = sin T t (k + 1 -A ) “ |sin tta | f’or k < A ^ k+1 >

116

II.

FIRST MAIN THEOREM FOR MEROMORPHIC CURVES

Hence (8 .7 ) for k )> 0 and the corresponding inequality (8 .9 )for k = 0 yield

$ r~(A+1 }log M(r) -dr £ \^ r0

nx i^Tlan l~A

(k>°» k R la^KR ®ta |bn l + lo® [J ^ Rfi(k;z/bn )

lo® f (z >

w ith R — » o o uniformly for \z\ ^ r. The proof of our theorem is thus reduced to the limit equation (9.4).

First compare the sum

§9*

121

HADAMARD »S SECOND THEOREM

(9.5)

\Z - = - r (k+1 *

^1—

I^KR

am

wi t h ^ i^

1 = n (R).

R

0

r

Since |(R2 /Sm l > R and |z| £ r, (9.5) is leas than

V

R>

(R-r)k+1 Repeat for A = k + 1 the argument used to prove Lemma 3. A:

_ v * >

_

1

2k+r

(k+1 )Rk+1

w

r2R d N 0 (r)

r2R R

r^+2

, N 0 (2R) - N 0 CR) y N 0 (2R) Rlc+1

R

r ^ +1

m

(2R)k+1

But N Q (R) £ T(R) + const., and hence

T(R)/Rk+1 — + 0 Implies n 0 (R)/R^+1 — ► 0 for R — > oo . It remains to discuss the integral in jtR (z).

Observe

that Hog

If H

- log+ |f| + log"*"11/f |.

Hence the absolute value of that Integral cannot exceed or

(9-6)

g (R-r)

2ir f'J 0

2tr

log+ |f(C)l • d * + $ 0 (; - R e 1*).

log+ |l/f«;)| • d*|.

122

II.

FIRST MAIN THEOREM FOR MEROMORPHIC CURVES

Because of log"*"If I £ log Vl + |f |2, log+(i/|f|) £ log V 1+ 11/f |2 and equations (3*3), both integrals in (9*6) are less than T(R) + const.

CHAPTER III THE SECOND MAIN THEOREM FOR MEROMORPHIC CURVES §1. The formula of the second main theorem We return to the general theory of meromorphic curves and propose to develop the analogue of Plucker^ formulas. Let us form the second differences of the equations (IT, 2.10). Because dP+i{zo) - 2W

+

= W

- 1'

the valence in the circle of radius R of the stationary points of rank p, (1.1)

Vp (R) - Z z(vp (z)-1)^(R;z)

enters into the resulting equation (1.2) Vp («fi!) + |Tp+l(R)-2Tp (R)+Tp_1(R)| = ^ ( r ) ] ? in which

- ) = x^. Then^ will have a definite value F(x!|,... ,xn ) at with respect to the coordinates x^ - x|, a fact which we indicate by the equation ) = P(x],...,x^) or

^ (*) = P(x1,...,xn ).

The relation (3. 2Q) carries over to all points )> suffi­ ciently near pQ : (3.2)

/d *\

F*(x*,...,x^) • det faxrj “ P(x1,... ,xn ).

vp is said to be continuous at p q if its value F ^ , . . , ^ } is continuous at the origin. Because of (3.2) this con­ dition is independent of the coordinate system. The product of a density and a function is a density. To every continuous density defined on the whole manifold and vanishing outside a compact part of it, we can assign an integral which satisfies the following two axioms: (i) J( 2> = S * 1 + S * 2(ii) If ^ vanishes outside a block P, (3 . 1 ), while having the value F(x1,... ,xn ) inside the block with respect to its coordinates x^, then is identical with the Riemann integral in 11 S n ••• I _ F(x.j,... ,3^) dx1 ... dxn . n “1i

§3 ._ON INTEGRATION IN GENERAL

133

Proof. Consider only those continuous which van­ ish outside a given compact part G of the manifold. En­ close each point in a block P(*0 ) = P represented in terms of its coordinates x^ by (3.1 ). Let e be a fixed positive number < 1. The block P contains the smaller block P6 defined by -6lj_ £ x^ £ el^. Because G is com­ pact we can ascertain a finite number of points (Q = 1,2,...) in G such that the contracted blocks P) £ 1 every­ where is said to be a probability function. Given an admissible block P, (5.1), we can easily construct a con­ tinuous probability function A(p) which equals 1 in the shrunk P and vanishes outside P. Take a continuous probability function A(x) of the real variable x which has the constant value 1 for -e £ x £ e while it vanishes outside the Interval (-1,1) and set A(lO=A^y^

inside P.

If we wish we can make A(p) as smooth as we like, in particular take care that it has continuous first deriv­ atives (is of class D 1 ). We simply have to let the basic function A(x) be of class D1. In this way we construct a function Agtv) for each of our covering blocks P^ and then argue as follows. a, b being the probabilities of two statistically in­ dependent events (o £ a, b £ 1 ) the one or the other event will happen with the probability a v b ~ a + b

- ab.

Notice the inequalities a v b £ 1,

a v b ^ a, ^ b, a fortiori ^ 0,

and the fact that the probability sum a v b m a y be w r i t ­ ten as a n ordinary sum a + b f w i t h b 1 « b - ab, o £ b f £ b.

The probability sum A 1 v A 2 v .... of our probability

functions A^ associated w i t h the blocks

equals 1 every­

where in G because at a ny point of G at least one of the A q « 1.

Write it In the form of a n ordinary sum ^

+ ji2

§5-

ON INTEGRATION IN GENERAL

135

+ ... with “ 'S’

“ Aq _Aq(A1 v ••• v A q-! )*

and thua complete the construction of the Dieudonnd * factors Mq. Computation of an integral by dissecting the domain of integration into disjoint pieces G^, each covered by a coordinate system of its own,would mean decomposition of the density^ to be Integrated, \I/= 5 ~ cr>J/ t by means of the characteristic functions cr of the parts G : q

cr

q

= 1 inside, '

^

q

= 0 outside G . q

In the Dieudonn^ procedure these discontinuous cr are re­ placed by the smooth Its advantage for the integra­ tion of continuous functions is obvious. In integrals extending over parts of a plane with the real coordinates x, y it is customary to write the ele­ ment of integration as dx dy; this corresponds to the idea that the plane is cut up into infinitesimal rectan­ gles of sides dx and dy parallel to the axes of coordin­ ates. However, when dealing with a complex variable z = x + iy and analytic transformations, it is more rea_ o sonable to write dz dz = |dz| instead; this is the area of an infinitesimal square of arbitrary orientation spanned by the vectors dz and i-dz. Indeed, conformal transformation carries an infinitesimal square into a square, but in general the orientation of rectangles parallel to the real and imaginary axes is destroyed. Our form of writing is suggestive of the law of transformation of Integrals (3.

j* dw dw -

|^| 2dz dz

under a one-to-one conformal transformation w = w(z). If Sometimes it Is convenient to use spheres instead of parallelotopes, but the definition of a multiple Riemann integral is simpler for parallelotopes.

136

III.

THE SECOND MAIN THEOREM FOR MEROMORPHIC CURVES

If w(z) is any regular analytic function in a compact part G of the z-plane of definite Jordan area, the map­ ping z -► w effected by it is generally speaking not oneto-one. But it may be made into a one-to-one mapping if the w-plane is replaced by a suitable covering Riemann surface (G is turned into it by the convention that covers the point w Q of the w-plane if w(zQ) = w Q). The area of this Riemann surface will then be given by the integral

If the Riemann surface has n(w) points over the point w of the w-plane we thus obtain the equation (3 .5 )

£n(w)dw dw = I l|ifl2dz G

In determining areas and integrating over limited parts of the manifold we have been unfaithful to our principle of integrating only continuous functions over the whole manifold, to which Dieudonnd's procedure is adapted. We return ruefully to it by introducing a continuous func­ tion ^(z) in the whole z-plane vanishing outside a com­ pact part G. The function w(z) is supposed to be regular in G, but G need not have a definite Jordan area. De­ noting by ^(zQ;w0) the order In which w(z) - wQ vanishes for z = zQ and setting ^(z;w) we find instead of (3 .5 ) the formula (3 .6 )

^N^(w)dw dw » j"^(z)||^| 2dz dz.

Thi§ Improvement of (3 .^) can be neatly demonstrated by the Dieudonn^ partition as follows. (Continuity of N^(w) will result as a byproduct of the proof.) Let zQ be any point of G, w(zQ) = w Q, and suppose

83.

ON INTEGRATION IN GENERAL

m Kind

w(z) -w Q to vaniah at z = zQ to the order v(zQ;w0 ) V y 0. Locally the solution of the equation w(z) = Kindform may he obtained in the w - w 0 = t1',

t =

= w

b 1(z-zQ ) + ...(b^O).

In a certain circle K(zQ), \z-z0 \ < pQ, the function t(z) will have an analytic inverse and z t will map K(zQ) one-to-one conformally upon a certain region in the t-plane. Choose a positive 6 < 1 and form the contracted A circle K {zQ ), \z-zQ \ < 6pQ. Since G may be covered by a finite number of such contracted circles K6(zQ), the Dieudonn6 partition reduces the proof to the case where the continuous #(z) vanishes outside K(zQ). Then by means of the one-to-one mapping z ** t and with 0(z(t))‘ = fit),

JV(z) l ^ l 2dz dz = J$*(t)|-^| 2dt dt. Introduce polar coordinates, t = re , and the latter in­ tegral will easily be changed Into the integral (w)dw dw where N^(w) Is the sum of the values of at the V roots t = ... , t^ of the equation t* = w - w Q. The formula (3.7) _jN^(w)iMw)dw dw = J0(z)ip(w(z))|-g^| 2dz dz involving an arbitrary continuous function ^(w) of w is a trivial generalization of (3.6). One has simply to re­ place ^(z) by ^(z)-ib(w(z)). This is the service which Dieudonn61s partition always performs: reducing a global to the corresponding local statement about an integral. When dealing with Integration over several complex variables z^, ..., zn we write the element of integration as dz- dz- ... dz dz . What is its law of Kind transformaKind • • n n tion under an analytic transformation w^ ® w^(z1,... ,zn ) Kind of the variables? Their differentials undergo a linear

138

III. THE SECOND MAIN THEOREM FOR MEROMORPHIC CURVES

transformation with the coefficients (aw^)/(a z^. sider therefore any linear transformation

Con­

and denote its determinant det (a^) by A . Splitting into real and. imaginary parts, zk = x^. + iy-^., wk = uk + iv^, we want to know what the determinant of the linear transformation is which carries the 2n variables x^., yk into the 2n variables u^, v^. Instead of x = -^(z+z), y = —-j(z-z') we may use z, z and. then see at once that that determinant equals A A . This result finds its ex­ pression in the general formula for analytic transforma­ tions dw1 dw1 ... dwn dwn = A A • dz^ dz’1 ... dzn dLzn ,

_

Observe that the factor A A = |A|

p

is never negative

§*+. Average of the N-term We are now well prepared to integrate the term N(R;a) over the unit sphere in a-space. Indeed, the formula (3.6) refers exactly to such a quantity as N(R;a). Let x^ = x^(z) be a reduced representation of our non­ degenerate curve £ for the neighborhood of the point z = zQ, and x ^ Z q ) ^ o. The points z of intersection with the plane (a) are determined by the equation (^.1 )

Qtr x 1(z) + ... + an*xn (z) = °

which for points z sufficiently near to zQ may be written in the form (^.2)

a,

a2 x2(z) + ... + an xn (z) x. (z)

AVERAGE OF THE N-TERM

§k.

139

For fixed a2, ... , the right-hand side is a function a (z). Its derivative is computed from (^.1): dot x1 dz

da + (a x ’+ ...-kx x *) = 0 , 1 1 n n

(xfa) = ~

• x1

dz

Apply the local formula (3-7) by letting p of w and choosing ) = exp(-|a- | ):

a- play the role '

~la i I2 i (b.3) jN^a) e 1 da^ d ^ = ^ ( Z )J1 X g Jl !x < ||

~la i I2 e 1 dz dz.

Here 0(z) is any continuous function vanishing outside a certain circle lz-zQ| < around zQ, and N^(a) =

z)W z ;

a)

with v,(z0 ;a) as in II, §2 denoting the order of vanishing at z = zQ of the linear form (^.1 ). Consequently f...p j(a )-e

$ 0 (z)| $ ...J e

-|a |2- . . . - | a J 2 da, da,

- |a | - .. 1

. . . dc*n do^ =

|a | |(■v 1rv\ I ^ _ _ _ n Ux da2 da2. . . d a n dan idzdz

I

X ^|

The left side equals In the inner integral on the right side stands for the quantity computed, from (^.2 ) in terms of z; a2, ... ,an . Let x = (x.j,...,x ) be a given vector, x 1 ^ o, and der an infin­ itesimal area of the plane defined in a-space by the equar tion x ^ + ... + x ai^ = o and dag da2 ... dc*n do1 its perpendicular projection on the coordinate plane a 1 = 0 . Then we have the elementary geometric formula (if.5 )

der =

- da Ix1I2

da .

... da

da.

Therefore we may write the inner integral at the right side of (4.^) as

UO

III. THE SECOND MAIN THEOREM FOR MEROMORPHIC CURVES

(u.6) (x a)= 0

|x|2

The formula then holds independently of the restriction x^Zq) ^ 0 . It will presently be shown that for any two vectors x ^ 0 and x 1 the integral (4.6) has the value

7^-1 i i s i i i ! . i*r If we now again interpret x^ as x^(z) and xj as its derivative, the expression

C.7)

SU) - g ll” ';ia |X| 4

'

is clearly independent of the gauge factor, and according to Dieudonnefs procedure the resulting formula (4.8)

roaN^(a) = I^jy(z)S(z) dz d?

holds not only locally but also globally, 1 . e. for any continuous function {z) vanishing outside a bounded part of the z-plane. Before proceeding further we ought to complete the argument by proving (4.5) and evaluating (4.6). In (4.5) assume |x| = 1 . Ascertain a unitary transformation llx^H of which x 1 1 = x 1 ^ 0 , ... , x 1n = xn is the first row: =

(i,j - i,...,n).

If ct is determined as function of c*2, • "by P 1 = 0 , then the determinant D of the substitution (a2,... ,an ) -► (P2 *•••

-^3 D =

x ^ arises from

ij

l,j — 2 , ..., n.

§4.

Kind

AVERAGE OP THE N-TERM x n , x 12,

x ln

xg1, x22,

x2n

xn1’ xn2’

,, x. 'nn

by subtracting from the 2nd, ..., n^*1 columns the first column multiplied by x 12/x1, ... , x ln/x1 respectively. Hence D = - • d e t l x1 } ) l i } . ,.... n and DD =

or x ixv

dp0

... 6ftn 6ftn =

~2 da2da2

dan dV

as we had maintained. In verifying the equation -|at

•• - ^ n 1

S-"S e

(xa)=0

|(x'a)|2 dda2 d a 2

dan dan .

142

III.

THE SECOND MAIN THEOREM FOR MEROMORPHIC CURVES

V-l 1 7tn _ l S

...f o

( t _+ . • •+ t ) 2 n dt

t e o 2

The e x p r e s s io n S ,

=

VI 1 1

tP

n

s e e t h e p r o o f o f Lemma 2 . B .

( 4 . 7 ) , h a s a s im p le g e o m e t r ic

The s q u a r e o f th e d is t a n c e

n e a r p o in ts

dt

2

h a s b e e n c o m p u te d b e f o r e ;

fic a n c e .

...

x a n d x + dx on th e

s ig n i­

o f a n y tw o i n f i n i t e ^

curve , dx = x 'd z ,

is

g iv e n

by ( I u 9 ) | x : x +dx||2 = . l t e i 3 a i a = -1 f e q | 2dz d i _ l S ( 2 ) . d Z d ¥ .

ixr T h is tw o

fo rm o f th e (r e a l)

d im e n s io n s

c o n f o r m a lly o n to ~ $ S (z )d z se n ts

"lin e

th e

\xr

e le m e n t " f o r t h e m a n i f o l d r e v e a ls

cu rv e ,

c o u ld be d e s c r ib e d a s |z| £ R o n t h e

We now s u b s t i t u t e b in in g th e

*

THEOREM . \z\

( 4. 10) T (r)

H e re , c ir c u la r th is

r e s u lt w it h

w h ic h r e v e a l s

£

z -p la n e

€

is

of

m apped

in te g r a l

o f th e

z -p la n e

re p re ­

o f £ w h ile

dz dz

th e v a le n c e

cu rv e .

th e p o t e n t ia l o f th e

are a

th e

c o r r e s p o n d in g p a r t

^ (R ;z )S (z )

t io n :

th a t

a n d h en ce th e

e x t e n d in g o v e r a n y p a r t

th e a r e a o f th e

2

as

in

o f th e Ch.

c ir c u la r are a

II,

0 (R ;z )

cond en ser o f r a d i i

p o t e n t i a l f o r i n

( 2 . 7 ) we a r r i v e

d e n o te s r Q a n d R.

( 4 . 8 ).

Com­

a t a p r o p o s itio n

a n e n t i r e l y new a s p e c t o f t h e o r d e r f u n c -

T rT (R ) R o f th e

is

th e v a le n c e o f th e

cu rv e £,

= - ^ ^ ( R ; z )S (z )d z

in

c ir c u la r

fo r m u la

d"z w i t h

S (z ) =

2 ^ x ^ 1^

.

T h e p a p e r [11 ] b y H . a n d J . W e y l c o n t a i n s t h e f o r m u l a I n l o g a r i t h m i z e S fo rm o n ly , a s i t i s n e e d e d t o e s t a b l i s h t h e c o n n e c t i o n b e t w e e n T a n d JQ ( s e e § 6 ) ; t h e e x a c t t h e ­ o re m a s g i v e n h e r e a n d i t s g e n e r a l i z a t i o n t o a r b i t r a r y r a n k p ( § 5 ) a r e d u e t o L . A h l f o r s [J_2]. F o r an o th e r a p p r o a c h s e e J . D u f r e s n o y , C . R . A c a d . S c i . P a r i s 211 f 1 9 4 0 , 5 3 6 -5 3 8 a n d 6 2 8 -6 5 1 .

§5. In c id e n t a lly a fu n c t io n

th is

th e o r e m o n c e m ore p r o v e s

in t e r e s t

is

i n n o n -d e g e n e ra te

re m o v a l o f th e h y p o t h e s is

cu rv e s.

th a t

fo r

( II,

4 . 8 ) p ro ve s

d e g e n e ra te

o f r b e c a u se

(th o u g h n o t t o t a l l y

th e a v e ra g e

th a t

of

Tm ( r ; a )

its

(^ )-sp a c e P O b se rve th e n

d e g e n e ra te )

cu rv e s

o f m (r;a )

t o b e In d e p e n d e n t

is .

(2.7)

The c o m p u ta tio n o f th e a v e r a g e h a s n o w h e re r e s o r t e d t o

Even

o f n o n -d e g e n e ra c y h a s

v a lu e b e c a u se th e a s s o c i a t e d c u rv e i n th e (d } P o f a l l p - a d s m ig h t w e l l be d e g e n e r a te .

§5 .

th a t T is

o f r e g u la r ty p e .

Our c h ie f so,

Kind

THE ORDER OF RANK p

Thus

o f N (R ;a )

th e h y p o t h e s is

in

c a r r ie s th is

o ve r.

s e c tio n

o f n o n -d e g e n e ra c y .

The o r d e r o f r a n k p THEOREM .

The o r d e r T

(R )

o f ran k p

( = 1, . . . , n - 1 )

o f a n o n - d e g e n e r a t e m e r o m o r p h ic c u r v e CD m ay b e e x ­ p re sse d as a

lin e a r

w it h p o s i t i v e

(5.1)

c o m b i n a t i o n o f $ ( R ;z ) o v e r z

w e ig h ts

Sp ( z ) ,

Tp (R) = ^ ( R ; z ) S p (z)dz d¥,

th e w e ig h t b e in g g iv e n b y

s P ( , ) - g i*p- v

- r 1|g-

ix,p r

P ro o f.

If

P* tp+1 = t, tp = s, we reduce our problem to the equation

J

f 0

A 1 M ± |1 e - t - a dt ds 0

(A+t)

where A = lap+2 ^2+ *•* + ^an^2' The 00 A

. r°° A dt+J o 2

o

slcie equals -tdt.

Transform the first part by partial Integration,

f00 «, A+■ de -^ 0 A+t

1 - r0 — 1 V 0 p eA dt r-_trn°0 (A+t)2

and thus find the value 1 for the sum (5.8). We have now established by induction the string of equations (5.5) for h = n - p - 1, ..., 0, ending with (5.1 )• This proof is longer than the first, but gives a deeper insight into the mechanism. The whole Chapter V will be based on a generalization of this argument. §6.

The compensating term for the stationary points The same expression Sp = Sp , whose zeros are the sta­ tionary points of rank p, occurs in the formulas (5-1 )> (1.3) for Tp and the quantity 2Qp = 2tt^ log 2Sz‘d^ This makes it possible to appraise in teiros of Tp . To any analytic substitution z = z(C) of the variable z the quantity Sp reacts as a density, z oPldzl2 _ op Sz|ld = C ’ where

is obtained from

§6.

THE COMPENSATING TERM FOR THE STATIONARY POINTS dx dC

151

dp~1: •1

dCp

aa Spz ia from Xp = Xp Kind Kind z . In particular, for the aubstitution Kind z - rQ-

(C - u+ii>,

u = log ~-~)

we find Sz “ ^ / r2»

lo8

= log Sg + log \r~Q- 2U.

Along a circle r ® const, differentiation with respect to 'gj*, equals -jWe modify our definition of by setting from now on (6.1)

where x'

dx d£

dp~1x *" djP'1

for z = re

, r = const.,

xfV-lxP+V iyP i^

Then the Second Main Theorem assumes the form iH

R

p + (T_, p+1 - 2Tp + Tp-11 ) = IPrX e P rQ - log e rQ .

In (5.1) we split off the integral over the nucleus while outside we replace S^dz dz b y S^dC d£T = sj|?d£ du.

Introducing the positive constant

we thus obtain

152

III. THE SECPHD MAIN THEOREM FOR MEROMORPHIC CURVES oo

(6.2)

T-c_log5-+$ P

P

r0

^du 11=0

with (6-3)

QP = - 1 ^ S > » .

We now introduce u = log instead of the radiua r aa the independent variable (Sogarithmic scale) so that T(u) means what has previously been-denoted by T(rQeu ). Sim­ ilarly for all otherfunctions of the radius. In all quantities defined by integration with respect to like m(r;a) or QP(r), the function of z under the integral is changed into one of ^ by the substitution z = rQ-eu+i (u^o). The potential assumes the simple ex­ pression = 0 for u ^ U,

* U - u for 0 ^ u ^ U,

® U for u ■ CT>U + S (U-u)QP(u) du,

0

V

or if one prefers, u T (u) = c u + § (u-uMQ^tu1) du!. P P o This Integral equation Is equivalent to the differential relations:

dT T p

- o,

— E du

for u « o; p

(6.4) d T. tf(u). du2 The formula of the Second M a i n Theorem n o w reads

V P * < v' “ p

“ V o” •

§6.

THE COMPENSATING TERM FOR THE STATIONARY POINTS

155

Once we have completely switched over to the logarithmic scale of the radius we may use the old letter r for u. With due apologies for any confusion thus caused, this will be done from now on; hence r designates the radius measured in the logarithmic scale. Because log is a concave function, the logarithm of the mean of any quantities exceeds the mean of their logarithms. Applying this remark to (6 .3 ) we find (6 .5 )

I log s|-d£= 20 p,

log qP 2

and (6.4) and (6 .5 ) imply the fundamental relation 1

V

r) * ? 108

■

Would the standard function Tp Itself, rather than its second derivative, appear on the right side, then we could rightly claim that-Q^ is essentially negative and thus attain the goal we set ourselves at the end of §1 . But one knows how precarious conclusion from a function on its second derivative isJ With what qualifications then, we must ask, is it true that the logarithm of the second derivative is majorized by the logarithm of the function itself? From the outset it is clear that what matters for our purposes is not the equation but the inequality R 0 R + £ (R-r)QP(r)dr £ T (R), V

y

0

which changes by means of (6 .5 ) into R 2C1 (r) (6 .6 ) c R + 5 (R-r)-e » dr £ T (R). ¥ 0 P It implies the remarkable inequality (6.7)

Tp (R) ^ cpR.

There is, however, another way of realizing that Tp grows at least as fast as R. Write the First Main

15k

III. THE SECOND MAIN THEOREM FOR MEROMORPHIC CURVES

Theorem in the form Np (R;A) + iflp(RjA) = Tp (R) + ®J(A). Choose a point zQ inside the nucleus \z\ £ rQ, adjust the coordinates to the branch of our curve at zQ, see (I, 7 .1 ) and take an A = A Q whose component A Q(l,...,p) vanishes. Then vp(z0'A o^ ^ 1 811(1 tllus Np(R^*A o^ ^ R * Denoting the constant n^(A 0) by a, we get (6.8)

Tp (R) ^ R - a.

Hence 1 + ( i - O R £ T-p(R ) aa soon as

R ^ RQ =

cp We add this to (6.6): R 2Q (r) (6 .9 ) 1 + R + J (R-r)-e p dr £ 2Tp (R) (for R^RQ). (6.6) has the advantage of holding for all R, whereas (6 .9 ) holds only from a certain R = RQ on (depending on the curve £ ). But (6 .9 ) has the advantage that the constant cp in (6.6) which also depends on the curve. Is replaced by 1. The factor 2 in the right member of (6 .9 ) will hurt nobody. We shall presently see why we care to have the additional term 1 in the left member of (6 .9 ). The double aspect represented by (6.6) and (6 .9 ) will be encountered in all our dealings with this situa­ tion, but the latter will prove the more important. In any case these considerations show that the combination of the two terms on the left side of (6.6) is of a some­ what artificial character. The question raised above is answered by the following LEMMA 6. A.

Let two functions f(r), F(r) (r^o)

be given such that (6 .1 0 )

1 + r + £ (r-p)e**^dp £ F(r)

(for r>0).

§6.

THE COMPENSATING TERM FOR THE STATIONARY POINTS Choose numbers ve > 1 and d.

(6 . 1 1 )

f(r) ^

155

Then

log F(r) + d

for all v y 0, except in an open set of measure less than (6.12)

M = _§T e-d ',

d'-jf*.

A qualification r > RQ might be added to both hypo­ thesis (6 .1 0 ) and conclusion. This lemma is an immediate consequence of LEMMA 6 . B. Let f(r) be a function with continu­ ous non-negative derivative and f(0 ) ^ 1 . Then

% £ ed' - ( f ( r ) ) K-, except in an open set of measure less than

1

- d T

e

.

Indeed, if in an interval

(6*15)

I f > ed' - ( f ( r ) ) %

then the integral over that interval satisfies the rela­ tion ^ r < e - d' ^ , and thus it follows that the integral Jdr over the finite or infinite number of open intervals in which (6 .1 3 ) holds is less than (6.14)

y’Kdy «

e~dI.

Returning to Lemma A, we set

1 + r. + J (r-p)-ef (p)dp = g(r), 0

30 t!hat g = 1, -g£ = 1 for r = o and

156

III. THE SECOND MAIN THEOREM FOR MEROMORPHIC CURVES $*£ _ ef(r). dr2

Applying Lemma A first to -g® and then to g itself, one gets the inequalities ef(r) = dfg^ ed,( ^ ) \

v» **"

^ VI

each holding except in an open set of measure less than (6.14). Hence ef(r) ^ e d,.(g(r))K2, except in an open set of measure less than (6 .1 2 ). The hypothesis of the lemma g(r) £ F(r) results in (6 .1 1 )* Lemma A is of Immediate application to the inequality (6 .9 ), yielding 2Qp(r) £ k log Tp(r) + d

for r > RQ

except in a set of measure less than M =

e~dt where now (1 + k.)d1 = d - K-2log 2 .

(This upper bound of the measure does not depend on the curve, whereas RQ does.) But let us not get lost in such subtleties, and rather be content with the following weaker statement which is considered an integral part of the Second Main Theorem: THEOREM. inequality (6 .1 5 )

Given any numbers

.>

k

1 and d, the

2Qp (r) £ ic-log Tp (r) - d

holds almost everywhere, i. e. with the exception of an open set of finite measure.

§7.

INEQUALITIES FOR ORDERS AND STATIONARITY VALENCES 157

§7.

Inequalities for orders and statlonarlty valencea of all ranks Before drawing simple qualitative consequences from this proposition we make a few obvious remarks about in­ equalities that hold almost everywhere. LEMMA 7 . A.

If each of the Inequalities

f*i(r) £ o, ... , fm (r) £ o holds almost everywhere, then they do so simultan­ eously. Proof. The join of m sets of measure less than , ... , Mjh Is of measure less than M = M 1 + ... + — This lemma makes it possible to combine "almost every­ where Inequalities" by addition and pass from f1 f2, f2 £ f3 to f1 £ fy LEMMA 7 . B.

Under the hypothesis of Lemma A one can ascertain a sequence r1 < r2 < ... , r ^ ^ oo with v —* oo, such that f^r^) >0 (is of class D1 ). Rotation by 90 ° carries the line element (dx, dy) into one with the components . d d(f>1 d

(grad ^-grad ) ------- + ------ - * z dx

dx

dy

dy

behaves like a density with respect to analytic transform­ ations of the parameter z = x + iy. Its integral, the right member of (1.8) is the Dirichlet integral ^qA = Dg [0 i,0 ], and thus we may write that formula as follows: d*=

d g [(M

].

r In (1.9) both and ^ are supposed to be harmonic, in (1 .8 ) only = const. = R in K Q and hence di>= 0 along the in­ ner bank of 7 . Whereas dtf has different values on the inner and outer banks of 7 , the differential d ^ is the same along both. Hence the result is the following con­ denser formula (2 .6 ) ^ l o g l f l - d * - -l^loglf|-d# = with the sum extending over the entire Riemann surface !R. The analogy to the corresponding principle In the theory of meromorphic functions in the z-plane is obvious: the circle of radius R has been replaced by the arbitrary region G, and d£ takes the place of the differential of the argument t of z. The proof of Green's formula as given In §1 Is applic­ able to the region H punctured by the little circular holes; but in carrying it through one has to cover the perimeters of these little circles by admissible blocks. It seems more natural to use the following slightly modi­ fied procedure for generalizing Stokes1s formula to the case where the differential has a finite number of singu­ lar points P 2, ... in G. We see to it that the

1 7 2 IV. FIRST AND SECOND MAIN THEOREMS FOR ANALYTIC CURVES

blocks assigned to points ^ never cover With this precaution we cover G by a finite number of admis­ sible blocks P^; those assigned to the singular points will of necessity be among them. We cut out the singular point which is the center of its rectangle P^, by a rectangular "window” so small that no other of the blocks Pq penetrates into It, and extend integration over the perforated instead of the complete P^. The salient point is that for the purpose of calculating Riemann in­ tegrals over this space one can divide it Into a regular pattern of arbitrarily small rectangles. This modifica­ tion of our procedure becomes Imperative if a singular

O

F)&. 2. point lies on the boundary r. Again we cut a little rectangular indentation over which integration does not extend into the half rectangle P n G as indicated by figure 2 , and then let the indentation close in on pQ. In this way we establish the condenser formula for the case where f has zeros or poles on 7 or T; the line inte­ grals are then to be interpreted as improper integrals. §3 . The first main theorem and, the valence integral for T Armed with the condenser formula we encounter no essen­ tial difficulty in carrying over the whole theory of mero­ morphic curves as far as it centers around the order function. Let £ be a non-degenerate analytic curve in n-space of type K with coordinates x^. The quotient of two linear forms (ax) = ^~a^x^ is a meromorphic function f on !R and application of the condenser formula to this function results in the

§5 .THE FIRST MAIN THEOREM AND VALENCE INTEGRAL FOR T 17S FIRST MAIN THEOREM.

Let

N(a) = Z^(*;oO0i(>>) be the valence in G of the points of intersection of the curve £ with the plane (a) and set “ («> = f e £ losns3nrd*

m°(a) -

1o« n s c r d*-

Then (3.1)

N(a) + {m(a) - m°(a)| = T

is independent of the intersecting plane (ct). If we want to emphasize dependence on G we write N[G;cl], m[G;ct], m°[G;oL], T[G] instead of N(

T =

where S is the density defined by (3.5)

s = 2 l [xx-'^l2 z IxT

(Xj ---- i) dz

in terms of any local parameter z. Because $q (p )/R increases with G the same is true for the quotients N[G;a]/R[G]

and

T[G]/R[G].

By means of the osculating element yP

z

_ r* ^ _

>

dz

• • «

9

dp~1x , -n - 1 J dzp 1

we form the density (3.6)

Sp = "

|XP'1|2- |XP+1|2 z z jxfT1

Its zeros are the stationary points of rank p. THEOREM. The order Tp of rank p is given by the integral (3*7)

Tp = - h ^ sP-

Let e - 1 = e(pQ) - 1 denote the order of the regular analytic differential dC at any point in H. The crit­ ical points toare those where this order is positive. A local parameter z may be introduced by C - CQ = ze • The level lines = const, of the potential in H are ana­ lytic curves, with one slight exception: the level line passing through a critical point of multiplicity e - 1 displays In the neighborhood of that point the rosette characterized by the equation !Rze = o. For any value r

§3 . THE FIRST MAIN THEOREM AND VALENCE INTEGRAL FOR T 175 in the interval 0 < r £ R we define a region Gp by the inequality )> R - r. It encloses the nucleus K Q, is part of G and is bounded by the level line r p, $ = R - r. The potential 0(r,y>) of the condenser Gp with the fixed inner conductor K Q is given by the equation 0 throughout (except at a critical point) and £ d^ = [ di$- = 2tr. rr > We denote by N(r;a) and T(r) the quantities N[G;a] and T[G] for G = Gp, in particular N(r;c*) =

i/(u;a)^(r;»)

and set

Then (3.8)

N(r;a) + m(r;a) - m°(ot) = T(r)

is independent of (ot), and T (r > = b

1 76 IV. FIRST AND SECOND MAIN THEOREMS FOR ANALYTIC CURVES

(Notice that although m°[G;a] depends on G, m°[Gr;a] = m°[G;qt] is independent of r.) Because of the properties of the elementary function {r;p) of r, N(r;a) and T(r) are likewise of regular type, I. e. non-negative, non­ decreasing, and convex. The corresponding formulas for rank p offer no difficulties. At a critical point the condition of analyticity Is violated for that level line rp which passes through In the neighborhood of \>Q we use C - CQ as the indepen­ dent variable in terms of which double integrals are expressed. The e tips of Gp grouped around pQ are each, when properly sliced off, represented by a rectangle o < K(C-C0 ) ^ a,

|3(C-C0)I £ b

on the positive side of the line !R(C“C0) = 0 . As - CQ is not a local parameter at \>Q we cut a dent into the rectangle around CQ as indicated by Figure 2 and then let the indentation close in on CQ. (This is all the more required if f happens to have a zero or pole at the crit­ ical point Either in this way or by continuity we realize that our formulas hold also for the "critical values" of r for which the contour r passes through a critical point. Nothing prevents us from using C “ C 0 aaKind indepen­ dent variable in the neighborhood of a,ny point of H, critical or not. Theii the part of the integral J0SP ex­ tending over H assumes the form (3 .9 )

H

a*.-

The integration with respect to ^ runs along the level line (p) = = const, where -gr equals i We there­ fore write

§4.

EXISTENCE OP THE CONDENSER POTENTIAL

X? -

'

177

for ^ " conat - R - r,

u p

'

stOpfr) - L 3jd,>. Then (3.9) changes into R 2ir^(R-r)Qp(r)dr. Adding the integral of

over K Q which equals R times

2,rcp “ ^K0sP we find (3.10)

R Tp « CpR + ^(R-r)Qp(r)dr.

Notice that the positive constant cp is independent of G. Application of the resulting formula to Gp instead of G gives Tp (r) - cpr + ^(r-f)Qp(p)df. The quantity r corresponds to the logarithm of the radius in the z-plane (Chaps. II and III), which, from Chap. Ill, §6 on, has been designated by the same letter. §4.

Existence of the condenser potential and relaxation of conditions for boundary At this juncture it seems appropriate to describe in broad outline how the potential $ of the condenser G - K Q may be constructed. We open a competition to all func­ tions of class D1 in the closure H* of H which assume the values 1, 0 on y and T respectively; that function vjr will win for which the energy JH (grad\J02 assumes mini­ mum value (Dirichletfs principle). In order to compute

1 7 8 IV. FIRST AND SECOND MAIN THEOREMS FOR ANALYTIC CURVES

the energy integral we cover H*by admissible blocks as described In §1 . Consider one such rectangle P associated with a bound­ ary point on r, In the plane of the corresponding local parameter z = x + iy and extend the function^, defined in the upper half y } 0 of P to the entire rect­ angle by the condition of "oddity" (4.1)

>Jr(z) = -¥(z).

Because ^ vanishes on r, the function thus continued be­ yond the boundary is still of class' D 1. The method des­ cribed in the author's "Die Idee der Riemannschen Flache", pp. 104-106, for the construction of a minimizing func­ tion, uses the device of replacing an admissible function Inside a circle by the harmonic function^ which coin­ cides with ^ along the periphery. With the "odd" contin­ uation this method works also for points on the boundary (Inside P). The harmonized'^'* will be odd, ^ ( z -) (z), because*^ is. Hence the result Is a harmonic function $ which can be extended as an odd harmonic func­ tion a little beyond the boundary. More precisely, this continuation beyond the open segment of the real axis con­ tained in P^ takes place in each of the boundary blocks Pq after $ has been expressed In terms of the correspond­ ing Zq. (That no contradiction results along those parts of the boundary where several P^ overlap is understand­ able on the ground of Schwarz's symmetry principle. But since the construction itself furnishes the boundary con­ dition = o on r, in this stronger form, an explicit ap­ peal to that principle, as in §2 , becomes superfluous.) On 7 the symmetry condition (4.1 ) is to be replaced by $(1F) = 2 - $(z). The construction could not start unless we were in possession of at least one admissible function"^. This point can easily be settled as follows. We assign to each point of KQ a block P(P0) of center \>Q which does

§4.

EXISTENCE OF THE CONDENSER POTENTIAL

179

not penetrate Into G, choose a fixed positive number 6 < 1, and select a finite number of points , ) p ... in K q such that the corresponding contracted blocks P cover K q. With a basic probability function A(x) of class D 1 we construct the Dieudonn6 factors ,2,...) for this covering, as described In Chap. Ill, §3. Then ^

=

(

=

A 1 V

A2

V

. . . )

equals 1 in K Q, 0 in G, and is of class D1 everywhere. We could be satisfied with solving the electrostatic problem for analytic contours, were it sure that the Riemann surface is exhaustible by regions G thus bounded. This Is certainly feasible with piecewise analytic con­ tours. For a first orientation assume that G in the neighborhood of a point on T is represented in terms of a suitable local parameter z by a sector o < arg z < Tfa of a small circle around z = 0 ("analytic comer"). By z = ia (i = it?) we map that sector upon a half circle y o and submit the competing functions ^ to the condi­ tion that they have continuous first derivatives with respect toj, t? even on the bounding diameter = o of the half circle, Including the center j = 0. Then we can continue into the lower half by the condition^ (7) = ~ "3f(l )> although the lower half is not the map of anythixg on the Riemann surface and is in this sense fictitious. We shall therefore obtain a solution J of the electro­ static problem which, when expressed in terms of j In the neighborhood of the analytic corner and continued into the lower half circle by the symmetry condition $GjT) = -$(*), is harmonic in the full j-circle. When for the investigation of the neighborhood of a critical point on the boundary rp of Gr we made use of the Independent variable C - C0, we already abandoned local parameters for the complete neighborhood of in favor of what may be termed a straightening parameter, a parameter for the Inside half of the neighborhood which

180 IV. FIRST AND SECOND MAIN THEOREMS FOR ANALYTIC CURVES maps the boundary onto a straight segment. Once having conceived this idea, one realizes that it works for any Jordan contour. A Jordan contour r on !R does not nec­ essarily divide R into two parts, but It has two banks. The exact formulation of this fact is given by the fol­ lowing construction which takes place in a conformal z-neighborhood 51 of a point 0 on r (see Appendix at end of chapter). One can draw two cross cuts A = AB and Af = A !B* which have no points In common with r except their end points A,B and A !,B* respectively. These lie on r, A and A 1 on one side, B and B ! on the other side of 0. A "path" In the z-plaiie is a finite sequence of oriented segments in which the starting point of one segment coincides with the end point of the preceding one while the' segments have no other common points. A and A 1 may be assumed to be paths of this elementary nature. In the z-plane the arc BOA of r together with A forms a Jordan contour the interior of which (A) lies in 7(; similarly A1 and (A1)-. The cross cuts A and Af are such that (A) and (A1) are disjoint and every point in a sufficiently small neigh­ borhood of 0 lies either in (A) or in (A!) or on r. If G is bounded by a finite number of disjoint Jordan con­ tours and 71 is free from points on the other contours, then (A) or (A1) or both will be part of G. (It could happen indeed that they both belong to G, in which case r does not separate G from TJ; this will necessarily be so if r is a Jordan arc rather than a Jordan contour.) We map (A) if it belongs to G, one-to-one conformally upon the interior of the upper half unit square -1 < f < 1> o O < 1 in the ( J * jr + it?)-plane. One knows* that the mapping is one-to-one and continuous even including the boundary, and we can see to it that A, 0, B map into I « -1, 0, +1, hence the arc AOB of r into the base of *Carath6odory, Math. Ann. 73 0913), 305-320; Conformal Representation, Cambridge Tract No. 2 8 , Cambridge, Eng., 1932.

§4.

EXISTENCE OP THE CONDENSER POTENTIAL

181

the rectangle. All this ia now applied to the vacuum H with its bounding Jordan contours r and -y. With every interior point of H we associate ah admissible block. We may then cover the closure H # of H by a finite number of such blocks and of boundary patches of the nature of (A). We impose upon the admissible functions ^ the conditions that they are of class D 1 in the interior of G, and that for each of the covering boundary patches as a func­ tion of the corresponding i has continuous first deriv­ atives in the representing rectangle including the open base ^ * o, - 1 < r < 1 . The existence of admissible functions is secured by our above example because ^ = vanishes at all points sufficiently near to r and has the constant value 1 at all points sufficiently near to 7 . We shall thus find a minimizing function which in each of the boundary patches can be continued as an odd harmonic function, $(7 ) = a little beyond the base into the fictitious lower half of the %-square. This will be so on r while continuation beyond the Jordan contours 7 takes place according to the symmetry condi­ tion $(7 ) = 2 ). ^ On T we have = 0, > 0 . Thus if we construct the analytic function F = $ - 10 in the unit square with the real part $ its derivative will not vanish along the real axis. (Incidentally F itself is a straightening parameter of the nature of j .) This enables us to as­ cribe a definite negative charge — f d 0 « - — Sf^d/ 2ir 2rr J dV to any sufficiently small arc, and thus^to any arc, of r. Moreover the differential dF, which is regular analytic In H, does not vanish near the boundaries and hence the number of its zeros Is finite We stick to the hypothesis that the fixed nucleus K Q is bounded by analytic curves separating K Q from X Q; It

182 IV. FIRST AND SECOND MAIN THEOREMS FOR ANALYTIC CURVES 13 the boundary T of G which we permit to consist of Jordan contours of non-analytic character. We may then normalize = ^ = R, in such a way that the charge 1 r — d£ on 7 becomes unity. Reversing the order we now put the formulas involving the voltage parameter r first. Since rp is analytic for 0 = f d# = 2it, Pr 7' (4.2)

N(r;a )

(4.3)

+ m (r;a)

T(r) =

- m °(a) =

T(r),

J *( r; > )S (* ).

With r tending to R, N(r;a) and T(r) tend to T =

N(»;Ap" 1 ) - 2Vp ( * ; A p ) + Vp + i (* * A P+1 | +

186 IV. FIRST AND SECOND MAIN THEOREMS FOR ANALYTIC CURVES fv (»-i !• dZ is also defined In K Q but dQ Is not. However because of the constant argument of dC along 7 we have Jd(3 log |2) = fd(3 log dZ),

y

^

7

and by our lemma this integral equals 2TtnK (dZ) - 2T0CQ. Addition of r*n^ (dZ) to the sum ^0(r;t>M*;dZ) extending over p € Hr results in the same sum extending over Gp or, what Is the same, over the entire Riemann surface. Hence we are led to introduce the valence Vp(r) of all stationary points of rank p in Gp, vp (r ) = ^ y ( r'p) fvp (f)-ii and the quantity ^(r) = z Qr +JTrf(r;p) fe(*)-l i

(pCKp).

In the definition of V (r) the summation over p is unre­ stricted, whereas In Tj(r) the restriction p £ or if one prefers p € H, has to be imposed. It is also legiti­ mate to write ^ (r) = X Qr + if, as usually, & runs over the critical points in H each counted with its proper multiplicity. With these notations (5 .2 ) changes into the equation !Np_1(r;Ap_1) - 2Np (r;Ap ) + Np+1(r;Ap+1)| + Vp (r) = (5

log|^|-dJ-J

r*—,

logl^l-diH + »j(r).

'

§5-

SECOND MAIN THEOREM

187

It Is not difficult to remove the hypothesis that dZ is free from zeros and poles on 7 and to extend the for­ mula (5.3) to the critical values of r. In this connec­ tion observe that the definition of r^(r) is indifferent to whether or not one includes in the sum the critical points on r , because $(r;p) vanishes there. The func­ tion Vp(r) Is of regular type. Now add ®p_1 (r;Ap~]) - 2mp(r;Ap ) + mp+ 1 (r;Ap+1) to both sides of the equation (5 .3 ) and introduce the quantities (5.10 2 flp (r) = b S

l0S sj-d*, 2 n ° = -1^5 log sg-d^. rr y A formula results from which the auxiliary A fs have dis­ appeared: (5-5r ) !Tp_1(r)-2Tp (r)+Tp+1(r)! + Vp (r) = !Cyr)-o£| +Y,(r). Finally let r tend to R. Then the three T(r) in (5*5r ) tend to T, V (r) to the valence of the stationary points in G, vp = X ^ o >

(y*> -

and Tj(r) to ^ = ■'itG] = X 0R + Y j W ’

the latter sum extending over all critical points to in H. (It is here that the finiteness of their number be­ comes important.) Therefore Hp(r) must tend to a limit flp = -Qp[G] and we get the FORMULA OF THE SECOND MAIN THEOREM: (5.5)

!Tp_, - 2Tp + Tp+11 + vp = ( n p - n j ) + v

188 IV. FIRST AND SECOND MAIN THEOREMS FOR ANALYTIC CURVES Two expressions for (5.6)

are available

i, = x 0R + X*(*») - x - R -

.They show that (5.7)

X 0 £ t)/R £ X.

Application of (5*5) to Gp instead of G carries us back to the equation (5.5p ). U p is the familiar compensating term for the station­ ary points of rank p. A new feature is the appearance of the term ^ which depends on the configuration K Q, G but neither on the curve nIs the same whether measured in the z- or the z -plane. The sum of all angles around p Q is 2% or rc> according to whether v0 is an inner or a boundary vertex. Hence by adding over all triangles equation (5 .8 ) leads to (5 .9 ) 2ltx=

d(arg dz) = irh2 - 2»rh0 - jrh^. A

But since every inner side belongs to two triangles, every boundary side to one, h^ + 2h 1 = 3 h g or h^ = 3h2 - 2h1. Thus (5.9) finally yields the well-known topological definition of the Euler characteristic, X = h 1 - h Q - h2. §6.

Positive and zero capacity. The "little" terms m° and -H-0 Once in possession of the formulas for the first and second main theorems we can turn to their evaluation.

190 IV. FIRST AND SECOND MAIN THEOREMS FOR ANALYTIC CURVES We shall find that our estimates depend on a fundamental distinction between two sorts of non-compact Riemann sur­ faces, those of positive and of zero capacity, or, as F. Klein would have said, of hyperbolic and parabolic types (leaving the term elliptic to describe the compact surfaces). In case the voltages R = R[G] for all admis­ sible G are bounded, let I be the least upper bound; in the opposite case set I = oo . For any positive € we can ascertain an admissible G^ such that R[G^] ^ I - € or 1/€ respectively. Then R[C] > I - € or > 1/€ for every G which encloses a certain compact set, namely the clos­ ure of G^. We express this fact by saying that R[G] tends to the limit I under exhaustion of the Riemann surface by G, R[G] — ► I

for

G — * R,

and speak of the two cases as those of positive capacity 1/I (I finite) and zero capacity (I = oo). A simply connected non-compact Riemann surface !R is conformally equivalent either to the whole z-plane or to its unit circle K 1; we maintain that this Is so according to whether !R is of zero or of positive capacity. Indeed, the unit circle K 1 has positive capacity, whatever com­ pact part of it the nucleus K Q occupies, because the electrostatic problem has a solution for the condenser K 1 - KQ in the z-plane. In order to show that the com­ plete plane is of zero capacity for any compact nucleus K q, enclose K Q by a circle K^, \z\ < pQ, and let G be any circle |z| < p of radius p > pQ. The voltage of. G - K A Is larger than the voltage log of G - K*, and u fo the latter tends to infinity with p — ► oo . It is desir­ Kind able, and should not be too hard, to prove that the dis­ tinction between positive and zero capacity is indepen­ dent of the nucleus K Q, even for multiply connected sur­ faces. But we shall not follow up this question here; our standpoint throughout has been that the structure

§6.

POSITIVE AND ZERO CAPACITY

121

underlying our investigation is a Riemann surface !R plus a definite nucleus K Q on it. ipQ[G] being a positive function of G and Jj[G;a] a function of G which may involve some parameters a, let the limit equation ip = o+(^Q)

(uniformly in a)

express that a number B and a compact set K can be ascer­ tained such that Hj[G;a] < B-ipQ[G] for all G z> K and all values of the parameters a. Notice the one-sidedness of the inequality] For a non-negative ip the symbol 0+ has the same meaning as 0. Similarly ^ = o (ip )

(uniformly in a)

states that for every € )> 0 there exists a compact set such that MG;a] and hence on G. Fortunately this dependence is controlled by the simple law

df° Dropping the index p we have

- s j 0. t°

log(l/S^) £ log(l/S Q) + 2 log R/RQ, consequently - 2a 0 [G] = - i ^ l o g O / S j M * £ ^log.(l/S

+ 2 log

The nrst part on the right side may now be treated as m°[G; K; T = T[G], R = R[G], s(r) s[Gr ]. Continuity of s(r) for 0 r < R is assumed. Clearly s = implies s1 = ^ ( T ) for any function s1 £ s and s2 = uig,(T) for s2 = s + c, c being any posi­ tive constant and B 1 = B*ec. We may now state the rela­ tion (7.1) in the abbreviated form (7.3)

2ilp = 0 for infinite I, and A(r) = —Vi (I-r) with some exponent h ^ 1 for finite I. Instead

§7.

THE FUNDAMENTAL INEQUALITY

197

of Lemma III, 6 . A. we now obtain LEMMA 7 . A. the inequality (7.4)

s ( r )

The relation s =

*-2log T(r) +

(1

+ *.) log

T ) implies that

A(r)

+ d

holds for all admissible G and 0 < r < R under the condition that Gp and r does not belong to a certain open subset of the interval 0 < r < R of measure less than M = ^ r-e_d' Here

k

| (1 + *c)d.t = d - *2log B|.

y 1 and d may be chosen at random.

Indeed, under the assumption f( 0 ) ^ 1 the inequality {g > ed 'will hold in an open set, the measure ^A(r)dr of which must be less than

Set 1 + r + [ (r-c>)e3^ d p = t(r) o 1 and apply this remark first to f(r) = -g- and then to f(r) * t(r). Observing that the hypothesis (7 .2 ) for Gp instead of G implies t(r) £ B-T(r) as soon as Gr ~=> K, we find that under this hypothesis, and excepting an open set O -H i e of measure less than , s(r)

K2log (B-T(r)) + (1 +K)(log A(r) + d !)

Given G, those r for which Gp K form a certain open interval Rg < r < R ending at R (which of course will be empty unless G=>K.) It is remarkable that the upper bound M for the measure of the exceptional set is independent

198 IV. FIRST AND SECOND MAIN THEOREMS FOR ANALYTIC CURVES of the choice of A(r), and in the application of the lem­ ma to (7.3) also from the curve Let us set ( (1 + K)*h log r (7.5) 2L(r) = (l+K)log A(r) = ] I (1+k) h log

(I = oo ), (I finite).

The quantity that matters in the formula for the second main theorem is the difference -fl - XlfJ, and since 0 P P -Xl [G] = o+(R) is all we can guarantee in the case of zero capacity, we lose little by choosing a high exponent for h in A(r) = rh , but we gain inasmuch as the sets of fixed A-measure M will be the smaller, the higher h is. In case of positive capacity, the term L(r) is a neces­ sary evil, and again a high h in A(r) = (I-r)”*1 is on the whole advantageous. After having fixed A^r) we introduce the phrase "a function f[G] satisfies the inequality f [G] £ 0 for almost all G" with the following meaning: There exists a posi­ tive number M and a compact set K such that for every ad­ missible G those r in the Interval R^ < r < R (defined by Gr 3 K) for which f [G ] )> 0 form an open subset of mea­ sure less than M. Inequalities holding for almost all G 0 will be marked by || . Replacing k. by n we have then proved: LEMMA 7 . B. any constants (7 .6 )

||

3 £

s =cj(T) implies the fact that for 1 and d the inequality k,

log T

+

L(R)

+

d

holds for almost all G. Because the join of several open sets of measures less than M 1, ... , ^ is one of measure less than M = M 1 + ... + Mjjj we can state the combination principle:

§7.

THE FUNDAMENTAL INEQUALITY

199

LEMMA 7 . C. Given a finite number of inequalities fj_[G] £ 0 the fact that each is satisfied for almost all G implies that the same is true for them simul­ taneously. Having proved an Inequality f(r) £ 0 for all r in the interval R^- < r < R except In a set of measure less than M, we should like to conclude that the Interval (R’,R) of measure p(RTR) = M contains values r for which f(r) £ 0. The interval (R',R) will exist as soon as ^i(OR) )> M. The only doubtful point is whether GR! contains K and thus R ’ y R^. We prove that this is so as soon as G is sufficiently large, I. e. as soon as G encloses a prop­ erly chosen compact part of !R. Take an admissible G° => K. The function $> ^ will have a positive minimum ) ^ (5 or 0q (P) 2

for G => G° and * € K.

In other words, G => G° implies that the "trimmed" G(i- G° and h ((1- M. Gr arises from G by trimming, the width of the rim taken off being p(rR). We may now state LEMMA 7. D. Suppose the inequalities f^EG] £ 0 (I = l,...,m) satisfied for almost all G. Then there exists a number M such that each sufficiently large admissible G can be changed by cutting off a rim of ♦ width less than M into a region G (=Gp, fi(rR) < M)

200 IV. FIRST AND SECOND MAIN THEOREMS FOR ANALYTIC CURVES for which all the inequalities f^[G ] ^ 0 are satis­ fied. If G varies so as to exhaust 1R e. g. by run­ ning over a sequence G^1 \ ... , then the * I** trimmed Kind KindG exhausts a . (The higher the exponent h of the measure function A(r) the less perceptible will be the trimming for large G.) LEMMA 7. E. Suppose that for any given positive constant € the inequality f[G] < € holds for almost all G. Then lim f [G] £ 0. Proof. Evident. This fact remains true even if we limit G to what we describe as a complete system 6 of admissible regions. Completeness requires (1) that for any compact K there exist regions G € €> containing K (IK is exhaustible by regions G of the system 6), and (2) that Gp (0 < r o. p ^

p

p

- .a° o = o p’ p

§7. o

201

THE FUKDAMENTAL INEQUALITY

is a quantity of type (o) and -O’ £ £1 , thus H * =

~to(Tp ) and (7.7)

n p - Xl° = (o) + |u>(Tp ).

Suppose we know that T grows faster than r|, and also faster than log .-jtr Kindthe case of finite I. Then (7.7) implies that, given any € ^> 0 , the left member of (5 -5 ) is less than for almost all G. As a matter of fact it is sufficient to assume the conditionjust mentioned Kind to be satisfied for almost all G. Hence we put down the following HYPOTHESIS ities (R1)

Given any positive €, the inequal­

*1 < €-Tp

II

(p =1 ,...,n-1 )

hold foralmostall G, and in case I is same istrue for the inequalities (fi")

II

1o S

i ^r

finite,

the

< €' V

We speak of the restricted hypothesis if we limit (R’) and (fi,f) to regions of G of a given complete system 6 . Since Tp is an increasing function of G, (fi") even in the restricted sense implies [oo]. As in III, §7, we conclude: THEOREM. Let € ) 0 be given. sis H the inequalities » TP+1 < (1 + P + €)V

Under the hypothe­

TP-1 < (1 +

Vp < (p(n-p) + €)Tp "

Vp + (TP+1 ♦ ' W

will hold for almost all G.

< (2 + €)TP

+

202 IV. FIRST AND SECOND MAIN THEOREMS FOR ANALYTIC CURVES Examples show that the hypothesis ft is actually essen­ tial (cf. Chap. V, §3). It is no wonder that the struc­ ture of the Riemann surface should reveal itself in some way in the behavior of its analytic curves; we learn here that it does so primarily by way of the one quantity r\ = r|[G]. The part (fi?) of hypothesis 5o is satisfied a fortiori if for any € )> 0 the Euler characteristic X = X [ G ] fulfills the inequality II

X < €— E R

for almost all G. Above we have pleaded for a high value of the exponent h In (7«5)* However the lowest possible value, h = o and h = 1 respectively, is indicated if we care for no­ thing but the fact that the resulting inequalities hold for almost all G, disregarding how big or small the ex­ ceptional r-sets turn out. Pure theorists that we are, we propose to abide by this convention hereafter, and thus write (7.8) L ^ = 0 (I =

cd

),

L (C=

log

(I finite).

The assumption that really mattered In the proof of our last theorem is the Inequality (7-9)p II

t] +

^

holding for almost all G after € ) 0 is arbitrarily fixed. In case of positive capacity it is obvious from the com­ bination principle that (7 -9)p ia Implied in the two simultaneous inequalitiea (fif), (fi"), but the converae ia leaa obvioua becauae we are not sure that *) is positive. However, from rj ^ x0R there follows that ^ where the constant t]Q = 0 if Xq ^ 0 and = Xqi if X q < 0. Hence (7«10)

log

0,

2 log

§8.

SPECIAL CASES OF ROTATIONAL SYMMETRY

205

as soon as R is sufficiently near its upper bound I, and then (7.9)p implies ||

T} < € Tp ,

f log

< € Tp .

It is therefore perfectly legitimate in either case of zero and of positive capacity to formulate the hypothesis = ftp for the rank p by (7-9 )p• A last remark is to the effect that, granted [0 0 ], the hypothesis ft for one p is sufficient; it will then be automatically satisfied for all p. Indeed, without any further assumption than [ 00], we have (7.11) || Vp + (Tp+1 + Tp.,) < (2 + €)Tp + r, + V Hence (7.9)p_1 implies

II

(1

-

€)Tp _1
^z0(RjzMz:a). The exponent A in Borel fs proposition betraya its origin within the frame of the old theory of entire functions of finite genus. It would be more satisfactory if, by shedding these remnants, one could prove Instead: There cannot be more than 2 distinct v a l u e s a for which (1.1)

N(R;a)/T(R) — ► 0

as R — ► oo .

R. Nevanlinna introduced the limit superior 1 - 6 (a) of the quotient (1.1) and called 6(a) the defect of the value a. Its vanishing indicates that the a-places of f do not permanently fall short of that density which the order T of f leads one to expect: Tim N(R;a)/T(R) = 1.

§2.

WEIGHTED AVERAGES.

THE BASIC GENERAL FORMULA 213

The extreme in the other direction is (1.1) or In case of zero and positive c a p a c it y respectively. In terms of the symbol u the resulting relation may be ex­ pressed as (2.7)

2 6 - w(T) - cjgtT)

216

V.

THE DEFECT RELATIONS

where the parameter B can be chosen as A + 1 for zero capacity, and as any number > A for positive capacity. (Remember that in the latter case the assumption [oo ] Chap. IV, §6, Is made once for all.) This is the general scheme which the exploitation of (2.5) will have to follow. We need not attempt to prove the existence of the Integral (2.6) in this generality. We shall presently specialize p so as to obtain for 0 a sum of' quantities and nip, and under these special circumstances there will be no doubt of the existence of © nor of the legitimacy of the entire procedure. Finding It hard to hit on the right choice of ^ right away, we follow the historical development by first studying the lowest case n = 2, 1. e. the meromorphic functions on IR. §5.

Defect relation for meromorphic functions on a Rie­ mann surface Assume f(a) to be homogeneous in the stricter sense that the equation p(T 2. Set k = k f/A and observe that one may so dispose of k 1 > 2 and A < 1 as to assign to k an arbi­ trary value y 2. We thus have proved: THEOREM, Assume the order T of the analytic curve to fulfill the hypothesis and k to be any number > 2 and let b range over any number q of distinct points. Then the inequality (3.6)

|[

V+]Tm(b) 1 ).

But by some simple prestidigitation one may eliminate the factor — jp*. Indeed, according to (IV, 7 •1 0 ) (3*9) II

+ log

< cT

would imply n

i log itr < ct

as soon as R is sufficiently near I, and multiplying the last inequality by K.-1 and adding it to (3 -9 ) one obtains

§3.

DEFECT RELATION FOR MEROMORPHIC FUNCTIONS

Given c < s -1 one may choose k .) 1 so that less than s -1. Hence if (3.10)

||

r^+ Hog

j Tr

kc

221

is still

! < (a - 1 - €)T

holds for some positive constant € and almost all G of a given complete system €>, then the non-constant meromor­ phic function f(^) of order T cannot omit more than 3 Q 2 ) values. The braces on the left indicatethat the term surrounded by them is present in case of positive capacity only. Picard*s theorem (s = 2 ) is obtained under the condition (3*1V)

tj

+ flog jZpi

(1 “ €)T.

The following two examples show that this condition may not be replaced by one appreciably weaker; they will thus clarify the role of the double-barreled hypothesis ft for Picard*s theorem. (1) The function f(z) - (1 - e2 )1/2 is single-valued on a Riemann surface 71 the two sheets of which spread over the z-plane with the ramification points z = 2irin (n = 0 , +1, +2 , ...). The function omits the 3 values +1, -1, 00 . Circles of radii rQ and R(>r0 ) around the origin are used as inner conductor K Q and exhausting region G respectively. Since fJ(z) = \ log -jlj-

for r Q £ IzI £ R,

the sum 2/*~< j>(to) extending over the critical pointa

to in

G - K Q is the valence in the z-plane of those zeros of f2 » 1 - ez which lie outside the circle K Q and hence, provided rQ < 2rr, ^l^(to) ~ ^ - log •£-,

0

t, -

Y.M - i log ^

~

h - 1o«

222

V.

THE DEFECT RELATIONS

On the other hand t

ff I - | r f f 2 | ~

In a similar fashion the algebroid function f(z) = Z 1 /3 (1 - e ) ' , which omits s + 1 values, illustrates the fact that the condition (3.10) is the sharpest of its kind, as far as the part ^ of its left member is con­ cerned. (2) In order to cover the f !-term in (3.11) and (3.10), consider the unit circle |o>| < 1 in a complex co-plane as our Riemann surface R and the modular function t(cj) as our meromorphic function f on R. It omits the 3 values o, 1, od • (As in §1, the upper half-plane has been transformed into the unit circle.) With the same choice for and G as in the first example (rn < R < 1 ) "R the voltage of G - Kn is u « log ■£- and hence tends to 1 0 I « log — with G — ► R. It is not difficult to prove r0 that the order T(R) of the modular function behaves asymp­ totically for R — ► 1 exactly as log JTR ~ log "JZy* More generally, the automorphic function tg(cj) which maps the unit circle in the cu-plane one-to-one conform­ ally upon the universal covering surface of the t-sphere pricked at s + 1 given points t » t^, ..., tg+1 omits these s + 1 values, and its order T(R) behaves asymptoti­ cally as -grf-log yrg- Cf. R. Nevanllnna [10], pp. 25^ and 2 5 9 .]]* *Gunnar af Mllstrtfm in an important paper, Acta Ac. Aboensis 12, No. 8, 1 9 ^0 , of whose existence I learned only after completion of the manuscript of this Study, deals with meromorphic functions for multiply connected regions in the z-plane for which a sort of limiting potential i exists. Both this and E. Ullrich’s investigation of algebroid functions quoted in Chap. IV, 58, point toward a* general theory of meromorphic functions on Riemann sur­ faces as developed here; the quantity t\ is clearly in evidence. H&llstram's paper is a source of highly inte-

§4. §4.

CONVERGENCE OP AN INTEGRAL

223

Convergence of an Integral

We have still to prove the integrability of the weight function (3.2).

To this end we make use of the triangle

property of distances in 2-space. || a:x|| + || b:x|| ^

LEMMA k. A. Proof.

|| a:b||

Supposing a.jb2 ~ a 2b l ^ 0 we W x i = 3ai + t b i

(n = 2). se^ ^1 55 19 2 ^*

Then

la^g - a ^ l = la^g - a^l-lsl, |bTx2 - b2x 1 |« la^g - a ^ M t l . With the normalization |a| =* |b| ® 1 we find

|x|2 « |a |2 + 111 2 + (a|b)sT + (b|a)¥t, and it remains to prove the inequality

(Is| + |t|)2 ^ |s|2 + |t|2 + (albjst’ + (b|a)¥t, which, however, follows at once from the trivial rela­ tions I(a|b)sT| The lemma w i l l be (*.i)

-

|(b|a)st | £ |sI - 111.

u s e d in the for m

lla$||+ || b5ll^

Hasbll.

Let now b run over a set ]> of q distinct points

b 1,

resting examples; e. g. there is one in w h i c h T is of lower order of magnitude than y\. — A different approach to Picard fs theorem, leading to a ver y significant gen­ eralization but away fr o m the analytic curves, is L. Ahlfors fs metric-topological study of the Riemann surface over the w-sphere of the inverse of a meromorphic Inunc­ tion w - f(z). See L. Ahlfors [£], a n d for nniltiplyconnected surfaces, J. Dufresnoy, C. R. Acad. Sci. Paris,

2\2 . 19M, 7^6-749.

22k

V.

THE DEFECT RELATIONS

... , b and 2d be the least of the mutual distances || t^rb^N (j ^ k). According to (4.1), all q distances II bjtt|| of the bj from an arbitrary a, except perhaps one, are greater than or equal to d. Denote by b 1 this excep­ tional b if it exists; if there is no exceptional b choose b- at random among the b.; Kind then j X jlog(||b.air2A) £ log(||b1air2A) + 2A (q-1 ) log d"1 £ log

"I ||bja|r2A + 2(q-1) log d"

The exceptional b 1 varies with a, but if we skip the intermediate member, our inequality holds for every a and therefore (4.2)

CTT-bIPaottl2 )“A £ (^)2{q"l)* Z b IIMr2A:

the product can be appraised in terms of the sum. Thus the integral of (3.2) will certainly converge if _p \ the integral of the individual term |[ba.|| does. In examining this we may take advantage of unitary invari­ ance, and assume b, = 1, bn = 0. Then Lemma III, 2.B, I C. —hi becomes applicable with n = 2, g(t) = t” , yielding the value I.-2A -|a|2 „ = _2r1.-A^ j‘||ba|r2A-e_,a' da, da, da^ d«2 = ir ^ t “ dt =

rr2

One now sees why the assumption A < 1 is essential. the constant A in (2 .5 ) we may choose

As

A A = -31-A 4 N T) 2(q_1 5 It depends on the exponent A and the configuration of the points b, but on nothing else. §5.

Starting the Investigation for arbitrary n The experience gathered for n = 2 encourages us to try our luck with a weight function of the simple type p («) = g( lfb«ll2 )

§5.

STARTING THE DJVESTIGATION FOR ARBITRARY n

225

where g(t) ia a positive function in the interval 0 £ t £ 1 and b any non-vanishing covariant vector which we normalize by |b| = 1. In order to compute S' for given vectora x ^ o and x ! we adapt the unitary coordinate ayatem to the data according to the acheme X =(x.|, (5*1)

^

0,

b = ( b 1,b 2 , =

(x2 ,

0, 0,

••• 9

0),

• • • f

X ^ , X*

. . . ,

x^).

Then 2

s*

r lxs'*2+ ---*xA“ n |S

J

~ ^ (lt>'Ahl)' ®?(Ah)' formed by means of the weigiht k_

it (

*5)

^ ? +1

/-LMbUll Ihl2

lb' I2 Vi

Vt

Apply the appraisal (IV, 6 .5 ) to [b!A ] rather than A . Vi The result is a relation mp {b,A j = (o) holding uniformly for all A*1, all normalized b perpendicular to fA*1!, and all steering functions J(s) satisfying conditions (I) and (II). We have thus obtained the inequality (6.6)

£,f?S-F(Ah ) £ en_p_h (Tp (Ah ) + (o))

for h = n - p - 1. We wish to free the result from the "j* hypothesis that b is perpendicular to [A i. To that end let b be an arbitrary normalized vector and % its com­ ponent perpendicular to fAh |, |b| = |[bAh ]| 1 ia a fixed conatant, but 1 = 1[G] ao dependa on G aa to tend to infinity with G — ► !R. We have con­ vinced ouraelvea that the beat choice of 1[G] givea the aame reault ^ ( T 2 ) with the aame parameter valuea B > 2eCw2 aa before. Thia aeema to indicate that Ahlfora^

§9.

DISCUSSION OP THE DEFECT RELATIONS FOR POINTS

2bj

estimate is an optimum which cannot be improved by de­ vising more refined steering functions J[G;s]. §9.

D is c u s s io n

o f th e

d e fe c t

r e la t io n s

f o r p o in ts

For points in "general position" we can choose w(a) = 1/p without violating the condition that no p-element carries a load exceeding unity. The inequality (8.2) then becomes (9.1)

II

+ ^ Z a (Sp(a)-mp_1(a))
, or that there ia no |XS ! with which more of them are in i-incidence. What we have to do ia to replace p by p ! in (12.3) and then to aum over p f = 1, ..., p. But before doing ao let it be observed that the aum roay form­ ally be written aa a aum extending over i from -oo to +oo , provided we define the binomial coefficienta for all i by the identity in x, o+*>n Summation by parta tranaforma the aum over i in (12.3) into ^ ■ ( p + i - l w n - p - i , _ ,p+iH n-p-i-1 ,|T i n h-I ’ 4 + 1 n h-i-1 ' ’ p+i'

g z a

and the expreaaion reaulting from aummation over p* = 1, . • . y P la c 2 - 5 j Z fl* p ( A >

-

1 ,3

< i + , > < £ : £ ! ) ' Ts -

v

Again the aum (i,a) ia a double sum over 1 and a with the range i < s ^ i + p

or

a - p

Z S ^ -p K 8! 1 > -

In this case it is a little easier to utilize the recur­ sion relations for the binomial coefficients by means of their generating function. (3^1 is the coefficient of t^x*1 in the polynomial (1 +tx)3”1(1 +x)n~3. Hence

Is the coefficient of t° in the Laurent series of the following function of t: Z £ s_pt'l f(1+tx)3"1(1+x)n'3 - t'1(l+tx)s(1+x)n~3'1 | +- _

=

( 9 —P

)

q _

1

----:— (1+tx) 1-t'1

-p|_ q _

1

_

(1 +X ) 3 1 f(l+x) - t

1

(1+tX) I

Thus the sum (12.6) equals (S—1 \(n-s-1 * vs-p' vh-s+py9 and the expression (1 2 .5 ) changes into (12.7)

Z ASp(A) + Z ^ S l p H h l s ^ a

" (h )Tp*

It is this combination which we know is essentially nega­ tive, provided the order T satisfies the hypothesis ft and the h-elements JAI are in general position. The result is of particular interest for h = n - p, because Sp(An”P) is the compensating term mp(Ap )lAP=*An”P] which appears in the First Main Theorem, and hence is a real defect, not merely a defect of a defect. What we find is the following

268

V.

THE DEFECT RELATIONS

THEOREM. Under the assumption of general posi­ tion and under the hypothesis ft the inequality (12.8)

||

H

m p(AP)