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Functional equations in the complex domain

Ludwig Reich

Contents

1. Introduction 12. The Herglotz Lemma: Applications to the theory of trigonometric

functions. Some generalizations of the Herglotz Lemma 43. Addition formulas: Characterizations of the cotangent function.

Special Eisenstein series 74. On the identity of C. G. Jacobi 115. Functional equations in the theory of the Gamma function 126. The transformation formula of the Theta function and systems of

linear functional equations in analytic number theory 157. Generalized Dhombres equations in the complex domain 19References 24

1. Introduction

In these lectures we want to show the role of certain functional equations inclassical analysis and to give one example of recent research on the gener-alized Dhombres equations which belong to the class of iterative functionalequations.In Section 2 the so called Herglotz Lemma in its additive form is used to con-struct the decomposition of the cotangent function into its partial fractionsand to prove a characterization of this function. The multiplicative versionof the Herglotz Lemma is used to prove the Eulerian product representationof the sine function by its duplication formula. We conclude this section bydiscussing several possibilities to generalize the Herglotz Lemma to more gen-eral linear functional equations as a uniqueness theorem for the consideredequations and to provide estimates of growth of possible solutions.While the Herglotz Lemma concerns functional equations in a single vari-able we study in Section 3 functional equations in two variables, namely so

2 Ludwig Reich

called addition formulas. The addition formula for the cotangent function isapplied to present two characterizations of this function within the set ofmeromorphic functions. The first one uses also a non linear differential equa-tion satisfied by π cotπz in the neighbourhood of the pole z = 0, which canbe transformed to a Briot-Bouquet differential equation. The second one isconnected with reducing the addition formula to the linear functional equa-tion which appears in the Herglotz Lemma.Another class of meromorphic functions where an addition formula (provedby A. Weil) play an important role are the special Eisenstein series

εk(z) =

∞∑ν=−∞

1

(z + ν)kfor k ≥ 2

and

ε1(z) = limn→∞

n∑ν=−n

1

z + ν.

From the addition formula one can derive two basic algebraic relations forEisenstein series, a recursive formula for the derivatives ε′k and the repre-sentation of εk (k ≥ 2) as polynomial in ε1. This contains the whole theoryof the special Eisenstein series and of trigonometric functions. We mentionalso the problem to characterize the functions εk (k ≥ 2) by certain additionformulas.In Section 4 we show that certain linear functional equations for the infiniteproduct (|q| < 1, z ∈ C \ {0})

A(q, z) =

∞∏ν=1

(1− q2ν)(1− q2ν−1z−1)

and for the Laurent series

J(q, z) =

∞∑ν=−∞

qν2

zν

lead to an elegant proof of Jacobi’s identity

J(q, z) = A(q, z).

In Section 5 we deal with the Gamma function Γ. The theory of this functionis based on its difference equation. Wielandt’s theorem characterizes Γ as thesolutions of its difference equation which is holomorphic the right half planeRe z > 0 and bounded in a certain strip parallel to the imaginary axis. UsingWielandt’s theorem one can obtain the duplication and the multiplicationformulas for Γ and Euler’s formula relating Γ and the Beta-function, as wellas Euler’s and Hankel’s integral representations. We also give a characteri-zation of Γ by a system of functional equations consisting of the differenceequation and the duplication formula which uses again the multiplicative Her-glotz Lemma from Section 2. We mention as an open problem, to characterizeΓ as solution of a system of functional equations consisting of certain cases ofthe multiplicattion formula, possibly together with the difference equation.

Functional equations in the complex domain 3

In order to indicate the importance of functional equations from the the-ory of theta functions we present in Section 6, following C. L. Siegel, thetransformation formula

ϑ

(−1

z, w

)=

√z

i

∞∑n=−∞

eπiz(n+w)2

(Im z > 0, w ∈ C) of the function

ϑ(z, w) =

∞∑n=−∞

eπin2z+2πiw.

The function g derived from ϑ(z, w) as

g(z) = eπiz12 ϑ

(3z,

1

2+z

2

)(Im z > 0)

satisfies the system of linear functional equations

g(z + 1) = eπi12 g(z)

g

(−1

z

)=

√z

ig(z)

which is also satisifed by the Dedekind η-function

η(z) = eπiz12

∞∏n=1

(1− eπinz) (Im z > 0).

A kind of uniqueness result on this system of functional equations, based on

the behaviour of the function z 7→ g(z)−η(z)η(z) if z goes to infinity in the upper

half plane, yields η = g, and hence Euler’s famous identy∞∏ν=1

(1− qν) =

∞∑ν=−∞

(−1)νq3ν2+ν

2 (|q| < 1).

In Section 7 we treat a different topic, namely the existence of analytic solu-tions of the generalized Dhombres functional equation

f(zf(z)) = ϕ(f(z)), (1.1)

where ϕ is the given function and f the unknown function. (1.1) belongs tothe class of iterative functional equations. The difficulties in such problemscome from the fact that the unknown function f or an expression involvingf (here zf(z)) is substituted into the unknown function.We describe two kinds of problems, namely the global problem where weare looking for holomorphic (or meromorphic) solutions f of (1.1) in a fixedregion D of C, and the local problem where we ask for solutions f of (1.1)which are holomorphic in a neighbourhood Df of z0 ∈ C, with initial valuef(z0) = w0. The region Df may depend on the individual solution f , and ingeneral this is the case. The case z0 = 0 is particularly important. We startby presenting results from the local (and formal) theory where we decribe thestructure of the set of all local analytic (or formal power series) solutions, and

4 Ludwig Reich

we give necessary and sufficient conditions for the existence of non constantsolutions in some cases of w0.From these local results one can deduce a characterization of those equations(1.1) which have non constant polynomial solutions in C. We obtain exactlythe equations (1.1) with

ϕ(y) = w0 + (y − w0)yk

with k ∈ N, w0 ∈ C. In this case all local analytic (or formal) solutions f arethe polynomials f(z) = w0 + ckz

k (ck ∈ C).We discuss some open problems related to the functional equations (1.1).

I am grateful to Dr. Jorg Tomaschek who has helped me very much to finishthe notes of these lectures.

2. The Herglotz Lemma: Applications to the theory oftrigonometric functions. Some generalizations of theHerglotz Lemma

We begin with

Theorem 2.1 (The additive Herglotz Lemma). Let g be an entire functionand assume that

g(z) =1

2g(z

2

)+

1

2g

(z + 1

2

), z ∈ C (2.1)

holds. Then g is constant.

For the proof, F. Schottky considered: Mg(r) = max {g(z)||z| ≤ r}, andapplies the maximum principle.According to G Herglotz, we differentiate (2.1) and then can consider Mg(r),without using the maximum principle.The functions π cotπz and

ε1(z) = limn→∞

n∑ν=−n

1

z + ν=

1

z+∑ν≥1

2z

z2 − ν2

are 1-periodic meromorphic functions,the poles are for both functions exactlythe points z = ν ∈ Z and the principal parts are 1

z−ν (ν ∈ Z). It is well known

that π cotπz satisifes (2.1), whenever z, z2 ,z+1

2 are not poles, ε1(z) is also a

solution of (2.1). To see this one shows that the partial sum∑nν=−n

1z+ν

satisfies (2.1) up to a rational remainder term which tends to 0 for n → ∞,for each fixed z. Hence π cotπz − ε1(z) can be considered as entire functionand it satisfies (2.1). By Theorem 2.1 it is a constant, and letting z → 0 onesees that

ε1(z) = π cotπz. (2.2)

This is the famous decomposition of π cotπz in its fractional parts.These arguments yield also the following characterization of π cotπz in theclass of meromorphic functions.


Theorem 2.2. Let g be a meromorphic function, having poles exactly in z =ν ∈ C, with the principal parts 1

z−ν , and assume that g satisfies

g(z) =1

2g(z

2

)+

1

2g

(z + 1

2

)if 2z, z, z + 1

2 is not a pole of g. Let g be an odd function. Then

g(z) = π cotπz.

The following is a multplicative version of the Herglotz Lemma.

Theorem 2.3. Let g be an entire function without zeros and assume that

g(z) = cg(z

2

)g

(z + 1

2

), z ∈ C (2.3)

holds, with a constant c 6= 0. Then

g(z) = c−1e−α2 eαz, (z ∈ C) (2.4)

with an arbitrary constant α.

For the proof we use that there exists an entire function h such thatg = eh. From (2.3) we deduce a linear equation for h which can be solvedusing the Herglotz trick.Now we give an application to the sine function. We have

sin 2πz = 2 sinπz · sinπ(z +

1

2

), z ∈ C, (2.5)

which is a functional equation of type (2.3). The infinite product

P (z) = z

∞∏ν=1

(1− z2

ν2

), z ∈ C (2.6)

is an entire function. It satisifes a functional equation of type (2.3). One can

prove this by showing that ε1(z) = P ′(z)P (z) satisfies (2.1), and hence(

P (z)P(z + 1

2

)P (2z)

)′= 0,

therefore there exists c 6= 0 such that

P (2z) = cP (z)P

(z +

1

2

), (z ∈ C). (2.7)

Since P (z) and sinπz have exactly the same zeros z = ν ∈ Z, both withmultiplicity 1, and from (2.5) and (2.7) we deduce that the quotient g(z) =P (z)sinπz is an entire function without zeros and a solution of

g(z) = cg(z

2

)g

(z + 1

2

), (z ∈ C)

of type (2.3). By Theorem 2.3 we get

P (z) =2

ce−

α2 eαz sinπz, z ∈ C.

6 Ludwig Reich

Since P and sinπz are odd, we have α = 0. From limz→0P (z)z = limz→0

sinπzz =

1 we get Euler’s product

sinπz = z

∞∏ν=1

(1− z2

ν2

), (z ∈ C).

These arguments also prove the following characterization of sine via its du-plication formula (2.5).

Theorem 2.4. Let f be an entire function whose zeros are exactly the pointsz ∈ C, all with multiplicity 1, and assume that

f(2z) = cf(z)f

(z +

1

z

)holds for all z ∈ C, where c is constant, c 6= 0. Then

f(z) =z

csinπz (z ∈ C).

The additive Herglotz Lemma refers to the functional equation (2.1)which is a so called linear functional equation and an equation in a singlevariable z. We will consider the Herglotz Lemma also in Section 3 in connec-tion with the addition formula for cot and in Section 5 in connection withthe duplication formula for Γ.We finish this section discussing some possibilities of generalizing the Her-glotz Lemma to more general linear equations. Combining the approaches ofF. Schottky and G. Herglotz one obtains

Theorem 2.5. Let g be an entire function, P a polynomial with degree ∂P = sor P = 0. Let αj , βj, j = 1, . . . , N be complex numbers such that the followingconsitions are satisfied:

a.) There is m, 0 ≤ m < N , such that |αj | = 1, βj = 0 for 1 ≤ j ≤ m.b.) If m + 1 ≤ j ≤ N , we suppose 0 < |αj | < 1. Let b1, . . . , bN be complex

numbers such that

|b1|+ |b2|+ . . .+ |bm| < 1,

and assume bj 6= 0, for at least one j with m+ 1 ≤ j ≤ N . Suppose that

g(z) = b1g(α1z) + . . .+ bmg(αmz) + bm+1g(αm+1z + βm+1) + . . .

+ bNg(αNz + βN ) + P (z) (z ∈ C) (2.8)

holds.

Then g is a polynomial. One can give an estimate of ∂g (if g 6= 0) dependingon the degree of P and on bm+1, . . . , bN , αm+1, . . . , αN .

If the inhomogeneity P is transcendental, then one can give an estimatefor the order ω(g) of g.

Theorem 2.6. Let g be an entire function, P be an entire transcendentalfunction. Let the complex numbers αj , βj, j = 1, . . . , N satisfy the conditionsa.) and b.) of Theorem 2.5, and assume that g fulfills (2.8). Then

ω(g) ≤ ω(P ).


Instead of constant ceofficients b1, . . . , bN on the right hand of (2.8)and 1 on the left hand side of (2.8) we may take polynomial coefficients. Theapproach of Schottky is still possible, whereas the Herglotz trick (appropriatedifferentiations) is not appropriate in this situation. We have

Theorem 2.7. Let g be an entire function, P be a polynomial. Let αj , βj,j = 1, . . . , N be complex numbers such that

a.) There is m, 0 ≤ m < N , such that |αj | = 1, βj = 0 (for 1 ≤ j ≤ m,0 ≤ m)

b.) If j ∈ {m+ 1, . . . , N}, then 0 < |αj | < 1.

Let p0, p1, . . . , pN be polynomials such that s = ∂p0 > 0, ∂pj ≤ ∂p0 (for j =1, . . . , N). There exists j ∈ {m+ 1, . . . , N} such that pj(z) 6= 0. Furthermore,

if pj(z) = b(j)s zs + . . ., then we assume

|b(0)s | − |b(1)

s | − . . . |b(N)s | > 0. (2.9)

Suppose that

p0(z)g(z) = p1(z)g(α1z) + . . .+ pm(z)g(αmz) + pm+1(z)g(αm+1z + βm+1) + . . .

+ pN (z)g(αNz + βN ) + P (z) (2.10)

holds for all z ∈ C.Then g is a polynomial.

One could give estimates for the degree of g.We conjecture that in Theorem 2.7 condition (2.9) can be weakened to

|b(0)s | − |b(1)

s | − . . . |b(m)s | > 0,

that means that b(m+1)s , . . . , b

(N)s are arbitrary.

It seems to be an open problem to prove existence results for equations (2.8)and (2.9), concerning entire solutions and meromorphic solutions. Also solu-tions of these equations which are holomorphic in a disc |z| < R would beinteresting.The reader may consult [2], 5.2.259 and [9], 11.2, [10], 1.2 for the basic resultsof this section.

3. Addition formulas: Characterizations of the cotangentfunction. Special Eisenstein series

In this section we deal with certain functional equations in several variables,namely with some addition formulas.We start with the addition formula

f(z + w) =f(z)f(w)− 1

f(z)f(w)(3.1)

where f is a meromorphic function and (3.1) is supposed to hold for all(z, w) ∈ C2 such that z, w, z+w is not a pole of f . It is well known that cot z isa solution of (3.1). The addition formula yields the following characterizationof cot.

8 Ludwig Reich

Theorem 3.1. Let g be meromorphic function in an open connected neigh-bourhood U of 0 The following holds true:g|U = cot if and only if g has at 0 the principle part 1

z and g fulfils (3.1) forall (z, w) sucht that w, z, w + z are not poles of g.

For the proof consider, if z is not a pole of g, the difference quotient1h (g(z+h)−g(z)), for small |h|. (3.1) we can use for g(z+h). Then, if h goesto 0, we obtain the differential equation

g′(z) = −g(z)2 − 1, z ∈ U \ {poles of g} . (3.2)

We have, by assumption g(z) = 1+B(z)z , where B(0) = 0, and B is holo-

morphic in a neighbourhood of 0. We get for B a so called Briot-Bouquetdifferential equation

zB′(z) = −B(z)−B(z)2 + z2

B(0) = 0. (3.3)

It is known (see [3]) that there exists a unique solution B of (3.3) withB(0) = 0, holomorphic in a neighbourhood of 0. But cot z leads by the abovetransformations to such a solution, hence g(z) = cot z for z ∈ U .Another characterization of the cotangent function, more precisely of ε1(z) =π cotπz uses also (3.1), but not the differential equation (3.2). We apply hereonce more the duplication formula (2.1).

Theorem 3.2. Let g be a meromorphic function which satisfies

g(w + z) =g(w)g(z)− π2

g(w) + g(z)(3.4)

whenever w, z, w+ z are not poles of g. Assume that g(

12

)= 0, g′

(12

)= −π,

and assume that g has at most three poles in |z| ≤ 1. Then

g(z) = π cotπz.

For a proof we put z = 12 into (3.4), use g

(12

)= 0 and deduce step by

step that g is 1−periodic, has poles at z ∈ Z, the principal 1z in z = 0 and

satisfies the duplication formula

g(2z) =1

2g(z) +

1

2g

(z + 1

2

)if z is not a pole of g. The next important step is to prove that Z is the setof all poles of g. This follows from the assumption that g has at most threepoles in |z| ≤ 1. We can apply Theorem 2.1 to z 7→ g(z) − π cotπz. (ForTheorem 3.1 the reader can consult [9]).There are other important addition formulas in the complex analysis, e.g. inthe theory of elliptic functions we have the well known addition formula forthe Weierstrass σ-function. M. Bonk (cf [1]) gave the general solution of thisaddition formula (solutions without any regularity conditions), based on theproperty of the σ-function itself.


We will here only consider an addition formula in the theory of special Eisen-stein series

εk(z) =

∞∑ν=−∞

1

(z + ν)k(k ≥ 2), (3.5)

ε1(z) = limn→∞

n∑−n

1

(z + ν)

=1

z+

∞∑ν=1

2z

z2 − ν2

=

∞∑ν=−∞

(1

z + ν− 1

ν

)

=1

z+

∞∑ν=−∞

(1

z + ν− 1

z − ν

).

(3.6)

It is obvious that in the definition of ε1 limn→∞∑n−n

1(z+ν) cannot be replaced

by∑∞ν=−∞

1z+ν , which does not have a meaning. All other series appearing in

(3.5) and (3.6) are absolutely convergent if z /∈ Z, and uniformly convergentin each compact set which does not contain an integer. Hence they are mero-morphic, 1-periodic functions. All periods are integers, ε1 has the principalparts 1

z−ν (ν ∈ Z) and εk the principal parts 1(z−ν)k

(ν ∈ Z), k ≥ 2.

Termwise differentiation and arbitrary arrangements of summands are al-lowed (except in limn→∞

∑nν=−n

1z+ν ). It is not too difficult to prove:

ε′1 = −ε21 − 3q2 (3.7)

with q2 = 2∑∞ν=1

1ν2 ,

ε1(z) =1

z−∞∑ν=1

q2νz2ν , 0 < |z| < 1 (3.8)

with

q2n = 2

∞∑ν=1

1

ν2n,

ε′k = −kεk+1 (k ≥ 1), (3.9)

εk =(−1)k−1

(k − 1)!ε

(k+1)1 (k ≥ 2). (3.10)

By direct tricky manipulations of series G. Eisenstein proved the additionformula

ε1(w + z) =ε1(w)ε1(z)− 3q2

ε1(z) + ε1(w)(3.11)

which we already know, if we use π2

6 =∑∞ν=1

1ν2 . We also have

ε3 = ε2 · ε1, (3.12)

10 Ludwig Reich

which can be obtained from the definitions.To sketch a theory of these special Eisensein series we follow now A. Weil[14], ch. 2 and present as basic result an addition formula involving ε1, ε2, ε3.

Theorem 3.3. We have

ε2(w)ε2(z)− (ε2(w) + ε2(z))ε2(w + z) = 2ε3(w + z)[ε1(w) + ε1(z)] (3.13)

if z, w, z + w is not an integer.

The proof is based on the identity

1

p2q2=

1

(p+ q)2

(1

p2+

1

q2

)+

2

(p+ q)3

(1

p+

1

q

)in the ring C(p, q) of rational functions in (p, q). In this identity we put, withintegers ν, µ, p = z+ν, q = w+ν−µ, sum up firstly over µ (where ν is fixed,if necessary we use the ”Eisenstein” summation, see (3.6)), then over ν, anduse several times the definitions of ε1, ε2, ε3 and their periodicity.From Theorem 3.3 we obtain two algebraic identities, involving ε1, ε2, ε3, ε4,namely

Theorem 3.4. We have

3ε4(z) = ε2(z)2 + 2ε1(z)ε3(z), (3.14)

ε3(z)2 = ε4(z) + 2q2ε2(z). (3.15)

We say a few words about the proof of (3.14). In the addition formulaof Theorem 3.1 we use for εk(w + z), (k = 1, 2, 3), for fixed, but arbitraryz the Taylor expansions with respect to w around w = 0 (these are rathereplicitly known by the differential formulas (3.9), (3.10)). But we also knowthe Laurent expansions of ε1(w), ε2(w), ε3(w) which can be calculated fromthe definitions. If we put this all into the addition formula of Theorem 3.1,arrange with respect to powers of w and compare the absolute term (coeffi-cients of w0) on both sides, we obtain (3.14).After some further, mainly algebraic calculations, one gets

Theorem 3.5. Each function εk is a polynomial with (constant) real coeffi-cients in ε, these polynomials can be recursively determined.

It is now not too difficult to see that the theory of trigonometric func-tions is essentialy the theory of our special Eisenstein series, as soon one

knows 2∑∞ν=1

1ν2 = π2

6 . Nevertheles, there seem to be still some open prob-lems, from the point of view of functional equations:

1. Are there intersting addition formulas analogues to (3.10) involving εkfor k ≥ 3?

2. Is it possible to characterize ε3 by (3.10)?

For the addition formula of cot and of the special Eisenstein series the readermay consult [9], 11.4.


4. On the identity of C. G. Jacobi

For each q with |q| < 1

J(q, z) =

∞∑ν=−∞

qν2

zν , z ∈ C× = C \ {0} (4.1)

is a Laurent series in C×, hence holomorphic.The infinite product (Abel’s triple product)

A(q, z) =

∞∏ν=1

(1− q2ν)(1 + q2ν−1z)(1 + q2ν−1z−1) (4.2)

is, by well known basic theorems of infinte products, for each |q| < 1 holo-morphic in C×.Applying certain functional equations for A(q, z), J(q, z) one gets an elegantproof of the following famous result of Jacobi.

Theorem 4.1.∞∑

ν=−∞qν

2

zν =

∞∏ν=1

(1− q2ν)(1 + q2ν−1z)(1 + q2ν−1z−1), (4.3)

for all (q, z) with |q| < 1, z ∈ C×.

The results on functional equations one may use to prove Theorem 4.1are summarized in

Theorem 4.2. a.) For all (q, z) ∈ C× ×B1(0) we have

A(q2z, q) = (qz)−1A(z, q), (4.4)

A(z−1, q) = A(z, q). (4.5)

b.) For all q with |q| < 1 we have

A(i, q) = A(−1, q4), (4.6)

J(i, q) = J(−1, q4). (4.7)

The linear functional equation (4.4) for A(q, z) (for an arbitrary q with|q| < 1) can be obtained by manipulations of the absolutely convergent in-finite product A, see (4.2). (4.5) is obvious, and also (4.6) and (4.7) are notdifficult. In order to prove Theorem 4.1 from Theorem 4.2 we consider theLaurent expansion of A(q, z),

A(q, z) =

∞∑ν=−∞

aν(q)zν , z ∈ C×.

Using the functional equations from Theorem 4.2, one after the other weobtain by induction a formula for aν(q) in terms of a0(q), and after theapplication of Liouville’s theorem, based on (4.6) and (4.7), we find Theorem4.1.

12 Ludwig Reich

This identity is the source of many others, which are important in numbertheory and in combinatorics. We mention only

∞∑ν=−∞

qν2

=

∞∏ν=1

(1 + q2ν−1

) (1− q2ν

),

the product formula for the Theta function on the left hand side. The studyof such identities is still a topic of research. In Section 6 we will use quitedifferent tools from the theory of Theta functions, including certain linearfunctional equations to prove Euler’s identity

∞∏ν=1

(1− qν) =

∞∑ν=−∞

(−1)νq3ν2+ν

2 .

For Theorem 4.1 and Theorem 4.2 the reader may consult [9], 1.5.

5. Functional equations in the theory of the Gamma function

We start, following Weierstrass, from the entire function

∆(z) = zCz∏ν≥1

(1 +

z

ν

)e−

zν (5.1)

with

C = limn→∞

(1 +

1

2+ . . .

1

n− lnn

).

Directly from this product representation we see

∆(z) = z∆(z + 1), π∆(z)∆(z − 1) = sinπz, (z ∈ C). (5.2)

If we introduce Γ(z) = 1∆(z) , then Γ is meromorphic in C, and has poles

exactly in z = −ν (ν ∈ N), all of first order. Γ satisifes the difference equation

Γ(z + 1) = zΓ(z) (5.3)

whenever z is not a pole.Γ can also be defined as

Γ(z) = limn→∞

n!nz

z(z + 1) . . . (z + n)(5.4)

with nz = ez lnn (Gauss). We have

Γ(z)Γ(1− z) =π

sinπz. (5.5)

Furthermore, Γ is bounded in the strip

{z|z ∈ C, 1 ≤ Re z < 2} . (5.6)

From (5.3) we get

res−nΓ =(−1)n

n!, n ∈ N. (5.7)

Many results in the theory of the Gamma function can be proved in an elegantway by applying the following uniqueness theorem for the difference equation(5.1).


Theorem 5.1 (H. Wielandt). Let F be holomorphic in the right half plane

H = {z ∈ C|Re z > 0}. Let F satisfy f(z+ 1) = zF (z) in H and assume thatF is bounded in S1 := {z|z ∈ C, 1 ≤ Re z < 2}. Then

F = F (1) · Γ, z ∈ H.

We will say a few words on the proof of this beautiful theorem soon.There is, however, another characterization Γ as a real function, namely

Theorem 5.2 (H. Bohr - B. Mollerup). Let F : R>0 → R>0 have the properties

(i) F (x+ 1) = xF (x), x > 0, and F (1) = 1.(ii) logF is convex in R>0.

Then F = Γ|R>0.

Theorem 5.1 seems to fit better to the frame of complex analysis.For the proof of Theorem 5.1 one considers v = Γ − F (1) · Γ. This functionsatisfies (5.3), and is bounded in the strip {z|z ∈ C, 1 ≤ Re z < 2}. By (5.3) vcan be meromorphically continued to C, with poles at most in z = −1,−2, . . .,at most of order 1. The residues can be calculated from (5.3), (5.7), and arein fact 0. Hence v is an entire function . From the assumed boundedness ofF in the strip S1 = {z|z ∈ C, 1 ≤ Re z < 2} and from (5.6) it follows that vis bounded in this strip, and by (5.3) also in S0 = {z|z ∈ C, 0 ≤ Re z < 1}.Then the entire function z 7→ v(z)v(1 − z) can be shown to be bounded ineach strip {z|z ∈ C,m ≤ Re z ≤ m+ 1} (m ∈ Z), with the same bound as inS0, hence it is bounded in C, and by Liouville’s theorem v = 0. This provesTheorem 5.1.Theorem 5.1 can be applied to show several famous classical results in thetheory of Γ. We mention some examples.1.) The multiplication formula.Let k ∈ N, k ≥ 2. Then

Γ(z)Γ

(z +

1

k

)Γ

(z +

2

k

). . .Γ

(z +

k − 1

k

)= (2π)

12 (k−1)k

12−kzΓ(kz).

(5.8)One applies Wielandt’s theorem to the function

F (z) = Γ( zk

)Γ

(z + 1

k

)· · ·Γ

(z + k − 1

k

)· 2π 1

2 (k−1)k12−z

which is holomorphic in the right half plane. A special case of (5.8) is theduplication formula

√πΓ(2z) = 22z−1Γ(z)Γ

(z +

1

2

), (5.9)

if 2z, z, z + 12 are not poles of Γ.

From (5.8) and (5.5) we get

sinπz = 2k−1 sinπz sinπ

(z +

1

k

). . . sinπ

(z +

k − 1

k

).

14 Ludwig Reich

2.) Euler’s integral representation

Γ(z) =

∫ ∞0

tz−1e−tdt, Rez > 0, (5.10)

with tz−1 = e(z−1) ln t (t ∈ R>0), where the integration path is the realhalfline.The existence (convergence) and regularity of the integral on the right handside has to be proved. This expression satisfies the hypothesis of Theorem5.1.A similar approach is also possible to prove the more sophisticated integralrepresentation of Hankel.3.) Euler’s identity for the Beta function.Define

B(w, z) =

∫ 1

0

tw−1(1− t)z−1dt,

then

B(w, z) =Γ(w)Γ(z)

Γ(w + z). (5.11)

Wielandt’s theorem can be applied to

F (z) = B(w, z)Γ(w, z)

in the right z−half plane, for each fixed w. One also needs the estimate

|B(w, z)| ≤ B(Re w,Re z).

The system of functional equations

F (z + 1) = zF (z)

√πF (2z) = 22z−1F (z)F

(z +

1

2

)(5.12)

which combines (5.3) and the duplication formula (5.9) can be used to char-acterize Γ.

Theorem 5.3. Let F be meromorphic in C, and assume that F takes positvevalues in R>0. Suppose that (5.12) holds for z such that the involved functionsare holomorphic. Then F = Γ.

The proof is based on the multiplicate Herglotz Lemma (Theorem 2.2),applied to F/Γ.An interesting problem, which seems to be open at least in the complex case,is the following:Is there a similar characterization of Γ using one (or several) cases of themultiplication (5.8) formula for k ≥ 3 (with or without using (5.3)), as The-orem 5.3? Very likely, certain generalizations of the multiplicative HerglotzLemma will be needed.For the contents of this Section the reader is referred to [10], ch. 2 or to [2],5.3.271-279. In the latter textbook Theorem 2 takes the role of Theorem 1.


6. The transformation formula of the Theta function andsystems of linear functional equations in analytic numbertheory

As already mentioned at the end of Section 4 the identity

∞∏ν=1

(1− qν) =

∞∑ν=−∞

(−1)νq3ν2+ν

2 , |q| < 1 (6.1)

plays an important role in analytic number theory. Following [11], §1, §2 wegive a sketch of an analytic proof of (6.1) which is based on the transformationformula of the ϑ-function and on a uniqueness theorem for a system of linearfunctional equations.If we write q = eπiz with z ∈ H = {z ∈ C|Im z > 0}, then (6.1) is

∞∏n=1

(1− eπinz

)=

∞∑n=−∞

(−1)neπiz(3n2+n)

(=

∞∑n=−∞

eπi3zn2+2πin( 1

2 + z2 )

).

(6.2)

In order to get closer insight in the right hand side of (6.2) we introduce theseries

∞∑n=−∞

eπiz(n+w)2 , z ∈ H, w ∈ C. (6.3)

This series is for each fixed z ∈ H absolutely convergent and also uniformlyconvergent in each subset of C, hence an entire function in w. It has period1. Therefore it has has a Fourier expansion

∞∑n=−∞

eπiz(n+w)2 =

∞∑n=−∞

ck(z)e2πikw (6.4)

where

ck(z) =

∫ 1

0

∞∑n=−∞

eπiz(n+w)2−2πikwdw, (6.5)

and the path of integration is [0, 1]. Using the periodicity and legitimatemanipulation of series leads to

ck(z) = e−πik2z

∫ ∞∞

e−πzi (w−

kz )

2

dw. (6.6)

A transformation of the variable of integration and application of Cauchy’sintegral theorem together with estimating the integrand in certain criticalregions leads to

ck(z) = e−πik2

z

√iz

∫ ∞−∞

e−πs2

ds = e−πik2

z ·√i

z(6.7)

16 Ludwig Reich

where this branch of√

iz has be taken which takes the value 1 at z = i.

Then (6.7) yields the Fourier expansion (6.4) in the form∞∑

n=−∞eπiz(n+w)2 =

√i

z

∞∑n=−∞

e−πin2

z +2πiw. (6.8)

The right hand side is directly connected with the Theta function

ϑ(z, w) =

∞∑n=−∞

eπin2z+2πiw. (6.9)

ϑ is for each fixed z ∈ H an entire function of w. (6.9) and (6.8) give imme-diately

Theorem 6.1 (Jacobi). For the function ϑ defined by (6.9) we have for z ∈ H,w ∈ C the transformation formula

ϑ

(−1

z, w

)=

√z

i

∞∑n=−∞

eπiz(n+w)2 (6.10)

where√

zi takes for z = i the value 1.

It is easy to see that the right hand side of (6.2) is

f(z) = ϑ

(3z,

1

2+z

2

), z ∈ H (6.11)

which is a holomorphic function of z ∈ H.From Theorem 6.1, (6.10) and applying some series transformations we de-duce from (6.11)

f

(−1

z

)=

√z

ieπi12 + πi

12z f(z). (6.12)

Introducing

g(z) = eπiz12 f(z) (6.13)

we find instead of (6.12)

g

(−1

z

)=

√z

ig(z), z ∈ H. (6.14)

Since f(z + 1) = f(z), we obtain for g

g(z + 1) = eπi12 g(z), z ∈ H. (6.15)

Now we consider Dedekind’s η-function

η(z) = eπi12

∞∏n=1

(1− e2πinz

)(6.16)

which is also holomorphic in H. η has its origin in the theory of ellipticfunctions. The definition (6.16) of η, (6.11) of f and (6.12) of g yield that(6.2) is the same as

η = g. (6.17)


In order to prove (6.17) we begin by showing that η is also a solution of thesystem of functional equations (6.14) and (6.15). The same behaviour of ηand g for z tending to∞ in H and a sort of uniqueness result gives us (6.17).Many interesting and rather delicate details are required to prove that η is asolution of (6.14), while (6.15) for η is almost immediate. Now we consider asketch of the proof of

η

(−1

z

)=

√z

iη(z), z ∈ H. (6.18)

There it is possible to take such a holomorphic logarithm of the function ηwhich is nowhere 0, that

log η(z) =πiz

12+

∞∑m=1

log(1− e2πimz

)(6.19)

and such that each term log(1− e2πimz

)is given by the logarithmic series.

If we do the same for η(− 1z

), then (6.18) is the same as

− πi

12z−∞∑m=1

∞∑n=1

1

ne−2πinm 1

z =1

2log

z

i+πiz

12−∞∑m=1

∞∑n=1

1

ne2πinmz, z ∈ H.

(6.20)In (6.20) we can interchange the order of summation, use the geometric seriesand introduce the functions cotπkz and consider partial sums, so that wehave eventually to prove

−πiz12− πi

12z+

n∑k=−n

i

4k

(cot ikz + cot

πk

z

)→

n→∞

1

2log

z

i. (6.21)

The proof of (6.21) is the most delicate part. One observes that the partialsum in (6.21) is the sum of residues of an appropriate meromorphic functionϕ(s)8s in a s-plane, taking the poles which lie in a parallelogram Pn (n ≥ 1)

such that the Pn tend in a certain way to infinity for n → ∞. This can beworked out in detail by choosing

ϕ(s) = cot s · cots

z

and using the theorem of residues when integrating along the boundary ofPn. We have to omit the details. One gets

Theorem 6.2. The functions

g(z) = eπiz12 ϑ

(3z,

1

2+z

2

)and

η(z) = eπiz12

∞∏n=1

(1− e2πinz

)

18 Ludwig Reich

satisfy both the system

ψ(z + 1) = eπi12ψ(z)

ψ

(−1

z

)=

√z

iψ(z), z ∈ H, (6.22)

of linear functional equations.

The Mobius transformations z 7→ z + 1 and z 7→ − 1z appearing on the

left hand side of (6.22) generate a groupM of Mobius transformations whichare holomorphic bijections of H. It can be shown that

M =

{L|L(z) =

αz + β

γz + δ(z ∈ H), α, β, γ, δ ∈ Z, αδ − βγ = 1

},

the modular group. Futhermore we need the result that

H =⋃L∈M

L(F ) (6.23)

where F ={z|z ∈ H, z = x+ iy, x2 + y2 ≥ 1,− 1

2 ≤ x ≤12

}. We consider now

the function

h :=g − ηη

(6.24)

which is holomorphic in H, since η is nowhere 0. Because of Theorem 6.2 wehave form (6.24),

h(z + 1) = h(z)

h

(−1

z

)= h(z)

for z ∈ H. Hence also h(L(z)) = h(z), z ∈ H, for all L ∈ M. By (6.23) weobserve that h takes all its values on H already in F . If we use expansions ofg and η in powers of eπiz, which can directly be deduced from the definitions,we see that

limz→∞, z∈F

h(z) = 0. (6.25)

Assume now h 6= 0, hence h non-constant in F . Because of (6.25) |h| takesits maximum in F in a subset of the form

x2 + y2 ≥ 1, −1

2≤ x ≤ 1

2, y ≤ K,

and according to the maximum principle, at a boundary point z0 of thissubset. Consider an open neighbourhood U of z0 in H. But what we havementioned above about the values of h z0 is also a point of maximum of |h|in U which contradicts the maximum principle. Hence h is constant, h = 0and (6.17) by (6.25). This proves (6.2) and (6.1).Also the identity( ∞∑

k=−∞

qk2

)4

= 1 + 8

∞∑n=1

nqn

1− qn− 8

∞∑n=1

4nq4n

1− q4n


(|q| < 1) where the left hand side is ϑ(z, 0)4 can be proved by similar ideas.Here one meets the system

f(z + 2) = f(z)

f

(−1

z

)= −z2f(z)

of linear functional equations for a function f which is holomorphic in H, anda similar uniqueness result. Solutions of this system are

8i

π

d

dzlog

η(z2

)η(2z)

and

ϑ(z, 0)4.

7. Generalized Dhombres equations in the complex domain

By a generalized Dhombres equation we understand, without being precisewith respect to the domains and ranges of the functions involved a functionalequation of the form

f(zf(z)) = ϕ(f(z)) (7.1)

where ϕ is given and f is unknown. (7.1) belongs to the class of iterativefunctional equations (cf. [4]). The typical difficulties arise from the fact thatthe unknown function or an expression containing the unknown function (herezf(z)) is substituted in the unknown function f .J. Dhombres started to study these equations by considering the example

f(xf(x)) = f(x2), x ∈ R>0,

f : R>0 → R>0

where f is a continous function: This equation describes a model in popu-lation dynamics but has its origin also in algebraic problems. Later on theresearch was continued by work of P. Kahlig and J. Smıtal on equations

f(xf(x)) = f(x)k+1, x ≥ 0 (7.2)

where k ∈ N, and f : R>0 → R>0 is continuous. We will meet equations ofform (7.2) at the end of our lecture again. It is natural to go further to equa-tions (7.1). The assumptions made in the work of J. Smıtal, M. Stefankovaand myself are, among others, that f and ϕ are continuous on appropriatereal intervals.It turned out that (7.1) is also an interesting object of study in the complexplane when ϕ and f are supposed to be holomorphc or meromorphic func-tions in certain regions.The main problem is the following.Global Problem

20 Ludwig Reich

Let ϕ : D → C be holomorphic in a region D of C. Let D be a region in C.Find non-constant holomorphic functions f : D → C such that

f(z(f(z)) = ϕ(f(z)) (7.1)

holds for all z ∈ D.More generally one could ask for meromorphic solutions f of (7.1) with

meromorphic given ϕ, etc.Furthermore, given w0 ∈ D and z0 ∈ D we could also ask for solutions

f of (7.1) satisfying

f(z0) = w0. (7.3)

This general problem seems to be rather difficult, and for arbitrary D,Dthere may not exist non-constant holomorphic solutions.Furthermore, we observe that because of the particular structure of the lefthand side of (7.1) the point z0 = 0 play a particular role.Following an approach which was used in the classical theory of ordinarydifferential equations in the complex domain we will start by asking for socalled local analytic (or even only formal) solutions of (7.1), and then try toget consequences concerning the global Problem. We begin with

Definition 7.1 (Local analytic solutions of (7.1) with f(z0) = w0). Let ϕ :

D → C be holomorphic and w0 ∈ D. By a local analytic solution f of (7.1)with f(z0) = w0 we understand a holomorphic function f : Df → C wherez0 ∈ Df , Df is an open connected neighbourhood of z0 and f(z0) = w0 andsuch that (7.1) holds for all z ∈ Df .

We emphasize that Df may depend an the individual solution f , i.e.we do not require that there is a common neighbourhood of z0 in which allconsidered local analytic solutions f with f(z0) = w0 are holomorphic andsatisfy (7.1). In fact, this situation really occurs.Such local analytic solutions exist and can be described in the followingsituations:

f(0) = w0(∈ C); f(z0) = 1, z0 6= 0; f(∞) = w0 ∈ C×; f(∞) =∞.

We will here only deal with the first case. In connection with the investigationof local analytic solutions we will also study formal solutions which are formalseries in z.The cases f(0) = 0 and f(0) = w0, w0 ∈ C× require a somewhat differentapproach. The subcases

f(0) = w0, w0 6= 0, w0 not a root of 1,

f(0) = 1,

f(0) = w0, w0 a root of 1, ord w0 ≥ 2,

also differ both with respect to methods and results.First of all we mention necessary conditions for the existence of non-constantlocal-analytic solutions.


A) If f(0) = 0, f 6= 0, then f(z) = ckzk + . . ., with k ≥ 1, ck 6= 0. Then

necessarily

ϕ(y) = yk+1 + dk+2yk+2 + . . . (7.4)

in a neighbourhood of y = 0.B) If f(0) = w0, w0 6= 0 and if f is a non constant local-analytic solution

of (7.1), then f(z) = w0 + g(z), g(z) = ckzk + . . ., k ≥ 1, ck 6= 0. We

get from (7.1) the necessary condition

ϕ(y) = w0 + wk0 (y − w0) + dk+2(y − w0)2 + . . .

in a neighbourhood of y = w0, hence

w0 = ϕ(w0), ϕ′(w0) = wk0 . (7.5)

If w0 is not a root of 1, then k is uniquely determined by ϕ. We also see thatalways

ϕ(y) = w0 + ϕ(y − w0), ϕ(0) = 0.

Assuming that the necessary conditions (7.4) and (7.5) are fulfilled we presentnow two theorems on the structure of the general local analytic solution f of(7.1) with f(0) = w0.

Theorem 7.2. Let ϕ(y) = yk+1 +dk+2yk+2 + . . . in a neighbourhood of y = 0,

with k ∈ N.

a) Then there exists a unique g0(y) = y + . . ., holomorphic at y = 0, suchthat the set of local analytic solutions of (7.1) is given by{

f |f(z) = g0(ckzk), ck ∈ C

}. (7.6)

The individual solution f is uniquely determined by ck. Furthermore

ϕ(y) = g0(ykg−10 (y)) around 0 (7.7)

where g−10 denotes the substitutional inverse of g0.

b) Let k ∈ N, g0(y) = y + γ2y2 . . . be given. Then there exists a unique

holomorphic ϕ (given by (7.7)) such that the set of all local analyticsolutions f of f(zf(z)) = ϕ(f(z)) with f(0) = 0 is given by (7.6).

In order to prove the structure (7.6) of all local analytic solutions weput without loss of generality f(z) = T (z)k, with T (z) = t1z + . . ., t1 6= 0,T−1 = U , define ψ(z) by ψ(z)k = ϕ(zk), ψ(z) = zk+1 + . . . and obtain from(7.1) the linear functional equation

zkU(z) = U(ψ(z)) (7.8)

which is locally equivalent with (7.1).Now we use that there exists a unique S, S(z) = z + . . ., holomorphic at 0,such that Bottcher’s equation

ψ(z) = S−1(S(z)k+1

)(7.9)

holds around z = 0. Using this S (the Bottcher function of ψ) we obtain

S−1(z)k · V (z) = V (zk) (7.10)

22 Ludwig Reich

with V = U ◦ S−1, hence V (z) = v1z + . . ., v1 6= 0. After some more partlytricky calculations we deduce (7.7) in Theorem 7.2.We mention that there also exist representations of the set (7.6) of all localanalytic solutions of (7.1) involving infinite products.If we choose, according to Theorem 7.2 b) the generator g0 as having aradius of convergence r with 0 < r < ∞, then the radus of convergence ofthe solution f(z) = g0(ckz

k) depends on |ck| and it tends to 0 if |ck| → ∞.Hence there cannot exist a common region where all local analytic solutionof (7.1) are holomorphic and satisfy (7.1).

Now we turn to the case f(0) = w0, w0 6= 0, w0 is not a root of 1.In this situation we have the necessary conditions (7.5) for the existence oflocal analytic solutions f of (7.1) with f(0) = w0. If |w0| 6= 1 or w0 is aSiegel number then these conditions are also sufficient for the existence ofnon-constant local analytic solutions. In all other cases (7.5) is sufficient forthe existence of non-constant formal solutions and we can give their completedescription. However, the problem of convergence of these solutions is stillopen. Before presenting our results we transform (7.1) into an equivalentform. By writting

f(z) = w0 + g(z), g(z) = ckzk + . . .

with k ≥ 1 and ck 6= 0, and using (7.5)

ϕ(y) = w0 + ϕ(y − w0) = w0 + wk0 (y − w) + . . . ,

(7.1) is equivalent with

g(z(w0 + g(z)) = ϕ(g(z)). (7.11)

Then the following result holds true.

Theorem 7.3. Assume that w0 6= 0, not a root of 1 and let ϕ, g be defined asabove.

(i) Then there exists a unique, in general only formal series g0(y) = y +γ2y

2 + . . . such that the set of all formal solutions f of (7.1) (i.e. g of(7.11)) is given by{

f |f(z) = w0 + g0(ckzk), ck ∈ C

}.

(ii) If |w0| 6= 1 or w0 is a Siegel number (cf. [12], §23), then these formalsolutions are convergent, hence yield local analytic solutions f of (7.1)with f(0) = w0.

The proof of Theorem 7.3 uses similar ideas as the proof of Theorem7.2. But instead of Bottcher’s functional equation we need here Schroder’sequation

ψ(z) = S−1(w0S(z)),

S(z) = z + . . . (7.12)

for the function ψ defined by ψ(y)k = ϕ(yk), ψ(y) = w0y+. . .. It is well knownthat under the assumptions of Theorem 7.3, (7.11), (7.12) has a unique for-mal solution S which gives a convergent ”generating” g0 in Theorem 7.3, (i).


If |w0| < 1, then there exist representations of the general solution f withf(0) = w0 of (7.1) involving infinite products.If w0 = 1 or w0 a root of 1 with ord w0 ≥ 2 then the theory of formal solutionsf of (7.1) with f(0) = w0 is much more complicate, but has recently beencompleted (see [13]). The set of necessary and sufficient conditions on ϕ forthe existence of non-constant solutions is much stronger and complicate thanthe corresponding conditions in Theorem 7.2 and Theorem 7.3, except therather simple case w0 = 1. However, these conditions can be fulfilled, and ifthey are, the set of all solutions may in certain cases depend on an arbitraryfunction (here an arbitrary formal series), not only on the coefficient ck inf(z) = w0 + ckz

k + . . . (k ≥ 1, ck 6= 0).Results on local analytic solutions seem to be very difficult to obtain, exceptin the case where the functions ϕ(u) can be linearized via conjugation. If ϕ(u)is a linearizable Mobiustransformation for which (7.1) has a non-constant so-lution (-such ϕ exist and can be described completely-), then the set of allformal solutions has a rather explicit description, and there are also localanalytic ones.We finish this section by presenting one result referring to the Global Prob-lem, but whose proof uses the local theory combined with some basic poly-nomial algebra and complex analysis.

Theorem 7.4. (i) Let ϕ be an entire function and assume that there existsa non-constant polynomial f0 such that f0(0) = w0 and

f0(zf0(z)) = ϕ(f0(z)), z ∈ C

holds.Then there exists k ∈ N and c

(0)k ∈ C× such that

f0(z) = w0 + c(0)k zk

ϕ(y) = w0 + (y − w0)yk.

If ck ∈ C, then f(z) = w0 + ckzk (z ∈ C) satisfies (7.1) (with the same

ϕ as above) for all z ∈ C. These polynomials are all formal solutions of(7.1) with f(0) = w0.

(ii) If conversely, w0 ∈ C, k ∈ N, and

ϕ(y) := w0 + (y − w0)yk, y ∈ C,

then the polynomials f(z) = w0 + ckzk, (ck ∈ C) are solutions of

f(zf(z)) = ϕ(f(z)).

According to a remark of M. Laczkovich (oral communication) if ϕ isan entire function then (7.1) cannot have an entire transcendental solutionin C.So far the global problem is widely open. Some results are known for equations(7.1) of the special form

f(zf(z)) = f(z)k+1

24 Ludwig Reich

(with k ∈ N) in C× and in annular regions {z|r1 < |z| < r2}.For this section one may consult [6], [7], [8], [5] and [13].

References

[1] M. Bonk, The addition theorem of Weierstraß’s sigma function. Math. Ann.298: 591–601, 1994.

[2] C. Caratheodory, Funktionentheorie. Erster Band. Birkhauser Verlag, Basel1960.

[3] E. Hille, Ordinary Differential Equations in the Complex Domain. JohnWiley, New York 1976.

[4] M. Kuczma, B. Choczewski, R. Ger, Iterative Functional Equations. Ency-clopedia of Mathematics and its Applications, vol. 32, Cambridge UniversityPress 1990.

[5] L. Reich, J. Smıtal, On generalized Dhombres equations with nonconstantpolynomial solutions in the complex plane. Aequat. Math. 80: 201–208,2010.

[6] L. Reich, J. Smıtal, M. Stefankova, Local Analytic Solutions of the general-

ized Dhombres functional equations I. Osterr. Akad. Wiss. Wien, Math.-natKl. Sitzungsberichte Abt. II 214: 3–25, 2006.

[7] L. Reich, J. Smıtal, M. Stefankova, Local Analytic Solutions of the general-ized Dhombres functional equations II. J. Math. Anal. Appl. 355: 821–829,2009.

[8] L. Reich, J. Smıtal, M. Stefankova, The holomorphic solutions of the gen-eralized Dhombres functional equation. J. Math. Anal. Appl. 333: 881–888,2007.

[9] R. Remmert, Funktionentheorie I. Springer Verlag, Berlin 1984.

[10] R. Remmert, Funktionentheorie II. Springer Verlag, Berlin 1991.

[11] C. L. Siegel, Analytische Zahlentheorie II. Mathematisches Institut der Uni-versitat Gottingen, Berlin 1964.

[12] C. L. Siegel, Vorlesungen uber Himmelsmechanik. Springer Verlag, Berlin1956.

[13] J. Tomaschek, Contributions to the local theory of generalized Dhombresfunctional equations in the complex domain. Grazer. Math. Ber., to appear.

[14] A. Weil, Elliptic Functions according to Eisenstein and Kronecker. SpringerVerlag, Berlin 1976.

Ludwig ReichKarl-Franzens-Universitt GrazInstitute of Mathematics and Scientific ComputationsHeinrichstraße 368010 Graz, Austriae-mail: [email protected]

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