Conformal Self Mappings of the Fundamental Domains of Analytic

Functions and Computer Experimentation

ANDREI-FLORIN ALBIŞORU1, DORIN GHIŞA2

1Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, ROMANIA

2Department of Mathematics, Glendon College, York University, Toronto, CANADA

Abstract: - Conformal self mappings of a given domain of the complex plane can be obtained by using the Rie-

mann Mapping Theorem in the following way. Two different conformal mappings φand ψof that domain onto

one of the standard domains: the unit disc, the complex plane or the Riemann sphere are taken and then ψ−1◦φ

is what we are looking for. Yet, this is just a theoretical construction, since the Riemann Mapping Theorem does

not offer any concrete expression of those functions. The Möbius transformations are concrete, but they can

be used only for particular circular domains. We are proving in this paper that conformal self mappings of any

fundamental domain of an arbitrary analytic function can be obtained via Möbius transformations as long as we

allow that domain to have slits. Moreover, those mappings enjoy group properties. This is a totally new topic.

Although fundamental domains of some elementary functions are well known, the existence of such domains for

arbitrary analytic functions has been proved only in our previous publications mentioned in the References sec-

tion. No other publication exists on this topic and the reference list is complete. We deal here with conformal self

mappings of fundamental domains in its whole generality and present sustaining illustrations. Those related to

the case of Dirichlet functions represent a real achievement. Computer experimentation with these mappings are

made for the most familiar analytic functions.

Key-Words: - Conformal mappings, Fundamental domains, Möbius transformations, Euler Gamma function,

Riemann Zeta function, Dirichlet functions, Computer experimentation

Received: April 26, 2023. Revised: July 28, 2023. Accepted: August 21, 2023. Published: September 26, 2023.

Dedicated to the memory of Professor Gabriela Kohr

1 Introduction

Let f(z)be a holomorphic function in Cwith the ex-

ception of isolated singular points, which can be poles

or essential singular points. It is known, [1], that

C=∪n≤∞

k=1 Ωk,where Ωkare open connected sets,

Ωk∩Ωj=∅,when j=kand every Ωkis conformal

(hence bijectively) mapped by fonto C\Lk,where

Lkis a slit, or a cut, i.e., a Jordan arc or a Jordan in-

finite curve. We will treat slits mostly as point sets.

If E⊂Cis any point set we will denote by f(E)the

image of Eby f, i.e.

f(E) = {f(z)|z∈E}

and by f−1(E)the pre-image of Eby fi.e.,

f−1(E) = {z|f(z)∈E}.

This convention cannot produce any confusion. The

uniqueness theorem of analytic functions guarantees

that for a non constant analytic function f(z)the pre-

image by f(z)of a point wis a discrete set of points

{zn}such that f(zn) = w.

A slit exhibits two distinct edges, [2], and a point

of the slit can be n on one edge or on the other. The in-

verse function f−1

|Ωk,which exists for every k, in view

of bijectiveness of fin Ωk,fails to have a continuous

extension to Lk,since for sequences of points tend-

ing to the same point on Lkfrom the sides of dif-

ferent edges the function has different limits. How-

ever, the function f|Ωkcan be extended by continu-

ity to the boundary ∂Ωkof Ωkand it maps ∂Ωkonto

Lk.This fact is granted by the Riemann-Caratheodory

Theorem of boundary correspondence in the confor-

mal mapping [3] and [4]. The same theorem allows

the extension of f−1

|Ωknot to Lk,but to the two edges of

Lkwhich then are mapped one to one by the extended

function onto ∂Ωk.

Ahlfors, [2], called the domains Ωkfundamental

regions of f. When fis a rational function of degree

m, then n=m, [5].

Fundamental domains are known also for analytic

functions having non isolated singular points, as for

example the modular function or the infinite Blaschke

products. For the modular function every point of the

real axis is a singular point (obviously, non isolated),

while for the infinite Blaschke products such points

are the cluster points of the poles.

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Examples of fundamental domains for different

classes of analytic functions were given in [6], [7].

We have n there that they are not uniquely determined,

yet some points of their boundaries must be the same

for any partition of the complex plane into fundamen-

tal domains and these are the branch points of the

function f. The branch points of fare the zeros of

derivative of f, the multiple poles and the essential

singular points of this function. Since in any neigh-

borhood of such a point the function fails to be injec-

tive, the branch points cannot belong to any funda-

mental domain and therefore they should be located

on the boundaries of these domains. This simple re-

mark, as well as the simultaneous continuation tech-

nique, [5], [6], [7], [8], [9], allowed us to implement

procedures of finding fundamental domains for dif-

ferent classes of analytic functions.

It is known that for a Blaschke product w=B(z)

of degree nthe equation B(z) = 1 has exactly ndis-

tinct solutions ζkall located on the unit circle and for

every kthe image by B(z)of the arc of the unit cir-

cle between ζkand ζk+1 is the whole unit circle. The

equation B′(z) = 0 has no solution on the unit circle,

[5], [7], [8]. Then the pre-image by B(z)of the point

1is the set of points {ζk}and when wmoves from

1towards 0on the real axis, npoints zkwill move

each one from the corresponding ζkinside the unit

disc describing some Jordan arcs. These arcs can only

meet each other at the branch points of B(z).When

two adjacent arcs starting from consecutive points ζk

and ζk+1 on the unit circle meet each other at ak, they

bound together with the arc of the unit circle between

ζkand ζk+1 a domain which is conformal mapped by

B(z)onto the unit disc with a slit alongside the real

axis from B(ak)to 1.The symmetric of this domain

with respect to the unit circle is conformal mapped

by B(z)onto the exterior of the unit circle with a slit

alongside the real axis from 1to 1/B(ak)and there-

fore the union Ωkof the two domains and of the com-

mon part of the boundary is conformal mapped by

B(z)onto the whole complex plane with a slit along-

side the real axis from B(ak)to 1/B(ak).The do-

main Ωkis a fundamental domain of B(z)and we

obtain the remaining n−1fundamental domains of

B(z)in a similar way.

Although we use the words move and describe this

is a static situation, where arcs are mapped bijectively

onto some intervals of the real axis.

Let us notice that the topological facts listed

above have as corollary a surprising algebraic result,

namely, that the image by B(z)of the roots of B′(z)

are all real.

Fig. 1, Fig. 2, Fig. 3 and Fig. 4 illustrate this affir-

mation for B(z)having the triple zeros a, −awith |a|

<1and 0.Different stages of the construction of the

Figure 1: Building fundamental domains of a

Blaschke product of degree 9

fundamental domains are exhibited in these figures.

This result has been generalized in [7] to infinite

Blaschke products and we have found that in every

neighborhood of a cluster point of poles (which is

necessarily on the unit circle) of an infinite Blaschke

product there are infinitely many fundamental do-

mains of that function. This is a completion of the

Big Picard Theorem in the sense that such a point is

not an isolated singular point (being limit of poles)

and instead of infinitely many points having the same

image it states that infinitely many domains have the

same image.

We notice also that every neighborhood of an iso-

lated essential singular point of any analytic function

fintersects infinitely many fundamental domains. In-

deed, let abe such a point and let Vbe a neighborhood

of a. If wis a non omitted value of f, then there are

infinitely many points zk∈Vsuch that f(zk) = w.

Each one of the points zkis either an interior point of

a fundamental domain Ωkor a finite number kmof

fundamental domains meet in zk. As the set (km)is

infinite, infinitely many fundamental domains fin-

tersect V.

The exponential function f(z) = ezhas as funda-

mental domains horizontal strips of width 2π, as for

example

Ωk={z|2kπ < ℑz < 2(k+ 1)π}

where k∈Z.

Each one of these domains is conformal mapped

by f(z)onto the complex plane with a slit alongside

the positive real half axis.

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Figure 2: Building fundamental domains of a

Blaschke product of degree 9

Figure 3: Building fundamental domains of a

Blaschke product of degree 9

Figure 4: Fundamental domains of a Blaschke prod-

uct of degree 9

Let us notice that f(z) = ezmaps one to one the

interval

{z=iy|0≤y < 2π} ⊂ Ω0

onto the unit circle. Moreover, since

ex0+iy0=ex0(cos y+isin y)

is the equation of a circle centered at the origin and

of radius ex0, when 0≤y < 2π,f(z)is a conformal

mapping of the half of Ω0on the left side of the imagi-

nary axis (corresponding to x0<0) onto the unit disc

(since ex0<1) with a slit alongside the real axis from

0 to 1. It is also a conformal mapping of the half of Ω0

at the right side of the imaginary axis onto the exterior

of the unit disc (since ex0>1) with a slit alongside

the real axis from 1 to ∞. This remark gives a good

idea of the geometry of the conformal mapping of Ω0

by f(z) = ez. We have a similar situation for every

Ωk. Fundamental domains of the exponential func-

tion are illustrated in Fig. 5.

The fundamental domains of f(z) = cos zare

bounded by vertical lines ℜz=kπ and ℜz= (k+

1)π, k ∈Z.This function realizes a conformal map-

ping of each one of these strips onto the complex

plane with a slit alongside the real axis complemen-

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Figure 5: Fundamental domains of ez

Figure 6: Fundamental domains of cos z

tary to the interval (−1,1).Indeed, we have

cos z=1

2(eiz +e−iz) = 1

2(eix−y+e−ix+y)

=e−y

2(cos x+isin x) + ey

2(cos x−isin x)

=cos x1

2(ey+e−y)−isin x1

2(ey−e−y)

=cos xcosh y−isin xsinh y.

Thus,

cos(kπ +iy) = (−1)kcosh y,

which shows that every vertical line z=kπ +iy,

k∈Zis mapped two to one by f(z) = cos zonto the

interval of the real axis from −∞to −1when kis odd

and from 1to +∞when kis even. Therefore every

vertical strip bounded by two consecutive such lines

is a fundamental domain for f(z) = cos z, which is

mapped by f(z)onto the complex plane with a slit

alongside the real axis complementary to the interval

(−1,1).Fundamental domains of the cosine function

are given in Fig. 6. All the other trigonometric func-

tions have similar fundamental domains.

The Euler Gamma function is an extension to

the whole complex plane of the arithmetic function

Γ(n) = (n−1)!.For ℜz > 1the extension is given

Figure 7: Fundamental domains of the Euler Gamma

and the Riemann Zeta function

Γ(z) = Z∞

e−ttz−1dt (1)

Integrating by parts, we find that Γ(z)satisfies the

functional equation zΓ(z) = Γ(z+ 1) which allows

its extension to the half-plane ℜz≤1.The pre-image

by Γ(z)of the real axis displays fundamental domains

of this function as shown in Fig. 7. The red curves

are the pre-image of the positive half axis and the

black ones that of the negative half axis. The domains

bounded by consecutive curves of the same color are

fundamental domains of Γ(z).They are conformal

mapped by the function onto the complex plane with

a slit alongside some intervals of the real axis. On the

right side of Fig. 7, the preimage of the real axis by

the Riemann Zeta function is illustrated. It is similar

to that of the Gamma function, yet not all the domains

bounded by its components are fundamental domains.

The modular function λ(τ)has been built, [2],

starting with a domain Ω1bounded by the half-lines

ℜτ=±1,ℑτ≥0and the half-circles |τ±1/2|=

1/2,ℑτ≥0.The Riemann Mapping Theorem states

that there is a unique conformal mapping λ(τ)of the

domain Ω1onto the complex plane with a slit along-

side the real axis from −∞ to −1and from 1to +∞

such that τ= 0,1,∞corresponds to λ= 1,∞,0,

[2]. Using the Schwartz Symmetry Principle this

mapping can be extended analytically to the whole

upper half plane. The domains obtained by iterated

symmetries with respect to the half-lines and half-

circles are all fundamental domains of λ(τ),which

are illustrated in Fig. 8. In every neighborhood of a

point of the real axis there are infinitely many such

domains and therefore the real axis is a singular line

of λ(τ).

The fundamental domains mentioned above are

obvious, yet for most of the classes of analytic func-

tions they have to be found. On the other hand, there

is no hope to compute the values of λ(τ)since the

Riemann Mapping Theorem states only the fact that

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Figure 8: The fundamental domains of the modular

function

such a function exists and is unique. However, it has

this special feature that the conformal mappings of the

fundamental domains one of each other are known,

namely those generated by τ→τ+1 and τ→ −1/τ.

We will show later that for any analytic function such

mappings can be found as long as we allow funda-

mental domains to have slits.

The Weierstrass ℘function has been defined, [2],

by the formula

℘(z) = 1

z2+X

ω=0 1

(z−ω)2−1

ω2(2)

where ω=mω1+nω2,(m, n)∈Z×Z\ {(0,0)}

with ω1and ω2arbitrary complex numbers having non

real ratio ω1/ω2.It is a doubly periodic function with

the periods ω1and ω2.The parallelogram determined

by ω1and ω2is divided by the diagonal from ω1to

ω2into two triangles which are fundamental domains

for ℘, [8]. The conformal mapping of these triangles

made by ℘onto the complex plane with three slits is

illustrated in Fig. 9.

2 Fundamental Domains of Dirichlet

Functions

The conformal mappings by Dirichlet functions have

been studied in [10], [11], [12], [13] and the way the

fundamental domains of these functions can be re-

vealed has been explained. However, for a better un-

derstanding of this topic we will repeat some of the

findings in those papers.

Figure 9: The fundamental domains of ℘and their

conformal mapping onto the whole complex plane

with three slits

Since the Dirichlet L-functions have been imple-

mented in different packages of software and for illus-

tration purposes we only make use of these functions,

it is necessary to present them separately.

ADirichlet character modulo qis an arithmetic

function χ(n)which is periodic of period qand such

that if qand nare not relatively prime, then χ(n) = 0

and if they are relatively prime then χ(n)is a root of

order φ(q)of the unity, where φis the Euler totient

function. For every qwe have χ(1) = 1.

Let us take q= 7.Then φ(q)=6and the six

Dirichlet characters modulo seven are given in Table

1, where ω=eπi/3.

χ\n0 1 2 3 4 5 6

χ1(n)0 1 1 1 1 1 1

χ2(n)0 1 ω2ω−ω−ω2-1

χ3(n)01−ω ω2ω2−ω1

χ4(n)0 1 1 -1 1 -1 -1

χ5(n)0 1 ω2−ω−ω ω21

χ6(n)01−ω−ω2ω2ω-1

Table 1: Dirichlet characters modulo 7

ADirichlet function is obtained by performing an-

alytic continuation to the whole complex plane of the

sum of a Dirichlet series

ζA,Λ(s) =

∞

n=1

ane−λns(3)

where Λ = (λn)is a non decreasing sequence of pos-

itive numbers, A= (an)is an arbitrary sequence of

complex numbers and s=σ+it is a complex vari-

able. It make sense to deal only with normalized se-

ries (3) in which a1= 1 and λ1= 0.For every nor-

malized Dirichlet series ζA,Λ(s)we have

lim

σ→+∞ζA,Λ(σ+it) = 1.(4)

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This apparently trivial property has surprising conse-

quences regarding the geometry of the mappings by

Dirichlet series. We list here a few of them.

The pre-image of the real axis by ζA,Λ(s)has

infinitely many components which are of the three

types:

a) Γ′

k,k∈Zextending for σfrom −∞ to +∞

and which are mapped by ζA,Λ(s)bijectively onto the

interval (1,+∞)of the real axis. These curves do

not intersect each other and Γ′

kand Γ′

k+1 form infi-

nite open strips Skwhich are mapped by ζA,Λ(s)not

necessarily one to one onto the whole complex plane

with a slit alongside the interval (1,+∞)of the real

axis. We count them from −∞to +∞such that Sk+1

is above Skand 0∈S0.

b) Every strip Sk, k = 0 contains a unique com-

ponent Γk,0of the pre-image of the real axis which

is mapped by ζA,Λ(s)bijectively onto the interval

(−∞,1) of the real axis. This component also ex-

tends for σfrom −∞ to +∞.

c) Every strip Sk, k = 0 contains a finite number

of components Γk,j ,j= 0 of the pre-image of the real

axis which are mapped bijectively by ζA,Λ(s)onto the

whole real axis. These components extend for σfrom

−∞ to some finite values. They are parabola shaped

curves with the branches extending to −∞.The strip

S0has infinitely many components Γ0,j .

The strip Sk, k = 0 having mcomponents Γk,j of

the pre-image of the real axis can be partitioned into

msub-strips, the interiors of which are fundamental

domains of ζA,Λ(s).The strip S0contains infinitely

many fundamental domains.

For k= 0 the fundamental domains of every func-

tion ζA,Λ(s)are bounded by components of the pre-

image by ζA,Λ(s)of the interval (1,+∞)of the real

axis to which components of the pre-image of the

segment between 1and ζA,Λ(vk,j )is added, where

ζ′

A,Λ(vk,j ) = 0.We have shown in [10] that such a

construction is always possible.

There is number σc≤ ∞ such that the series (3)

converges locally uniformly for ℜs > σcand diverges

for ℜs < σc.The number σcis called the abscissa of

convergence of the series (3).

When an=χ(n)are Dirichlet characters of some

module qand λn=log nthen we have a Dirichlet

L-series:

L(χ, s) =

∞

n=1

χ(n)

ns(5)

The character whose values are only 0and 1is

called principal character. Every Dirichlet L-series

defined by a non principal character has the abscissa

of convergence 0,while the Dirichlet series defined

by principal characters have the abscissa of conver-

gence 1.

When q= 1 the only character is the principal one

and the corresponding series is the Riemann series,

whose abscissa of convergence is known to be 1. The

analytic continuation of this series:

ζ(s) =

∞

n=1

ns(6)

is the famous Riemann Zeta Function.

The Riemann Zeta Function is one of the most

studied analytic functions, in view of its many appli-

cations in number theory, algebra, complex analysis,

statistics, as well as in physics.

The pre-image of the real axis by ζ(s)shows in-

finite strips, Fig. 7, which can be divided into sub-

strips representing fundamental domains of this func-

tion. The way it can be done is described in [10].

We have found later, [11], [12], [13], that this

property is common to the whole class of Dirich-

let functions to which the Riemann Zeta function be-

longs. The Dirichlet functions are analytic continu-

ations to the whole complex plane of Dirichlet series

(3). We will keep the notation ζA,Λ(s)for such a func-

tion.

The components of the pre-image by ζA,Λ(s)of

circles centered at the origin and of radius rare of

three types depending on the values r.

When r < 1, these components are all closed

curves around some zeros of the function. For rsmall

enough each one of these curves contains just one

zero.

When r= 1,(the unit circle) every strip Sk, k =

0contains exactly one unbounded component of the

pre-image of the unit circle and some bounded ones.

The unbounded component has the branches tending

asymptotically to the boundary ∂Skof Sk.

When rincreases over one, the unbounded com-

ponents fuse all into one unbounded curve expanding

from t=−∞ to t= +∞.The bounded components

expand also and two of them can fuse into one con-

taining the zeros of both of them.

These components and the pre-image of the real

axis form an orthogonal net of quadrilaterals which

are conformal mapped by ζA,Λ(s)on half-rings cen-

tered at the origin with two opposite sides on the real

axis and included respectively in the upper and the

lower half-planes.

When anare all real, then the real axis is included

in the pre-image by the respective Dirichlet function

of the real axis. Otherwise, such an inclusion does

not take place. This is obvious for the Riemann Zeta

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Figure 10: The pre-images by two Dirichlet L-

functions of an orthogonal net formed with circles

centered at the origin and the real axis

function, Fig. 7, but also for the Dirichlet L-functions

illustrated in Fig. 10 and Fig. 11.

3 Fundamental Domains with Slits

The list of the classes of analytic functions we have

visited is not exhaustible, yet it gives us an idea of

what happens in general. When dealing with any an-

alytic function we can focus on a particular funda-

mental domain Ω. The function performs a conformal

mapping of Ωonto the complex plane with a slit L. If

we do also a Möbius transformation Mof the image

plane, the function M◦fis a conformal mapping of

the domain Ωonto the complex plane with two slits:

Land M(L).One of them is the image by M◦f

of a slit in Ω,thus we need to deal with fundamental

domains with slits. This is even more obvious when

we study conformal self mappings of Ωof the form

f−1

|Ω◦M◦f.

More exactly, there is a slit L′in the complex plane

carried by Monto Land which is the image by fof

a slit LMof Ω.Therefore, fcarries LMinto L′, M

carries L′into L,f−1

|Ωcarries the two edges of Linto

∂Ω,thus

χM=f−1

|Ω◦M◦f(7)

carries LMinto ∂Ω.The inverse function of χM,

where defined, is

χ−1

M=f−1

|Ω◦M−1◦f.

There is a slit L′′ of the complex plane, the image of

Lby M−1and which is also the image by fof a slit

Figure 11: The pre-image of the real axis by two

Dirichlet L-functions defined by χ2(n)and χ4(n)

modulo 7

LM−1of Ω,thus fcarries LM−1into L′′, which is

carried by Minto Land Lis carried by f−1

|Ωinto ∂Ω.

Theorem 1. The function χMis a conformal map-

ping of Ω\LM−1onto Ω\LM.The boundaries

∂Ω∪LM−1and ∂Ω∪LMof the two double con-

nected domains correspond one to each other through

χMin the following way: ∂Ωis carried into LMand

LM−1is carried into ∂Ω.

Proof: The functions f, M and f−1

|Ωare analytic

functions in their domains, except for the simple pole

of M, if there is one. The function M−1◦fis a con-

formal mapping of Ωand carries ∂Ωinto L′′,which is

carried by f−1

|Ωinto LM−1,hence χMis a conformal

mapping of Ω\LM−1which carries LM−1into ∂Ω.

On the other hand, the function χMdoes not take any

value belonging to LMsince it is injective and χ−1

carries LMinto ∂Ω.By the boundary correspondence

theorem LMmust be carried by χMinto LM−1.This

completely proves the theorem.

It is known that for two arbitrary Möbius transfor-

mations M1and M2the function M1◦M2is also a

Möbius transformation. On the other hand, where all

the functions are defined, we have:

χM1◦M2(s) = f−1

|Ω◦(M1◦M2)◦f(s)

= (f−1

|Ω◦M1◦f)◦(f−1

|Ω◦M2◦f(s))

=χM1◦χM2(s).

(8)

Thus, the composition law in the family {χM}of

transformations of Ωwith slits defined by all Möbius

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transformations Mthrough the formula (7) is an in-

ternal operation. It is obvious that if M0is the identity

mapping, them χM0is the identity transformation of

Ω.Also, for every Mthe transformation χM−1is the

inverse mapping of χM.The associativity of the com-

positions of mappings χMis a direct corollary of the

same property of the mappings M. Thus, we have:

Theorem 2. Any subgroup of Möbius transforma-

tions defines through the formula (7) a group of trans-

formations of any fundamental domain Ωwith slits of

an analytic function f.

Proof: Let Gbe a subgroup of Möbius transfor-

mations. We can think at any subgroup well stud-

ied in the literature, as for example the subgroup of

Möbius transformations which represent the unit disc

onto itself, or those with real coefficients etc.. Let

GΩ={χM|M∈G},where χMis defined by the

formula (7). It doesn’t harm to use the same sign for

the composition in Gand in GΩ.We need to prove

that GΩwith the assumed composition law is a group

of transformations of Ω.If χM1, χM2∈GΩ,where

M1, M2∈G, then by (8) we have: χM1◦χM2(s) =

χM1◦M2(s),thus the composition in GΩis an inter-

nal operation. If M0is the unit element of G, i.e. for

every M∈Gwe have M◦M0=M0◦M=M,

then χM◦χM0=χM◦M0=χMand χM0◦χM=

χM0◦M=χM,therefore χM0is the unit element of

GΩ.Finally, χM◦χM−1=χM◦M−1=χM0and the

conclusion is that GΩis indeed a group of transfor-

mations of Ωwith slits.

4 Fixed Points of Self-Mappings of

the Fundamental Domains

Suppose that for s0∈Ωwe have that f(s0)is

a fixed point of the Möbius transformation M, i.e.

M(f(s0)) = f(s0).We notice that f(s0)does not

belong to the slit Lcorresponding to Ω.We have

χM(s0) = f−1

|Ω(M(f(s0))) = f−1

|Ω(f(s0)) = s0,

therefore s0is a fixed point of χM.Reciprocally, if s0

is a fixed point of χMthis means

f−1

|Ω(M(f(s0))) = s0,

M(f(s0)) = f(s0),

therefore f(s0)is a fixed point of M, which obviously

does not belong to L. Moreover:

Theorem 3. The fixed points of the transforma-

tion χMof Ωare those points sfor which z=f(s)

are fixed points of M.

When Mhas a fixed point zlocated on L, then

to this point correspond two fixed points s1and s2of

the extended χMsuch that χM(s1) = χM(s2) = z

( Fig. 13). If no fixed point of Mis located on L,

then there is a one to one correspondence between the

fixed points of M(z)and those of χ(s).

It is known, [15], that any Möbius transformation

Mhas either just one fixed point (the parabolic case),

or two fixed points in which case the configuration of

the mapping by Mcan be of three kinds: elliptic, hy-

perbolic or loxodromic. All depends on the so-called

multiplier of M, which is a number µsuch that

M(z)−ξ1

M(z)−ξ2

=µz−ξ1

z−ξ2

(9)

where ξ1and ξ2are the fixed points of M. The number

µis associated to ξ1.

The multiplier associated to ξ2is 1/µ, [16].

If s1and s2are the fixed points of χM(s)corre-

sponding to the fixed points ξ1and ξ2of M(z), then

the formula (9) translates into a similar formula for

χM(s), namely:

f(χM(s)) −f(s1)

f(χM(s)) −f(s2)=µf(s)−f(s1)

f(s)−f(s2).

An important property of Möbius transformations

is that of transforming circles into circles, where by

circle we understand a proper circle or a straight line

(circle of infinite radius). We will call Ω-circle the

pre-image by f|Ωof a circle from the complex plane.

We notice that if for a circle Cwe have C∩L=∅,

then the pre-image CΩof Cis a closed curve in C.

Otherwise, if Ctraverses Lthen CΩends up at two

different points on the boundary ∂Ωof Ωand if C

is tangent to Lthen CΩis a closed curve tangent to

∂Ω.These are obvious topological properties of the

conformal mapping f|Ω.

It is known that to every non-parabolic Möbius

transformation with fixed points ξ1and ξ2we can

associate two families of circles, namely all circles

passing through ξ1and ξ2and all circles orthogonal

to them centered on the line between ξ1and ξ2and

containing just one of the points ξ1or ξ2. These last

circles are called the Apollonius circles. Together the

two families of circles form the so called Steiner net.

The pre-image by f|Ωof a Steiner net will be called

Ω-Steiner net, which is a net of orthogonal curves in

Ωsince f−1

|Ωis a conformal mapping.

The Steiner net of Mdescribes the way Mmoves

the points of the complex plane and this depends on

the value of µ. If µ=eiθ, θ ∈R(the elliptic case)

the points are moved alongside the Apollonius circles

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counterclockwise around ξ1and clockwise around ξ2

and the orthogonal circles move one into the other.

If µ∈R, µ = 1 (the hyperbolic case) the points are

moved such that the Apollonius circles expand around

ξ1and they shrink around ξ2.When µ=ρeiθ, θ ∈R,

ρ= 1 (the loxodromic case) the motion of the points

by Mis a combination of the previous two, resulting

in trajectories of double spirals issuing from ξ1and

entering in ξ2.When the transformation is parabolic,

i. e., it has a unique fixed point ξthen we have two

families of orthogonal circles passing through ξand a

combination of elliptic and hyperbolic motions.

Theorem 4. The pre-image by f|Ωof the Steiner

net generated by a Möbius transformation Mis a net

of orthogonal Ω-circles, the Ω-Steiner net. This net

describes the motion of the points sby the self map-

ping χMof Ωin the same way the Steiner net of M

does with the points f(s).

Proof: Suppose that Mis non parabolic, there-

fore it has two distinct fixed points ξ1and ξ2. If none

of them belong to the slit associated to Ω, then they

are the image by fof two distinct points s1, s2∈Ω,

which are fixed points of χM.If one of them belongs

to the slit, then it is the image by the extended fof

two points on ∂Ω.

The pre-image by f|Ωof every Apollonius circle

around ξkis a Ω-circle around sk, k = 1,2,the Apol-

lonius Ω-circle. The pre-image by f|Ωof every circle

orthogonal to the Apollonius circles is a Ω-circle or-

thogonal to all the Apollonius Ω-circles. Since f−1

|Ω

is a conformal mapping, the sense of the motion of

the points on the Ω-circles by χMis the same as that

of their f|Ω-images by M. So, in the elliptic case the

points smove on the Apollonius Ω-circles around sk

in counterclockwise around s1and clockwise around

s2.In the hyperbolic case, the Apollonius Ω-circles

around s1expand while those around s2shrink. Fi-

nally, in the loxodromic case the motion of the points

sis on a double spiral issuing from s1and entering in

s2.

In the parabolic case, where Mhas a unique fixed

point ξthere are four families of orthogonal Ω-circles

passing through s0=f−1

|Ω(ξ).The transformation χM

of Ωwill move the points sin the same way as M

moves the points zof the complex plane in the corre-

sponding Steiner net.

The computer experimentation with Dirichlet L-

functions revealed some surprising facts. Since the

formula (4) suggests that the components of the pre-

image of circles passing through z= 1 tend to ∞

as σ→ ∞ one could conclude that the function χM

corresponding to M(z) = (2z−1)/(2 −z)has only

one fixed point, which is attractive, while ∞is re-

pelling. However, in Fig. 19 appeared some other

repelling points. We realized that these are points on

the boundaries of the fundamental domains where the

respective Dirichlet L-function takes the value z= 1.

Fig. 19 portrays the conformal self mapping of sev-

eral fundamental domains whose boundaries cannot

be n in the image. What we know for sure is the fact

that these boundaries are necessarily between compo-

nents of the pre-image of Apollonius circles asymp-

totically tangent at infinity.

5 Some Subgroups of the Group of

Transformations of Ω

The subgroups of the group of transformations of

Ωcorrespond to subgroups of the group of Möbius

transformations. The most familiar subgroup of

Möbius transformations is

G=Ma,θ



Ma,θ(z) = eiθ z−a

1−az ,|a| = 1, θ ∈R

(10)

which leaves invariant the unit circle and transforms

the unit disc into itself when |a|<1and onto the ex-

terior of the unit disc when |a|>1.Also the exterior

of the unit disc is transformed into itself when |a|<1

and onto the unit disc when |a|>1.

It can be easily checked that Ma,θ ◦Mb,η =Mc,ϕ,

where

c=a+e−iθb

1 + eiθab and ϕ=θ+η,

and thus the composition in Gis an internal law. The

unit element of Gis obtained for a= 0 and θ= 0.

The inverse element Mb,η of Ma,θ is obtained for b=

−eiθaand η=−θ. It is obvious that the elements

of Gwith |a|<1form a subgroup of G. This is not

true for the elements of Gwith |a|>1since the unit

element of Gis not among them.

The interval (1,+∞)of the real axis is trans-

formed by Ma,θ into an arc of a circle and if a∈R

and θ= 0 it is transformed into an interval of the

real axis. This remark will be useful for the examples

which follow.

Let us deal first with the Möbius transformation

M(z) = 2z−1

2−z(11)

which is an element of Gabove with a= 1/2 and

θ= 0.This is a non-parabolic Möbius transforma-

tion with the fixed points 1and −1.We will clas-

sify the circles of the corresponding Steiner net tak-

ing into account their position with respect to the slit

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Figure 12: The Steiner net of the Möbius transforma-

tion (11) with the slits corresponding to the exponen-

tial function

L= (1,+∞)associated to a fundamental domain Ω

of a Dirichlet function ζA,Λ(s). We notice that the

Apollonius circles corresponding to this transforma-

tion are of two kinds: those around 1,which intersect

the slit in one point and therefore their pre-images

by ζA,Λ(s)are arcs having the ends on each one of

the components of ∂Ωand those around −1which

do not intersect the slit and therefore their pre-images

are closed Ω-circles. All these Ω-circles fill the do-

main bounded by the pre-image of the imaginary axis,

which is a parabola-like curve with the branches tend-

ing asymptotically towards ∂Ωat σ=−∞.The re-

maining part of Ωis filled by the pre-images of circles

intersecting the slit. Each point of ∂Ωis the end point

of such an Ω-circle.

Let us notice that since Mhas real coefficients, it

maps the real axis onto itself and so does M−1.Thus,

for L= (1,+∞)we have that L′and L′′ are parts

of the real axis. Since M(1) = 1, M (2) = ∞and

M(∞) = −2,we have L′= (−∞,−2) ∪(1,+∞).

Also, since M−1(1) = 1 and M−1(∞) = 2,we have

that L′′ = (1,2).Since L′′ ⊂L, for these functions

we have LM−1⊂∂Ω,thus χM: Ω →Ω\LM.

Since Mhas the fixed points 1and −1,the function

χMshould have the fixed points s1and s−1for which

ζA,Λ(s1) = 1 and ζA,Λ(s−1) = −1.Yet, there is no

such s1∈Ωsuch that ζA,Λ(s1)=1.We only have

limσ→+∞ζA,Λ(σ+it) = 1.We cannot say that ∞

is a fixed point for χMsince χMis not defined at

∞. However, ∞is repelling for χMand χMhas the

attracting fixed point s−1.

This is a hyperbolic Möbius transformation with

the multiplier µ= 3.Let us notice that if a circle C

of the net passing through −1and 1has the center at

ih, h ∈R,then its radius is r= (h2+ 1)1/2.For any

Apollonius circle orthogonal to it of center x∈Rand

radius ρwe have

r2+ρ2=x2+h2=x2+r2−1,

Figure 13: The conformal self-mapping by χMof the

fundamental domain of the exponential function and

the corresponding Ω-Steiner net when M is given by

the relation (11)

hence x2=ρ2+ 1,which means that xand ρde-

pend on each other, but not on hand r, confirming the

known fact that the Apollonius circles are all orthogo-

nal to every circle Cpassing through the fixed points

of the Möbius transformation. These relations help us

to illustrate the corresponding Steiner net by choos-

ing conveniently the parameters of the two families

of circles. In the picture above we have chosen for

hthe values: ±1/2,±1and ±3/2.Correspondingly,

we have for rthe values: √5/2,√2and √13/2.If

we let ρ=r, then we have for xrespectively: ±3/2,

±√3and ±√17/2.Now we can write the equations

of the 12 circles chosen to represent the Steiner net

and use a software to draw the respective net and its

pre-image by any analytic function into any funda-

mental domain of that analytic function.

Fig. 12 is so simple since the Möbius transfor-

mation M(z)we have taken has real coefficients and

therefore the real axis is mapped by M(z)onto itself

and LMand LM−1are located on the preimage of the

real axis, Fig. 13, Fig. 15 and Fig. 17 . In particular,

when the slit is L= (1,+∞),which is the case of

Dirichlet functions and the fundamental domains R1

and R−1, then L′and L′′ are also on the real axis. The

situation is a little more complicated when the coeffi-

cients of M(z)are complex. Then the pre-images of

Lby Mand M−1are arcs of a circle, Fig. 14, which

are mapped by f−1

Ωinto some curves on the Ω-Steiner

net, Fig. 14 and Fig. 15. Let us deal with the Möbius

transformation:

M(z) = z−a

1−az , a /∈R,|a| = 1 (12)

This is a non parabolic Möbius transformation with

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Figure 14: The Steiner net of to the Möbius transfor-

mation (12) with the slits corresponding to the expo-

nential function

the fixed points ξ1,2=±eiθ,where θ=arg a. It can

be easily checked that if a=reiθ then the multiplier

of the transformation (12) is µ=1+r

1−r,therefore this

transformation is hyperbolic.

The function M(z)transforms the interval

(1,+∞)into an arc of circle with the ends at

1−a

1−a=e2iϕ,where ϕ=arg(1 −a)and −1/aand

this arc of circle is mapped by f−1

|Ωinto LM.Also,

M−1transforms (1,+∞)into an arc of circle with

the ends at 1+a

1+a=e2iψ,where ψ=arg(1 + a)and at

1/aand this arc of circle is transformed by f−1

|Ωinto

LM−1.

When the slit corresponding to f|Ωis (0,∞),as in

the case of the exponential function, we have M(0) =

−a, M(∞) = −1/a, M−1(0) = aand M−1(∞) =

1/a.

Let us take a=1

2(1 + i),hence arg a=π/4.We

get the non parabolic Möbius transformation with the

fixed points ±(1+i)/√2.It can be easily checked that

the image by Mof the interval (1,+∞)is an arc of

circle with the ends in −iand −(1+i)passing through

(−1/2)(1 + i)and the image by M−1of the same

interval is an arc of circle with the ends in (4 + 3i)/5

and 1 + ipassing through (9 + 7i)/10.When the slit

corresponding to f|Ωis (0,∞),then L′is an arc of

circle ending at −(1 + i)and (−1/2)(1 + i)and L′′ is

an arc of circle ending at (1/2)(1 + i)and 1 + i, Fig.

15.

Additional Steiner nets for different functions are

illustrated in Fig. 16 and Fig. 18.

Figure 15: The conformal self-mapping by χMof the

fundamental domain of the exponential function and

the corresponding Ω−Steiner net when M is given by

the relation (12)

Figure 16: The Steiner net of the Möbius transfor-

mation (11) with the slits corresponding to the cosine

function

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Figure 17: The conformal self-mapping by χMof the

fundamental domain of the cosine function and the

corresponding Ω-Steiner net when M is given by the

relation (11)

Figure 18: The Steiner net of the Möbius transforma-

tion (11) with the slits corresponding to the Dirichlet

L-function L(7,2,s)

Figure 19: The conformal self-mapping by χMof the

fundamental domain of the Dirichlet L(7,2,s) func-

tion and the corresponding Ω−Steiner net when M is

given by the relation (11)

6 Conformal Mappings of the

Fundamental Domains with Slits

One Onto Each Other

Let Ωkand Ωjbe two fundamental domains of the an-

alytic function fand let f|Ωkand f|Ωjbe conformal

maps of these domains onto the complex plane with

the slits Lkand respectively Lj.There is a slit Lk,j

of Ωkwhich is mapped by f|Ωkonto Ljand there is

a slit Lj,k of Ωjwhich is mapped by f|Ωjonto Lk, in

other words Lk,j =f−1

|Ωk(Lj)and Lj,k =f−1

|Ωj(Lk).

The function χk,j (z) = f−1

|Ωj◦f|Ωk(z)is a conformal

mapping of Ωk\Lk,j onto Ωj\Lj,k.These two double

connected domains have the boundaries ∂Ωk∪Lk,j

and respectively ∂Ωj∪Lj,k and χk,j carries ∂Ωkinto

Lj,k and Lk,j into ∂Ωj.when k=jthen Lk,j =∅

and χk,k is the identity mapping of Ωk.It is also ob-

vious that χ−1

k,j =χj,k.

We can only compose functions χk,j with χm,k for

different jand mand we obtain χm,j .

Each one of the domains Ωk,Ωjand Ωmhave now

two slits , namely Lk,j and Lk,m,respectively Lj,k

and Lj,m and finally Lm,k and Lm,j .

The mappings can be extended to the boundaries.

We notice that

(χk,j ◦χm,k )◦χl,m =χm,j ◦χl,m =χl,j

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and

χk,j ◦(χm,k ◦χl,m) = χk,j ◦χl,k =χl,j ,

thus the composition law is associative. However, the

set of these mappings fails to form a group since no

two arbitrary mappings can be composed.

Let us notice that when Lk=Ljthen Lk,j =

Lj,k =∅and χk,j is a conformal mapping of Ωkonto

Ωj.This mapping can be extended by continuity to

∂Ωkand the extended function maps one to one ∂Ωk

onto ∂Ωj.A lot of classes of analytic functions ful-

fill this condition, as for example the exponential, the

modular function, the trigonometric and hyperbolic

functions.

Finally, if fhas a finite number mof fundamen-

tal domains, as is the case of the rational functions of

degree m,then the number of functions χk,j is m2.

7 Conclusion

The study of the fundamental domains of analytic

functions initiated by Ahlfors, proved to be useful

in revealing global mapping properties of these func-

tions and in particular in the theory of distribution of

zeros of Dirichlet functions. In our opinion the fa-

mous Riemann Hypothesis related to the non trivial

zeros of the Riemann Zeta function is an aspect of a

deeper inside of these global mapping properties of

Dirichlet functions. This topic has been treated in

the literature only by number theory techniques. Us-

ing the powerful tool of conformal mappings has the

advantage of a global view. Moreover, this tool en-

abled a strategy of divide and conquer, in which it was

enough to deal with a specific strip containing a fi-

nite number of fundamental domains in order to draw

conclusions valid for the whole complex plane. The

graphics we included have shown that the behavior of

those mappings is like that of the Möbius transforma-

tions. They do to the fundamental domains the same

thing that the Möbius transformations are doing to the

whole complex plane. In this paper we have focused

on a particular property of fundamental domains of

any analytic function, namely that of allowing con-

formal mappings onto itself. These mappings are ob-

tained by combining the function with Möbius trans-

formations and the inverse function. We have also

studied the conformal mappings between two funda-

mental domains.

Acknowledgment:

The authors are thankful to the peer reviewers for

their comments.

References:

[1] Andreian-Cazacu, C., and Ghisa, D., Global

mapping properties of analytic functions,

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[2] Ahlfors, L. V., Complex Analysis, International

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[3] Nehari, Z., Conformal Mapping, International

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[4] Graham, I., and Kohr, G., Geometric Function

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[14] Barza, I., Ghisa, D., and Muscutar, F.A., On the

Location of Zeros of the Derivative of Dirichlet

L-functions, Annals of the University of

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Algorithm 1.1 and 1.2 in C++.

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exe-

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