Conformal Self-Mappings of the Complex Plane with Arbitrary Number

of Fixed Points

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

1andrei.albisoru@ ubbcluj.ro, 2dghisa@ yorku.ca

Abstract: There are known conformal self-mappings of the fundamental domains of analytic functions via Möbius

transformations. When two adjacent fundamental domains have a straight line or an arc of a circle as a common

boundary, the Schwarz symmetry principle can be applied for one of those mappings and what we obtain is a

conformal self-mapping of the union of those domains in which each one of the domains is mapped onto itself.

Repeating this operation until the whole plane is exhausted, we obtain a conformal self-mapping of the complex

plane in which every fundamental domain is conformally mapped onto itself. We prove in this paper that this

is true for any analytic function. Since the self-mappings of fundamental domains have each one at least one

fixed point, ultimately, for the self-mapping of the complex plane, we obtain at least as many fixed points as

is the number of fundamental domains. When dealing with a rational function, this number is finite, otherwise

we obtain infinitely many fixed points. Computer experimentation allows the illustration of these concepts for

most of the familiar classes of analytic functions. There are known applications of the Möbius transformations in

physics via the Lorentz group. Relating those application to the present work may contribute to the advancement

of the knowledge in that field.

Key-Words: Conformal mappings, Fundamental domains, Möbius transformations, Dirichlet functions,

Computer experimentation, Steiner net

Received: March 29, 2023. Revised: November 21, 2023. Accepted: December 11, 2023. Published: December 31, 2023.

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 conformally (hence bi-

jectively) mapped by fonto C\Lk, where Lkis a slit,

or cut, i.e., a Jordan arc or a Jordan infinite 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 f, i.e.,

f−1(E) = {z|f(z)∈E}.

This convention cannot produce any confusion.

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

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

tion f(z)is defined on every Ωk, except for the es-

sential singular points and it maps the boundary ∂Ωk

of Ωkonto the slit Lk. However, the inverse func-

tion f−1

|Ωk, which exists for every kin view of bijec-

tiveness of f|Ωkfails to have a continuous extension

to Lksince for sequences of points tending to the

same point on Lkfrom the sides of different edges

the function has different limits. Yet, f−1

|Ωkcan be

extended to the two edges of Lkand it maps those

edges onto ∂Ωk.This fact is granted by the Riemann-

Caratheodory Theorem of boundary correspondence

in conformal mapping, [3] and [4]. Ahlfors has called

the domains Ωkfundamental regions of the function

f(see, e.g., [2]). They prove to be useful in revealing

global mapping properties of analytic functions and

in particular in the theory of distribution of zeroes of

Dirichlet functions.

We have shown in [5] that to every Möbius trans-

formation Ma conformal self-mapping of every such

a domain Ωcan be associated. Suppose that Ωis con-

formally mapped by fonto the whole complex plane

with a slit L. The function Mmoves the slit Linto a

slit L′, which is the image by fof a slit LMof Ω. On

the other hand, M−1moves Linto a slit L′′,which is

the image by fof another slit LM−1of Ω. We have

proved in [5]:

Proposition 1. The function

χM=f−1

|Ω◦M◦f

is a conformal mapping of Ω\LM−1onto Ω\LMin

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which the boundary ∂Ωof Ωis carried into LMand

LM−1is carried into ∂Ω.

This means that χMcan be extended by continuity

to ∂Ωand the image of ∂Ωby the extended function

is LM.

Also χMcan be extended by continuity to the sides

of LM−1and the image by the extended function of

LM−1is ∂Ω.

Proposition 2. The fixed points of the mapping

χMand those of the Möbius transformation Mare

related in the following way: z=f(s)is a fixed point

of Mif and only if sis a fixed point of χM.

It does not mean that the two functions have nec-

essarily the same number of fixed points. Indeed, if

z∈Lthen there can be two points sand s′such that

f(s) = f(s′) = z, hence to a fixed point of Mit may

correspond two fixed points of χM. An example has

been given in [5], where such a situation effectively

occurs.

The purpose of this paper is to prove that for any

fundamental domain Ωof fand for any Möbius trans-

formation M, the mapping χMcan be extended to

the whole plane in such a way that every fundamen-

tal domain of fis mapped by the extended function

onto itself. Then it can be concluded that the respec-

tive mapping has at least as many fixed points as the

number of the fundamental domains of f.

2 The Case of Rational Functions

Global mapping properties of rational functions re-

lated to their fundamental domains have been studied

in [6]. For the rational functions of the second degree

we have shown that:

Proposition 3. Every rational function Rof the

second degree can be written under the form R=

M2◦T◦M1,where M1and M2are Möbius transfor-

mations and T(ζ) = ζ2.

The function ζ=M1(z)transforms the z-plane

into the ζ-plane in such a way that a line or a circle L

from the z-plane goes to the real axis into the ζ-plane.

The function η=ζ2transforms each one of the upper

and lower half-planes from the ζ-plane into the whole

η-plane with the slit L′which is the positive real half-

axis. Finally, the function w=M2(η)transforms the

η-plane with the slit L′into the w-plane with a slit

which is an arc of a circle or a half-line L′′. Sum-

ming up, the rational function w=R(z)transforms

each one of the two domains determined by Linto

the whole w-plane with the slit L′′ which is an arc of

a circle or a half-line.

Example. For the function

R(z) = (18 −i)z2−6(2 −i)z+ 2 −9i

(2 + 9i)z2−6(2 + i)z+ 18 + i,(1)

we have

M1(z) = 3z−1

3−z, M2(η) = 2η−i

2 + iη (2)

and Lis the real axis. We have M2(0) = −i/2,

M2(1) = 3/5 −4i/5 and M2(∞) = −2i. Thus,

w=R(z)conformally maps each one of the upper

and the lower half-planes onto the whole w-plane with

a slit alongside the arc of the circle determined by the

points −i/2,−2iand 3/5 −4i/5.

The pre-image by R(z)of the Steiner net, [5], il-

lustrates a conformal self-mapping of the complex

plane in which the upper and the lower half-planes

are each mapped onto themselves.

To find this pre-image we need an expression of

the multi-valued function R−1(w), which is M−1

1◦

T−1◦M−1

2. We have

M−1

1(ζ) = 3ζ+ 1

ζ+ 3 ,

M−1

2(w) = 2w+i

2−iw ,

T−1(η) = √η,

thus

R−1(w) =

3√2w+i

2−iw + 1

√2w+i

2−iw + 3

The two branches of T−1(η)provide conformal map-

pings of the complex plane onto itself, Fig. 1, in

which the upper, respectively lower half-plane are

mapped each one by the respective branch onto itself.

Due to Proposition 1, such a mapping can be ob-

tained for any second-degree rational function.

For higher degree rational functions different tech-

niques are needed.

However, for selected rational functions of higher-

degree a similar technique can be applied. For exam-

ple, for the function

R(z) = (54 + i)z3−9(6 + i)z2+ 9(2 + 3i)z−(2 + 27i)

(−2 + 27i)z3+ 9(2 −3i)z2−9(6 −i)z+ 54 −i,

we have three branches of the multi-valued function

R−1(w), namely,

R−1

k(w) =

3ωk3

√2w+i

2−iw + 1

ωk3

√2w+i

2−iw + 3

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Fig. 1: The conformal self-mapping of the complex

plane induced by a second-degree rational function.

k= 0,1,2,where ωkare the roots of order three of

the unity.

The function T(ζ) = ζ3maps conformally the

three sectors determined by the rays

ζ(t) = ωkt, t ≥0,

where ωk, k = 0,1,2, are the roots of order three of

the unity, onto the whole η-plane with a slit alongside

the positive real half-axis. The Möbius transforma-

tion w=M2(η)carries the real half-axis onto the slit

L′′ in the w-plane. On the other hand,

z=M−1

1(ζ) = 3ζ+ 1

ζ+ 3

carries the three rays into the curves

zk(t) = 3ωkt+ 1

ωkt+ 3 , t ≥0.

These curves are described in the next example, as

well as the domains bounded by them, which are fun-

damental domains for the function R(z). The confor-

mal mapping of each one of these domains onto itself

is described by the Ω-Steiner nets shown in Fig. 2

below.

The case of Blaschke products is special, since the

fundamental domains are circular and once we have

the mapping for one fundamental domain, the others

can be obtained by symmetries with respect to circles.

A second-degree rational function which is a

Blaschke product is a function of the form:

B(z) = z−a

1−az

z−b

1−bz eiθ,(3)

Fig. 2: The conformal self-mapping of the complex

plane induced by a third-degree rational function.

where 0≤ |a|<1,0≤ |b|<1, θ ∈R.

For illustration, let us take a= 1/3 and b=−1/3,

θ= 0.Then,

B(z) = 9z2−1

9−z2,(4)

with the fundamental domains the left and the right-

hand half-planes. The Möbius transformation

M(z) = 2z−1

2−z

with the fixed points −1and 1is illustrated by the

Steiner net in [5]. It induces a conformal mapping of

the complex plane onto itself in which the left and the

right hand half planes are mapped each one onto itself.

This transformation has four fixed points as seen in

the Fig. 3 below. Indeed, for any y∈R, we have

B(iy) = B(−iy) = −9y2+ 1

9 + y2,

which shows that B(z)maps the imaginary axis onto

the interval (−9,−1

9), symmetric points with respect

to the origin having the same image. Hence, each one

of the left and the right half-planes is mapped confor-

mally onto the whole complex plane with a slit along-

side the real axis from −9to −1

We dealt in [6] and [7] with the Blaschke products

of the form

Ba(z) = (a

|a|

z−a

1−az )n

, a =reiα,

0< r < 1, α ∈R, n = 2,3, ....

(5)

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Fig. 3: The conformal self-mapping of the complex

plane induced by a Blaschke product of the second-

degree.

An elementary computation shows that the equa-

tion Ba(z) = λn,where 0≤λ≤1has the solutions

zk(λ) = ωkλ+r

ωkλr + 1eiα, k = 0,1, ..., n −1,(6)

where ωkare the roots of order nof unity. The for-

mula (6) is that of a Möbius transformation in λand

therefore when λvaries from 0to 1, the point zk(λ)

describes an arc of a circle γkwith the end points in

zk(0) = aand zk(1) = ωk+r

ωkr+ 1eiα

on the unit circle. Two consecutive arcs γkand γk+1

(where γn=γ0) together with the arc of the unit

circle between them bound a domain which is con-

formally mapped by Ba(z)onto the unit disc with a

slit alongside the real axis from 0to 1. By the sym-

metry principle, the symmetric of this domain with

respect to the unit circle is conformally mapped by

Ba(z)onto the exterior of the unit disc with a slit

alongside the real axis from 1to infinity. The conclu-

sion is that Ba(z)partitions the complex plane into n

point sets, the interior of which are fundamental do-

mains of Ba(z)and each one of them is conformally

mapped by Ba(z)onto the whole complex plane with

a slit alongside the positive real half axis.

Fig. 4: The conformal self-mapping of the com-

plex plane induced by a Blaschke product of the third

degree.

Let us take a= 1/3 and n= 3. By using the

same procedure as previously, we obtain, Fig. 4, a

conformal self-mapping of the complex plane with 6

fixed points.

The boundaries of the fundamental domains of the

Blaschke product

B(z) = (3z−1

3−z)3

can be obtained by solving for zthe equation

B(z) = t3, t ≥0.

The solutions are

zk(t) = 1+3tωk

3 + tωk

, k = 0,1,2,0≤t≤ ∞,

which represent three arcs of circle γk, k = 0,1,2.

Since ω0= 1, we have

z0(t) = 1+3t

3 + t,

which is, for t≥0, the interval γ0= (1/3,3) of the

real axis. It meets the unit circle for t= 1 at the point

z= 1.

For k= 1,2, the two arcs meet the unit circle when

|zk(t)|= 1, i.e., when

(1 + 3tωk)(1 + 3tωk) = (3 + tωk)(3 + tωk).

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This equation is equivalent to t2= 1. Since t≥0,

the right solution is t= 1. We have

zk(1) = 1+3ωk

3 + ωk

thus γ1and γ2are arcs of circle with the ends in

1/3 and 3and passing through z1(1) and respectively

z2(1).

Thus, the fundamental domains of B(z)are the do-

mains: Ω1bounded by γ0and γ1,Ω2bounded by γ0

and γ2, and the unbounded domain Ω0bounded by γ1

and γ2.

Since B(zk(1)) = 1 and z= 1 is a fixed point of

the Möbius transformation

M(z) = 2z−1

2−z,

by the Proposition 2 the points zk(1) are fixed points

of the transformations

χk(z) = (B−1

|Ωk◦M◦B)(z), k = 0,1,2,

as shown in Fig. 4.

3 The Case of an Arbitrary Analytic

Function

Let f(z)be an arbitrary analytic function in Cwith the

exception of isolated singular points and let Ω1and Ω2

be adjacent fundamental domains of fwhich are con-

formally mapped by fonto the complex plane with

the same slit L. Any Möbius transformation M(z)

defines conformal mappings

χk(s) = f−1

|Ωk◦M◦f(s)

of Ωk\L′

konto Ωk\Lk, k = 1,2, where Lkand L′

are slits in Ωk. Let Γ = ∂Ω1∩∂Ω2and let us extend

by continuity every f|Ωk(s)to Γkeeping the same no-

tations for the extended functions.

Theorem 1. For every s∈Γwe have

f|Ω1(s) = f|Ω2(s) = f(s),

hence

f|Ω1(Γ) = f|Ω2(Γ) = f(Γ).

Moreover, f−1

|Ω1(w)and f−1

|Ω2(w)exist for every w∈

f(Γ) and they are equal.

Then, χk(s)are extended to Γ, such that χ1(s) =

χ2(s)for every s∈Γ.The extended function to (Ω1∪

Ω2∪Γ)\(L′

1∪L′

2)is a conformal mapping χ1,2of this

domain onto (Ω1∪Ω2∪Γ)\(L1∪L2)such that its

restriction to every Ωk\L′

k, k = 1,2is a conformal

mapping of Ωk\L′

konto Ωk\Lk.

Fig. 5: Illustration of Theorem 1 for an arbitrary

Dirichlet function ζA,Λ(s)and a real Möbius trans-

formation M.

Proof: By the boundary correspondence theorem

in a conformal mapping, for k= 1,2, the functions

f|Ωkcan be extended by continuity to the boundaries

∂Ωkof Ωkand both extensions map ∂Ω1∩∂Ω2onto

the same edge of L. Indeed, the function fis locally

injective at every point s0where f′(s0)= 0 and s0is

not a multiple pole or an essential singular point. In

particular, this happens at every point s0∈Γ. There-

fore, in a small neighborhood Vof f(s0)the function

f−1(z)exists and it coincides with f−1

|Ωkwhere both

are defined.

Consequently, f−1

|Ωkcan be extended to f(Γ) where

they are equal, hence they are extensions of each

other and therefore χk(s)are extensions of each other.

The function χ1,2(s)which coincides with χk(s)in

(Ωk∪Γ)\L′

k, k = 1,2maps conformally(Ω1∪Ω2∪

Γ)\(L′

1∪L′

2)onto (Ω1∪Ω2∪Γ)\(L1∪L2)in such a

way that Ωk\L′

kis conformally mapped onto Ωk\Lk

for k= 1,2.

If it happens that L′

k⊂∂Ωk(as in [6]) then χk(s)

is defined in Ωk.

In Fig. 5, two adjacent fundamental domains Ω1

and Ω2with the common boundary Γ′

kof an arbitrary

Dirichlet function ζA,Λ(s)are exhibited. A big cir-

cle of radius rcentered at the origin is the image by

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ζA,Λ(s)of an infinite curve η1,while a small circle

of radius ϵcentered at the point z= 1 is the im-

age of another infinite curve η6and some arcs η′

3and

η′′

3around the points u′

0and u′′

0for which we have

ζA,Λ(u′

0) = ζA,Λ(u′′

0) = 1.The domain Dϵ,r bounded

by the two circles is the conformal image of the two

domains Ωϵ,r and Ω′

ϵ,r included in the fundamental

domains Ω1and respectively Ω2of the Dirichlet func-

tion ζA,Λ(s).When ϵ→0and r→+∞the two

domains become Ω1and respectively Ω2, while Dϵ,r

becomes the whole complex plane. We have

ζA,Λ(s′

0) = ζA,Λ(s′′

0) = 0,

ζA,Λ(u′

0) = ζA,Λ(u′′

0) = 1,

ζ′

A,Λ(s1) = ζ′

A,Λ(s′

1) = 0,

lim

σ→+∞ζA,Λ(σ+it) = 1,

where σ+it belongs to Γ′

kor to η5or to η′

5. The

four points s0, ζA,Λ(s0),M(ζA,Λ(s0)) and χ1(s0) =

χ2(s0), as well as their neighborhoods are portrayed.

This figure illustrates not only the affirmations of

Theorem 1, but also the way a Dirichlet function is

conformally mapping its fundamental domains onto

the complex plane with some slits. Every slit is along

the interval (1,+∞)of the real axis and along an in-

terval from z= 1 to z=ζA,Λ(sk)where ζ′

A,Λ(sk) =

Regarding the sensitivity of the method, it is ob-

vious that any change in a fundamental domain will

trigger changes in all fundamental domains, perturb-

ing the whole landscape of the Fig. 5. However, the

branch points will remain the same.

We need to point out that the famous Riemann Zeta

function is the particular Dirichlet function in which

λn=log nand an= 1 for every n. It is known that

the Riemann Zeta function has important applications

in physics. The results above may draw a new light

on this application.

Theorem 2. (The Main Theorem) To every func-

tion f, which is analytic in Cwith the exception of

isolated singular points, and to every Möbius trans-

formation M, a conformal mapping of the complex

plane, with the singular points of fand some cuts re-

moved, can be associated such that every fundamen-

tal domain of fwith a cut removed is conformally

mapped into itself.

Proof: It is known that C=∪n≤∞

k=1 Ωkwhere Ωk

are fundamental domains of f. Suppose that Ω1and

Ω2are adjacent and construct χ1,2as in Theorem 1.

Now, suppose that Ω3is adjacent to Ω1∪Ω2and let

χ1,2,3a conformal mapping of (Ω1∪Ω2∪Ω3)\(L′

1∪

L′

2∪L′

3)obtained in the same manner as χ1,2.It maps

conformally each one of Ωk\L′

k,k= 1,2,3onto

Ωk\Lk. We continue in this way up to n, if nis finite,

or indefinitely if nis infinite. What we obtain is a con-

formal mapping of the complex plane with the singu-

lar points of fand some cuts removed in which every

fundamental domain of fis conformally mapped into

itself. If L′

k⊂∂Ωkthere is no cut to be removed from

Ωk.

We illustrate next this theorem for some of the

most known classes of analytic functions.

4 Illustrations For Some Classes of

Analytic Functions

The exponential function has infinitely many funda-

mental domains which are horizontal strips of width

2π. We have shown in [5] how each one of these strips

can be conformally mapped onto itself through the in-

termediate of a Möbius transformation. We will use

throughout in what follows the Möbius transforma-

tion

M(z) = 2z−1

2−z.(7)

Let

Ωk={z=x+iy|2kπ < y < 2(k+ 1)π},

for k∈Z. The Ω0-Steiner net in [5], portrays the

conformal self-mapping χMof the strip Ω0defined

χM(s) = Log(M(es)),

where Log is the principal branch of the multival-

ued function logarithm. To obtain a conformal self-

mapping of an arbitrary fundamental domain of ezwe

need to use the corresponding branch of the logarithm.

Yet, this can be obtained by making consecutively

symmetries of the Ω0-Steiner net with respect to the

lines z=x+ 2kπi. Fig. 6 below illustrates a confor-

mal self-mapping of the complex plane in which ev-

ery strip Ωkis mapped onto itself. We notice that this

conformal mapping has infinitely many fixed points,

namely z=kπi. For keven they are repelling fixed

points and for kodd they are attracting.

For the cosine function the fundamental domains

are vertical strips

Ωk={z=x+iy|kπ < x < (k+ 1)π}, k ∈Z.

The Ω0-Steiner net is shown in [5].

If we take the symmetric of this net with respect

to the lines z=kπ +iy we obtain a net covering the

whole complex plane and portraying a conformal self-

mapping of the complex plane with the fixed points

z=kπ,k∈Z, in which every fundamental domain

Ωkof the cosine function is conformally mapped onto

itself. Fig. 7 illustrates this mapping.

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Fig. 6: The conformal self-mapping of the com-

plex plane by the function χ(s)which coincides with

χM(s)in every fundamental domain of the exponen-

tial function.

Fig. 7: The conformal self-mapping of the com-

plex plane by the function χ(s)which coincides with

χM(s)in every fundamental domain of the cosine

function.

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 by the formula

Γ(z) = ∫∞

e−ttz−1dt. (8)

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 func-

tion Γ(z)has no zero and it has infinitely many poles

which are z= 0 and z=−n,n∈N. The pre-image

of the real axis by Γ(z)displays the fundamental do-

mains of this function. These domains are confor-

mally mapped by Γ(z)onto the complex plane with

slits alongside some intervals of the real axis. Fig. 8

below represents the pre-image by Γof the Steiner net

of the Möbius transformation (7) which is a collection

of Ω-Steiner nets, each one illustrating a conformal

self-mapping of the respective domain Ω. These do-

mains are bounded by the curves colored red for the

pre-image of the positive real half-axis and blue for

the pre-image of the negative real half-axis and they

are mapped conformally by the function w= Γ(z)

onto the complex plane with slits alongside some in-

tervals on the real axis. Each one of these domains

contains one fixed point in the interior, which is re-

pelling, and two attracting fixed points on the bound-

ary. Both, attracting and repelling fixed points are at

the intersection of the pre-images by Γ(z)of the real

axis in the wand of the unit circle (colored green).

These domains, put together, illustrate a conformal

self-mapping of the complex plane in which every

fundamental domain is mapped onto itself.

A Dirichlet function is obtained by performing an-

alytic continuation to the whole complex plane of a

Dirichlet series

ζA,Λ(s) =

∞

∑

n=1

ane−λns,(9)

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. Any Dirichlet series can be normalized such that

a1= 1 and λ1= 0 and we deal only with normalized

Dirichlet series. For such a series we have

lim

σ→+∞ζA,Λ(σ+it) = 1,(10)

uniformly with respect to t. When, for anwe have the

values of a Dirichlet character χand λn=log n, [5],

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DOI: 10.37394/23206.2023.22.106

Andrei-Florin Albişoru, Dorin Ghişa

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Fig. 8: The conformal self-mapping of the complex

plane by the function χwhich coincides with χM(s)

in every fundamental domain of the function Γ(z).

then the series (9) becomes

L(χ, s) =

∞

∑

n=1

χ(n)

ns(11)

and it is called Dirichlet L-series.

The fundamental domains of Dirichlet functions

have been studied in [8]. They are bounded by the

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

val (1,+∞)of the real axis and by pre-images γkof

the segments from z= 1 to z=ζA,Λ(sk),where

ζ′

A,Λ(sk) = 0.For the Dirichlet L-functions, the loca-

tion of these zeroes of the derivative has been studied

in [9].

For any Dirichlet function (9) the fundamental do-

mains are infinite strips which are mapped confor-

mally by ζA,Λ(s)onto the whole complex plane with

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

followed by a slit alongside the segment from z= 1

to z=ζA,Λ(sk).When ζA,Λ(sk)=0,the point sk

is a double zero of ζA,Λ(s)and the corresponding slit

is the positive real half-axis. The existence of double

zeroes of Dirichlet functions has been proved in [10]

for linear combinations Dirichlet L-functions satisfy-

ing the same Riemann type of functional equation.

We have also shown in [11] that ζA,Λ(s)cannot

have any zero of a higher order than two.

Since the interval (1,∞)of the real axis is in-

cluded in the slit of every fundamental domain Ωkof

ζA,Λ(s),the pre-images by ζA,Λ(s)of the Apollonius

circles around z= 1, of the Steiner net of the Möbius

transformation (7), are orthogonal to the pre-image

of that interval. Some of them may cut also under

different angles the curve γk.The components of the

Fig. 9: The conformal self-mapping of the com-

plex plane by the function χwhich coincides with

χM(s)in every fundamental domain of the Dirichlet

L-function L(7,2, s).

pre-image of the orthogonal circles to the Apollonius

circles, which pass through −1and 1, can be divided

into two categories: the unbounded ones, which ap-

proach asymptotically ∂Ωkwhen σ→+∞and the

bounded ones. Each one of them is mapped onto it-

self by the corresponding function χM(s).By Theo-

rem 2, there is a conformal mapping χ(s)of the com-

plex plane which coincides with χM(s)in every fun-

damental domain Ωkof ζA,Λ(s). Fig. 9 illustrates this

affirmation when ζA,Λ(s)is L(7,2, s).

Fig. 9 and Fig. 5 need to be seen together for a bet-

ter understanding of the conformal self-mapping of

the complex plane generated in Fig. 9 by the Dirichlet

L-function L(7,2, s)and the Möbius transformation

M(z) = 2z−1

2−z.

While Fig. 5 has been conceived by imagination, Fig.

9 is a computer-generated graphic. It shows that the

fundamental domains of L(7,2, s)are indeed those

illustrated in Fig. 5 and that their conformal self-

mappings given by

χM(s) = L−1

|Ω◦M◦L(7,2, s)

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DOI: 10.37394/23206.2023.22.106

Andrei-Florin Albişoru, Dorin Ghişa

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978

Volume 22, 2023

in every fundamental domain, Ωof L(7,2, s), pro-

duce together a conformal self-mapping of the com-

plex plane with some slits. The pre-image by

L(7,2, s)of the orthogonal circles to the Apollonius

circles of M(s)shown in Fig. 9 prove that the η-

curves from Fig. 5 are real. If we could add to Fig. 5

the fixed points of χ(s)and the Ω-Apollonius circles,

then the complete description of the conformal self-

mapping of the complex plane with slits would result

also in the case of the function illustrated by Fig. 5.

5 Conclusions

Up to now, the only known conformal self-mappings

of the complex plane were the Möbius transforma-

tions. Moreover, it has been proved, [12], that these

are the only possible such transformations. In the

previous paper, [5], we dealt with conformal self-

mappings of the fundamental domains of analytic

functions. In this paper, we succeeded to extend this

idea to the whole complex plane. We have proved

that to any analytic function fin Cwith the excep-

tion of isolated singular points and to any Möbius

transformation, a conformal self-mapping of the com-

plex plane with some slits can be associated, such that

it maps conformally every fundamental domain of f

onto itself. Computer experimentation has been used

to illustrate this result for the most familiar classes of

analytic functions.

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,

Progress in Analysis and Applications,

Proceedings of the 7th ISAAC Congress, World

Scientific Publishing Co. 2010, pp. 3-12.

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

Series in Pure and Applied Mathematics, 1979.

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

Series in pure and Applied Mathematics, 1951.

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

Theory in One and Higher Dimensions, Marcel

Decker Inc., New York, Basel, 2003.

[5] Albişoru, A.F., and Ghisa, D.. Conformal

Self-mappings of the Fundamental Domains of

Analytic Functions and Computer

Experimentation, WSEAS Transactions on

Mathematics, Volume 22, 2023, pp. 652-665.

[6] Ballantine, C., and Ghisa, D., Global Mapping

Properties of Rational Functions, Progress in

Analysis and its Applications, Michael Ruzansky

and Jens Wirth edits., London, UK, 2010, pp.

13-22.

[7] Ghisa, D., Fundamental Domains and the

Riemann Hypothesis, Lambert Academic

Publishing, Saarbrücken, 2012.

[8] Ghisa, D., Fundamental Domains of Dirichlet

Functions, in Geometry, Integrability and

Quantization, Mladenov, Pulovand, and

Yoshioka edits., Sofia, 2019, pp. 131-160.

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

Location of the Zeros of the Derivative of

Dirichlet L-functions, Annals of the University

of Bucharest, No. 5 (LXIII), 2014, pp. 21-31.

[10] Cao-Huu, T., Ghisa, D., and Muscutar, F. A.,

Multiple Solutions of the Riemann Type of

Functional Equations, British Journal of

Mathematics and Computer Science, No. 17 (3),

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[11] Ghisa, D., The Geometry of Mappings by

General Dirichlet Series, Advances in Pure

Mathematics, 7, 2017, pp. 1-20.

[12] Nevanlinna, R., Analytic Functions,

Springer-Verlag, Berlin, Heidelberg, New York,

1970.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.106

Andrei-Florin Albişoru, Dorin Ghişa

E-ISSN: 2224-2880

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Volume 22, 2023