The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains

The Cauchy integral formula expresses the value of a function ( ) f z , which is analytic in a simply connected domain D, at any point 0 z interior to a simple closed contour C situated in D in terms of the values of ( ) 0 f z z z − on C. We deal in this paper with the question whether C can be the boundary ∂Ω of a fundamental domain Ω of ( ) f z . At the first look the answer appears to be negative since ∂Ω contains singular points of the function and it can be unbounded. However, the extension of Cauchy integral formula to some of these unbounded curves, respectively arcs ending in singular points of ( ) f z is possible due to the fact that they can be obtained at the limit as r → ∞ of some bounded curves contained in the pre-image of the circle z r = and of some circles 1 z a r − = for which the formula is valid.


Introduction
We make reference to [1] for elementary knowledge in complex analysis used below. It is known (see [2]) that for every rational function ( ) R z of degree n the complex plane can be partitioned into n sets whose interior are fundamental domains of ( ) R z , i.e. they are mapped conformally (hence bijectively) by ( ) R z onto the whole complex plane with some slits. A similar partition takes place for transcendental functions (see [3]), except that for those functions the Although integrals on unbounded contours have been used frequently in complex analysis (see [1], page 214), they have never appeared in the context of Cauchy integral formula. The main novelty of this paper is that it makes possible such an undertaking. The famous Cauchy integral formula is in this way upgraded from a rather local instrument to a more global one. Moreover, it shows that the functions we are studying are completely determined by the values on the boundaries of their fundamental domains.
The integral on ∂Ω of ( ) namely they consist in isolating the singular points and z = ∞ by the pre-image of some circles whose radii are let tend to zero, respectively to infinity, then in applying the Cauchy integral formula to the bounded sub-domains of Ω obtained in this way and making sure that the integrals on the boundaries of the complementary domains tend to zero when the radii tend to zero or to infinity.
As ( ) The contours we used for integration needed to be illustrated and most of the graphics are computer generated by the software Mathematica. When this was not possible, we used illustration by hand made drawings. However, they are pictures of known fundamental domains (see [1], page 268 and 282). One of the most studied classes of meromorphic functions is that of Dirichlet functions and it can be considered as a prototype in many aspects. Let us start then with this class.

General Properties of Dirichlet Functions
The Dirichlet functions are obtained by analytic continuation of general Dirichlet series across the line of convergence. The family of general Dirichlet series includes that of well known Dirichlet L-series defined by Dirichlet characters. These last series can be all extended as meromorphic functions in the whole complex plane. The extended functions are called Dirichlet L-functions. They are implemented in Mathematica and some affirmations about general Dirichlet functions are illustrated by using Dirichlet L-functions. However, the interest in more general functions is obvious and we have recently devoted to them a lot of publications (see [2]- [15]). An account of recent advances in this field can be found in [8].
A to a meromorphic function in the whole complex plane. We keep the notation ( ) , A s ζ Λ for the extended function when it exists and we call it Dirichlet function. Following Speiser [16], who studied the Riemann Zeta function, we have used in [2]- [15] the pre-image of the real axis by takes real values. For every Dirichlet function it is a family of analytic curves whose structure has very profound implications on the value distribution of that function. Figure 1(a) illustrates the pre-image of the real axis by a Dirichlet L-function defined by a complex Dirichlet character and Figure  1(b) by a real one. Details about Figure 1(c) are found in Section 3.
We have proved (see for example [8]) that for any Dirichlet function  can be an unbounded curve On the other hand the origin of such a curve must be a point are embraced curves (see [8]) and when 0 k = .
The curve  when is the case, as in Figure 2). These are strips unbounded to the right and to the left when   boundaries of fundamental domains bounded to the right. It is known (see, for example [13]) that every k S -strip, can be partitioned into a finite number of sets whose interior are fundamental domains of ( ) S -strip contains infinitely many fundamental domains. The way they are mapped conformally onto the complex plane with some slits by the Riemann Zeta function is illustrated in Figure 2 (see [13], Figure 6).

Cauchy Integral Formula for Fundamental Domains and S k -Strips of the Function
The Cauchy integral formula has the form: where the function Proof: Let us take ′ <   . Then the pre-image of the circle intersected with Ω is formed with two arcs 3 η′ inside 3 η and 6 η′ at the right of 6 η . The arcs 6 6 , η η′ and the curves curvilinear quadrilateral whose conformal module is the same as that of the quadrilateral determined by 6 6 , γ γ′ , the real axis and the segment from , which in turn is less than the conformal module of the ring domain It is known (see [17], page 31) that the value of this last then this module is log 2 , which shows that the length of 6 η remains bounded as 0 →  , since otherwise the respective module would tend to ∞ , contrary to the fact that it remains Although the integrand tends to zero as r → ∞ , a limitation of the initial integral is problematic, due to the factor 2 r in the last term. So, as long as we cannot make sure that  and then at the limit as 0 →  this equality becomes (4) and the theorem is proved.
We notice that this theorem says that the values of ( ) We notice that is not injective in k S , hence the integrals (4) and (5) give us the same value for different points s in k S . If we would like to have a unique point corresponding to a given value, then we need to use the formula (3).
Also, taking into account the fact that the domain interior to every curve If the equation of the curve

The Distribution of the Values of a Dirichlet Function
The contour of integration in Theorem 2 is simpler than that appearing in Theorem 1. However, (3) has the advantage of representing a univalent function

Extension of Cauchy Integral Formula for the Derivatives of Dirichlet Functions
Following the known technique of computing for every natural number n.
It is known (see [4] and  . The same procedure can be applied to derivatives of any higher order and the theorem is completely proved. Figure 5 portraying the pre-image of the real axis by the Riemann Zeta function and by its derivative shows that their k S -strips and their fundamental domains overlap, but they do not completely coincide (see [11]). However, the  integral (8) gives the same value for is zero, by Cauchy Theorem. The same is true for the integrals on the boundaries of the corresponding fundamental domains of the two functions.

Extension of Cauchy Integral Formula to Fundamental Domains of Modular Function
By the Riemann mapping theorem there is a unique analytic function    This theorem tells us that the modular function is completely determined by its real values and by the values on the pre-image of an arbitrary big circle centered at the origin.

Extension of Cauchy Integral Formula to the Fundamental Domains of the Exponential Function
D. Ghisa