The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains ()
1. Introduction
We make reference to [1] for elementary knowledge in complex analysis used below. It is known (see [2]) that for every rational function
of degree n the complex plane can be partitioned into n sets whose interior are fundamental domains of
, i.e. they are mapped conformally (hence bijectively) by
onto the whole complex plane with some slits. A similar partition takes place for transcendental functions (see [3]), except that for those functions the number of fundamental domains is infinite. Every fundamental domain
of an analytic function
is either unbounded or
contains singular points of
, or both.
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
of
shall be treated as an improper integral the convergence of what remains to be investigated. This can be accomplished in different ways which apply to particular classes of functions; hence instead of trying to prove theorems valid for any analytic function, we must treat separately those classes of functions. However, the techniques used are in general similar; namely they consist in isolating the singular points and
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
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
is injective in every fundamental domain, if such a domain is mapped conformally by the function onto the complex plane with a slit; then for some values
there is a function
corresponding to that fundamental domain such that
,
. The function
is injective in the interval
and maps this interval onto an arc
included in that fundamental domain. Making the change of variable
in the integral
it becomes an integral on the interval
and it is possible that it tends to zero as
or
. This assertion should be checked for every particular class of functions.
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.
2. 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 general Dirichlet series
is defined by an arbitrary sequence of complex numbers
, the coefficients of the series and by a non decreasing sequence of non negative numbers
, the exponents of the series. It is given by the formula
(1)
We will deal only with normalized general Dirichlet series in which
and
. For such a series we have
uniformly with respect to t (see [8], Theorem 3). There is a number
, called the abscissa of convergence of
,
such that the series (1) converges for
and it diverges for
. The series converges uniformly on compact subsets of
and therefore it is an analytic function in that half plane. Denoting by
we have proved in [8] that if the abscissa of convergence of
is finite then the abscissa of convergence of
is zero and if
has only isolated singular points on
, then
can be continued across the line
to a meromorphic function in the whole complex plane. We keep the notation
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
. This is the set of points in the s-plane where
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
this pre-image is formed with unbounded curves (components) which fall into three categories. Namely, there are infinitely many curves
, which do not intersect each other and consecutive
and
(
below
) form infinite strips
extending for
going from
to
. The counting is such that
. Every curve
is mapped homeomorphycally by
onto the interval
of the real axis and therefore every
-strip is mapped (not necessarily one to one) onto the whole complex plane with a slit alongside this interval. For
every strip
contains a unique component
of the pre-image of the real axis which is mapped homeomorphycally by
onto the interval
of the real axis and a finite number of components
which are mapped each one homeomorphycally by
onto the whole real axis. The component
![]()
Figure 1. The pre-image of the real axis by Dirichlet L-functions.
extends for
going from
to
, while
are parabola shaped curves with a finite supremum of
, therefore we can distinguish the interior and the exterior of such a curve.
In the case of a strip
, if
then connecting
with
by a Jordan arc
the component of the pre-image of
passing through
can be an unbounded curve
, when for
we have
. On the other hand the origin of such a curve must be a point
on a curve
such that
. The curve
is bounded when its ends belong to different curves
and
. This is the case when
and
are embraced curves (see [8]) and when
. The curve
is mapped 2 to 1 by
onto
. Then we can form fundamental domains using parts of the curves
, the curves
(and
, when is the case, as in Figure 2). These are strips unbounded to the right and to the left when
is unbounded and they are bounded to the right when
is bounded. They are mapped conformally by
onto the whole complex plane with some slits alongside the interval
of real axis and some other slits alongside
.
In the case of the strip
, when the zeros of
are complex, the curves
are all bounded for
and together with
they form the
![]()
Figure 2. Conformal mapping of fundamental domains by
.
boundaries of fundamental domains bounded to the right. It is known (see, for example [13]) that every
-strip,
of
can be partitioned into a finite number of sets whose interior are fundamental domains of
. The
-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).
3. Cauchy Integral Formula for Fundamental Domains and Sk-Strips of the Function
The Cauchy integral formula has the form:
(2)
where the function
is analytic in a simply connected domain D containing the simple closed contour C and
is an arbitrary point inside C.
We would like
to be a Dirichlet function
and C to be the boundary
of a fundamental domain
of
or the boundary
of an
-strip. The problem is that
and
are not simple closed contours. However, we can show that the formula (2) can be extended to these curves.
The shape of the fundamental domains of
depends on the pre-image of the real axis and on the zeros of
. Since
is injective in every fundamental domain the zeros of
must be located on the boundaries of these domains. Figure 3 portrays a fundamental domain
of
bounded by a curve
, the part of the last curve
from
on which
vary from
to
, where we have
, as well as the pre-image of the segment determined by
and
where
is the zero of
the closest to
. The pre-image
of the circle
and the pre-image
of the circle
are also drawn, where r is big enough and
is small enough. Figure 1(c) illustrates computer generated pre-images of these circles for
(the orange curve)
![]()
Figure 3. A fundamental domain of
and its conformal mapping.
and
(the green curve). It has been worked by Florin Alan Muscutar. Due to the continuity of
at
and to the fact that
is a normalized Dirichlet series, the arc
squeezes to the point
and
with s on
tends to
as
. Also
with s on
tends to
as
.
The domain
is mapped conformally by
onto the complex plane with a slit alongside the subinterval
of the real axis and alongside the segment determined by
and
. Also, the domain
is mapped conformally onto the ring domain
determined by the two circles with the corresponding slit (see Figure 3). The function
is analytic in a domain containing
and therefore the Cauchy integral formula is valid for
.
Theorem 1. If we denote by
the infinite strip obtained from
as
, then for every
we have
(3)
Proof: Let us take
. Then the pre-image of the circle
intersected with
is formed with two arcs
inside
and
at the right of
. The arcs
and the curves
and
determine a curvilinear quadrilateral whose conformal module is the same as that of the quadrilateral determined by
, the real axis and the segment from
to
, 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 module is
. If we take
then this module is
, which shows that the length of
remains bounded as
, since otherwise the respective module would tend to
, contrary to the fact that it remains constant
. Let us evaluate
. Since
we have that
and since the length of
remains bounded we have
.
On the other hand, by Cauchy theorem
does not depend on
since
. Then we can let
in
and we obtain (3).
It is not clear what happens with
as
. Making the change of variable
, where
, which is allowed since
is injective on
, we get
.
Although the integrand tends to zero as
, a limitation of the initial integral is problematic, due to the factor
in the last term. So, as long as we cannot make sure that
tends to zero as
, the problem of extending the Cauchy integral formula to the whole fundamental domain remains unsolved.
Theorem 2. Let
be an arbitrary strip of the function
as defined in Section 2 and for r big enough let
be the part of the pre-image by
of the circle
included in
. We denote by
the part of the strip
bounded at the left by
. Then for every
we have:
(4)
Proof: For an arbitrary
and for the given r, let us build the domains
as in Theorem 1 corresponding to every fundamental domain
. The sum of the corresponding arcs
from these domains is
and the sum of the corresponding arcs
is an arc
connecting
and
. The arcs
squeeze each one to the respective points
when
, where
.
Since for each arc
we have
it results that
. Denoting by
the domain bounded by
and the arcs
we see that this domain becomes
when
. On the other hand, the Cauchy integral formula is applicable to
, i.e. for every
we have
and then at the limit as
this equality becomes (4) and the theorem is proved.
We notice that this theorem says that the values of
are completely determined by its values greater than 1 and those taken on an arbitrary circle of radius big enough. Moreover, the integral of the formula (4) is always convergent.
If
is the parametric equation of
such that
then the formula (4) becomes
(5)
We notice that
, hence the first integral in (5) is an improper integral. By Theorem 1 this integral is always convergent.
The function
is not injective in
, hence the integrals (4) and (5) give us the same value for different points s in
. 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
, which is not embracing curve, is mapped conformally and therefore injectively by
onto the upper or the lower half plane, a unique point s from that domain corresponds to every given value from the respective half plane. Then the following formula is true for every r big enough:
(6)
where
is the boundary of the domain bounded by
and the pre-image of the circle
.
If the equation of the curve
is
such that
then the formula (6) becomes
(7)
where
.
4. 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 in
. The pre-image of the circle
intersects several components of the pre-image of the interval
of the real axis situated in
, more exactly if
has m zeros in
, then this pre-image intersects exactly
such components, hence it traverses m fundamental domains. Each one of these domains contains a unique point
such that
. A point on the circle
should turn m times around the origin for its pre-image to traverse the m fundamental domains going from
to
. At every turn it assigns a unique point
where
takes the same value. Since
is univalent in
that value is completely determined by the values on
and
.
5. Extension of Cauchy Integral Formula for the Derivatives of Dirichlet Functions
Following the known technique of computing ![]()
we find that
(8)
Thus, as for
the values of
are completely determined in every strip
by the real values greater than 1 of
. By recursive computation we find that:
(9)
for every natural number n.
It is known (see [4] and Figure 4) that if a Dirichlet L-function
has m zeros in the strip
then
has
zeros in
(which are all simple zeros).
This figure illustrates the following:
Theorem 3. If
has m fundamental domains in
then every derivative
has exactly
zeros in
.
Proof: The pre-image by
of a circle
has m components in
which are disjoint if r is small enough. By letting r increase these components expand and after two of them touch each other, they fuse into a unique component including the corresponding zeros. When
a unique component becomes unbounded with branches tending to
with
. The remaining bounded components are outside this unbounded one and none of them can intersect
. Increasing r past 1 the unbounded components from all the strips
fuse into a unique unbounded component crossing these strips. After touching bounded components of the pre-image of the circle
with
(if such components exist in a given
), these last components are absorbed into the unbounded one and the corresponding zeros of
pass
![]()
Figure 4. The zeros of
and those of its first two derivatives for
.
to the right of it. For r big enough all the zeros of
from the strip
will be at the right of the unbounded component of the pre-image of the circle
. That value of r depends on
.
The points where two components of the pre-image of a circle
touch each other are the zeros of
. A complete binary tree can be formed having as leaves the zeros of
and as internal nodes these touching points. It is known from the graph theory that if a complete binary tree has m leaves then it has exactly
internal nodes. This proves that
has
zeros in
. The zeros of the second derivative are obtained in a similar way, yet since
, there will be m components of the pre-image by
of the circle
for r small enough, even if there are only
zeros of
. One of these components contains no such a zero. These components touch each other at the zeros of
and by the previous analysis there should be
zeros of
. 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
-strips and their fundamental domains overlap, but they do not completely coincide (see [11]). However, the
![]()
Figure 5. The pre-image of the real axis by
and
.
integral (8) gives the same value for
as
(10)
since if
is the corresponding strip of
the integral on
is zero, by Cauchy Theorem. The same is true for the integrals on the boundaries of the corresponding fundamental domains of the two functions.
6. Extension of Cauchy Integral Formula to Fundamental Domains of Modular Function
By the Riemann mapping theorem there is a unique analytic function
mapping conformally the domain D bounded by the half lines
,
,
and the half circle
onto the upper half plane
such that
,
and
. The function
can be continued by symmetry into the upper half plane as in Figure 6.
The symmetric domain
of D with respect to the imaginary axis is mapped conformally by
onto the lower half plane and
is mapped conformally onto the whole complex plane with the slit
, alongside the real axis, hence
is a fundamental domain of
. In fact the union any two adjacent domains bounded by the half circles above (the half lines can be considered half circles too!) and the common half circle is a fundamental domain of
. For example
is mapped conformally by
onto the whole complex plane with the slit
and
is mapped conformally by
onto the
![]()
Figure 6. Continuation by symmetry of the modular function.
whole complex plane with the slit
etc. So, Figure 6 exhibits a partition of the upper half plane into fundamental domains of
. Each one of these half circles ends up into a singular point of
, therefore these singular points form a dense subset of the real axis, which implies that
cannot be continued across the real axis and its full domain of definition is the upper half plane.
The way the function
has been constructed implies that for every s in the upper half plane we have
and
and then (see [1], page 280)
uniformly with respect to
. Hence the part of the pre-image by
of the circle
included in
is an arc
:
,
, where
uniformly with respect to
. In particular, denoting by
the inverse of
in
, we have
as
.
On the other hand, the part of the pre-image by
of the circle
included in
is formed for r big enough with two arcs
and
such that for
we have
as
and for
we have
as
. Let us denote by
the part of
obtained by removing the pre-image of the set
.
Theorem 4. For any fundamental domain
of the function
and any point
we have:
(11)
Proof: Let us deal first with the fundamental domain
. By isolating the point
with a small circle
and the point
with a big circle
and by removing from
the pre-image of the disc
and of the exterior of the disc
we obtain a bounded domain
for which the Cauchy integral formula is applicable: for any
we have
. Taking
and having in view the Cauchy theorem we conclude that
and since
, we can set
and get (11).
Now, if we take for example
and proceed similarly, the new
is the part of the pre-image of the circle
included in D, hence again.
and the extension of the Cauchy integral formula is true for the new unbounded domain
. We are brought to the same conclusion when
.
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.
7. Extension of Cauchy Integral Formula to the Fundamental Domains of the Exponential Function
It is known that the horizontal strips bounded by consecutive lines
are fundamental domains
of the exponential function
. The function
maps conformally each one of these strips onto the complex plane with the slit alongside the positive real half axis. The pre-image by
of the circle
is the vertical line
and if we denote by
the intersection of this line with any fundamental domain of
then
and at the limit as
we get an indetermination of the form
. Thus
might be divergent. However, we can prove:
Theorem 5. For any fundamental domain
of
and for any positive number r, if we denote
we have
(12)
where
is an arbitrary point of
.
Proof: The intersection of the pre-image by
of the annulus
and
is a bounded domain
and for every
we have
(13)
In order to obtain (12) it will be enough to show that
, where
is
. This integral is
and since the integrand is bounded and
we obtain at the limit the formula (12).
8. The Case of Trigonometric Functions
We illustrate this case by dealing with the function
(see [9], page 51). for which the fundamental domains are vertical half strips
determined by the lines
,
and
, symmetric of
with respect to the real axis. They are mapped conformally by the function
onto the complex plane with a slit alongside the interval
. For
, the pre-image of the circle
is the curve of equation
. An elementary computation shows that this equation is equivalent to
. This show that
in Ωj and
in
and
when
, respectively
when
. Hence the pre-image of the circle
is formed with two sinusoidal curves symmetric with respect to the real axis. Moreover,
in
and
in
uniformly with respect to x when
is on either one of these curves. Let us denote by
the part of this pre-image situated in
. We would like to evaluate
. For every domain
there is an analytic function
such that
. Then making the change of variable
in this integral we get
.
Let us notice that
and we need to chose the sine minus in the last term since if to
corresponds
then to
corresponds
, hence
is an odd function of
.
By making the change of variable
, we get
(14)
When r is big enough the term
varies very little as
varies from
to
. Assuming a constant instead of this term, what it remains to integrate between
and
is an odd function, hence the integral is zero. This doesn’t necessarily mean that
. However, the previous remark justifies the conjecture that this limit is true and therefore the Cauchy integral formula extends to the boundaries of the fundamental domains of the function
.
9. Extension of Cauchy Integral Formula to the Fundamental Domains of the Weierstrass
Function
The Weierstrass
function is defined (see [1], page 272) by the formula
(15)
where the sum ranges over all
, where
with
and
arbitrary complex numbers having non real ratio
. It is known that
is a doubly periodic function with the periods
and
. Hence it is sufficient to know the values of
into the (fundamental) parallelogram determined by
and
in order to be able to find its values anywhere in the complex plane. The series (15) converges uniformly and absolutely on any compact subset of
which does not contain points
, therefore it is a meromorphic function in the complex plane. The points
are double poles for
and hence they are triple poles for
.
It can be easily shown that
. Moreover, since
, by denoting
, we have
, thus z and
are symmetric with respect to the center of the fundamental parallelogram, hence if we know the values of
in one of the triangles determined by a diagonal of the parallelogram, then we know its values in the whole parallelogram. Also,
takes the same value at points symmetric with respect to the middle of each one of the sides of this triangle. Since the function is univalent in the triangle and maps each side two to one onto some curve originating in the image of the middle of the respective side and going to infinity (the ends of each side being poles) we conclude that these triangles are fundamental domains of
. Let us denote
,
and
. Then
maps conformally every fundamental triangle onto the whole complex plane with infinite slits originating at
and
. Figure 7 illustrates this situation. It shows also that each one of the domains
is mapped conformally onto the corresponding domain
with two sides of
going onto the slits. From the
![]()
Figure 7. Fundamental triangle of
and its conformal mapping.
differential equation of
(see [1], page 278) is obvious that
, which implies that the triangle with vertices
contains the origin, hence
has a zero in the domain
and, obviously another one in the symmetric domain with respect to
(These might have been unknown facts until now!).
Theorem 6. The Cauchy integral formula can be extended to any fundamental domain of the Weierstrass
function .
Proof: For r big enough the circle
is intersecting every slit of
and the pre-image by
of the domain
is formed with infinitely many connected open sets covering each one a vertex of the period parallelograms. The function
is analytic on the complementary set of this pre-image. In particular, the Cauchy integral formula is applicable to any fundamental triangle from which that pre-image has been removed. If we denote by
such a set, then we have:
(16)
for every
. Due to the univalence of
in
, it has an inverse function
defined in the disc
. Thus,
. With the change of variable
, the integral on the part of
belonging to the pre-image of the circle
becomes
(17)
Since the points
are triple poles
the term
tends to infinity as fast as
when
therefore the integrand in (18) tends to zero as
. If we denote by
any fundamental domain of
, then this limit guarantees the absolute convergence of the improper integral
and the fact that for every
we have:
(18)
which represents the extension of Cauchy integral formula to the fundamental domains of the Weierstrass
function. It asserts that the function is completely determined by its values on the boundary of any fundamental triangle.
Theorem 7. For any fundamental domain
of
and for every point
the value of an arbitrary derivative
at
is given by the formula:
(19)
Proof: Since the integral (18) converges absolutely, we can differentiate term by term in (18) with respect to
as many times as we want and we obtain the first equality in (19). For the second equality we write (17) with
instead
and notice that the corresponding term in (17) still tends to zero as
. Then a formula similar to (18) is true with
replaced by
, hence the improper integral
converges absolutely and we obtain the second equality in (19).
10. An Integral Formula for the Weierstrass ζ-Function
Weierstrass denoted the antiderivative of
(which is defined up to an additive constant) by
. Therefore
and if we normalize it so that it is odd (see [1], page 273) we get
. The series converges absolutely and uniformly on every compact set which does not contain any period point
. We obtain
by integrating on any path that does not pass trough the poles the function
from
to z, where we can take
such that
. Then having in view (18) we can write:
(20)
11. Conclusion
The concept of fundamental domain, as defined by Ahlfors (see [1], page 99), is crucial in understanding the geometry of the mappings by analytic functions. We realized that the Cauchy integral formula can be extended to the boundary of such a domain. However, this extension cannot be performed for an arbitrary analytic function and the process requires specific treatment for specific classes of such functions. We selected in this paper classes of functions we thought to be the most representative. The selection is far from exhaustive and a lot of work remains to be done.
Acknowledgements
We thank Aneta Costin for her support with technical matters.