## Abstract

By using conformal mappings, it is possible to express the solution of certain boundary-value problems for the Laplace equation in terms of a single integral involving the given boundary data. We show that such explicit formulae can be used to obtain novel identity for special functions. A convenient tool for deriving this type of identity is the so-called *global relation*, which has appeared recently in a wide range of boundary-value problems. As a concrete application, we analyse the Neumann boundary-value problem for the Laplace equation in the exterior of the Hankel contour, which appears in the definition of both the gamma and the Riemann zeta functions. By using the explicit solution of this problem, we derive a number of novel identities involving the hypergeometric function. Also, we point out an interesting connection between the solution of the above Neumann boundary-value problem for a particular set of Neumann data and the Riemann hypothesis.

## 1. Introduction

Simple boundary-value problems for the Laplace equation in two dimensions can be solved by the powerful technique of conformal mappings. However, the usual implementation of the conformal mappings presented in most text books is limited because it applies only to the case that the given boundary data is piecewise constant. In §2, we present a technique that can be applied to the general case of arbitrary boundary data (see also [1]). By using this technique, we construct in §2 a solution of the Neumann boundary-value problem of the Laplace equation in the domain *D*, which is the exterior of the so-called Hankel contour *H* (figure 1) defined by
1.1where
A solution of the above Neumann boundary-value problem is expressed in theorem 2.2 in terms of a single integral involving the three functions {*g*_{+}(*r*),*g*(*θ*),*g*_{−}(*r*)} defining the Neumann data. Using this explicit solution, we derive in §3 four integral identities. A particular case of these four identities can be derived by employing the explicit solutions
1.2In the general case, these identities are derived using the so-called *global relation*. We recall that this relation plays a crucial role for the construction of the solution of a wide class of initial boundary-value problems, for both evolution and elliptic PDEs [2–17]. Here, instead of employing the global relation to solve the given boundary-value problem, having already constructed the solution via conformal mappings, we use the global relation to obtain the integral identities mentioned earlier.

Regarding the global relation, we note that in addition to its basic role for the analytical solution of a large class of PDEs [18–20], it has also been used in the following contexts: (i) it yields a novel non-local formulation of the classical problem of water waves [21–25], (ii) it provides a useful approach to Hele–Shaw type problems [26], and (iii) it gives rise to novel numerical techniques for elliptic PDEs in the interior of a convex polygon [27–32].

We also show that the Riemann hypothesis is valid if and only if the above Neumann boundary-value problem, where the functions {*g*_{+}(*r*),*g*(*θ*),*g*_{−}(*r*)} have certain concrete representations (see equations (2.17) and (2.19)), does not have a solution that is bounded as .

## 2. A Neumann boundary-value problem for the Laplace equation

The Dirichlet and Neumann boundary-value problems for the Laplace equation in the exterior of the Hankel contour can be solved via conformal mappings. In this respect, we note that the usual implementation of the conformal mappings presented in most text books fails here because it only applies to the case of piecewise constant Dirichlet or Neumann data. For the case of arbitrary Neumann data, the following result is useful.

### Proposition 2.1

*Let q*(*r*,*θ*) *satisfy a Neumann boundary-value problem for the Laplace equation. Let ω*(*z*) *be the conformal mapping of the relevant domain to the upper half of the complex ω-plane, where*
2.1*Then*,
2.2*where*
2.3*and q*_{y}(*ξ*,0) *can be computed in terms of the given Neumann data using the identity*
2.4

### Proof.

Observe that 2.5Indeed, the r.h.s. of this equation equals which equals the l.h.s. of (2.5).

The conformal mapping implies
Furthermore, using *z*=*r* e^{iθ}, we find
Hence,
and equation (2.5) becomes (2.4).

Equation (2.2) is the well-known Poisson formula for the Neumann problem. □

### Theorem 2.2

*Let the domain D be defined by
*2.6*Let q(r,θ) solve the following Neumann boundary-value problem for the Laplace equation in the domain D:*
2.7a
2.7b
and
2.7c
*where the functions g*_{±}*(r) and g(θ) have appropriate smoothness and decay. A solution of this boundary-value problem is given by*
2.8a*where R(ρ) is defined by
*2.8b*and principal-value integrals are assumed if needed.*

### Proof.

It is straightforward to verify that the function *ω*(*z*), defined by
2.9maps the domain *D* defined in (2.6) to the upper half complex *ω*-plane. The points {*a* e^{iα},*a* e^{−iα},*a*} are mapped to the points {−1,1,0}, respectively. Also,
2.10The condition *y*=0 implies either *θ*=±*α* or *r*=*a*. Hence,
Thus, equation (2.2) becomes
2.11awhere
2.11b

Using as well as the boundary conditions (2.7), equation (2.11a) becomes the following: 2.12

By employing the identity equation (2.12) becomes equation (2.8a). □

### Remark 2.3

(Asymptotics) The large *r* asymptotic of the solution *q*(*r*,*θ*) defined by (2.8a) is given by
2.13where the constants *S* and are defined as follows:
2.14and
2.15Indeed, each of the brackets of the r.h.s. of (2.8a) can be written in the form
where *T* is given, respectively, for each bracket by
For large *r*, we find the following estimate:
2.16

### Remark 2.4

(A particular example) Let *u*(*z*) be a complex valued function such that *z*^{s−1}*u*(*z*) is bounded for all *z* on the Hankel contour *H*, . Suppose that we choose the functions {*g*_{+}(*ρ*),*g*_{−}(*ρ*),*g*(*φ*)} as follows:
2.17Using these functions in the definition (2.14) of *S*, we find
2.18Indeed, recall that *H* involves three integrals; making in the first, second and third integrals of the r.h.s. of (2.18) the substitutions
respectively, we find that the r.h.s. of (2.18) equals −*S*.

### Remark 2.5

(The Riemann function) Suppose we chose the functions {*g*_{+}(*ρ*),*g*_{−}(*ρ*),*g*(*φ*)} by (2.17), where
2.19Then, the function *S*(*s*) is proportional to the Riemann zeta function, namely,
2.20Hence, the Riemann hypothesis is valid iff there does *not* exist a solution of the Neumann boundary-value problem defined in theorem 2.2 with the functions {*g*_{+}(*ρ*),*g*_{−}(*ρ*),*g*(*φ*)} defined by equations (2.17) and (2.19), which is bounded as .

Unfortunately, we have not been able to use the above interesting connection between the Riemann hypothesis and the solution of the concrete Neumann boundary-value problem specified by the function *u*(*z*) defined in (2.19), in order to make any progress on the Riemann hypothesis. However, this connection does suggest a possible way of deriving certain identities involving the Riemann function, which will be presented in a future publication. Here, we illustrate the general approach of deriving such identities for the simpler case of the hypergeometric function.

### (a) The Dirichlet boundary-values

By evaluating the r.h.s. of equation (2.8a) at *r*=*a* and at *θ*=±*α*, we find the following expressions for the Dirichlet boundary-values:
2.21and
2.22Adding and subtracting equations (2.22)^{+} and (2.22)^{−}, we find the following equations:
2.23and
2.24

In the following, we assume that the given functions *g*_{±}(*r*) and *g*(*θ*) satisfy the constraint *S*=0.

Let *c* be a constant, then
2.25where *G*(*r*) is defined by
2.26Each of the two logarithmic terms on the r.h.s. of equation (2.23) grows logarithmically as ; however, the condition *S*=0 implies that these two terms cancel. Indeed, using the fact that *G*(*r*)*S*=0, equation (2.23) can be rewritten in the form
2.27Equations (2.24) and (2.27) are the basic equations needed for the derivation of certain integral identities.

## 3. The global relation and certain integral identities

The global relations for the Laplace equation in the domain *D* are the following two equations:
3.1The above restriction in *k* is the consequence of the large *r* behaviour of *q*, namely of the estimate
If the domain *D* is defined by (2.6), then equations (3.1)^{±} become
3.2Adding and subtracting (3.2)^{±}, we obtain the following basic equations:
3.3and
3.4Replacing, on the r.h.s. of the relations (3.3) and (3.4), *q*(*r*,−*α*)±*q*(*r*,*α*) by the r.h.s. of (2.24) and (2.27), we find the following equations, which are valid for Re *k*<*α*/2*π*:
3.5and
3.6where the functions are defined as follows:
3.7
3.8
3.9
and
3.10In the above equations, principal-value integrals are assumed if needed.

The function *g*_{−}(*ρ*)−*g*_{+}(*ρ*) is related to *g*(*φ*) through the equation *S*=0, whereas the function *g*_{−}(*ρ*)+*g*_{+}(*ρ*) is independent of *g*(*φ*). Hence, equations (3.5) and (3.6) imply the following basic identities:
3.11and
3.12where
and *f*(*k*) is some function of *k*.

The above equations will be verified explicitly in the next section, where it will also be shown that the function *f*(*k*) is given by
3.13

### Remark 3.1

Using the estimates
and
it follows that the first two integrals on the r.h.s. of equation (3.7) involving the above terms are well defined for Re *k*<*π*/2*α*. If *k* satisfies the stronger restriction Re *k*<0, then it is *not* necessary to subtract the term involving *G*(*r*). Actually in this case, we find
3.14

### Remark 3.2

If Re *k*<0, then equations (3.11) and (3.12) are replaced by the equations
3.15where and denote the expressions obtained from *F*_{1} and *F*_{2} by neglecting the terms involving *G*(*r*).

Equation (3.15) can be obtained *directly* as follows: the function
3.16is a solution of the Neumann boundary-value problem with
3.17Substituting and expressions (3.17) in equation (2.22)^{+}, we find
3.18Replacing *φ* with −*φ* and dividing by *a*^{k}/*π*, we find .

Similarly, substituting and expressions (3.17) in equation (2.21), we find .

The function
is also a solution of the Neumann boundary-value problem with
The above solution implies *F*_{3}=*F*_{4}=0.

### Remark 3.3

The first integral in the definition of *F*_{1} involves a singularity at *θ*=−*φ*. Making the change of variables *θ*=−*φ*+*x*, the relevant singularity is mapped at *x*=0 and the associated integral is . This singularity can be handled by Cauchy principal-value integrals. In this case, the contribution of this singularity vanishes. Indeed, this contribution involves
and using integration by parts, it follows that the contribution from *x*=*ϵ* vanishes.

If, instead of the principal-value integral, we use the limit from above and below, the relevant contribution is which clearly does not exist.

## 4. Verification of the four identities

In order to simplify the functions defined by (3.7)–(3.10), we introduce a change of variables,
4.1Then, can be written as
4.2
4.3
4.4
and
4.5where the functions are defined by
4.6
4.7
4.8
4.9
4.10
4.11
4.12and *γ*_{1} is depicted in figure 2.

In order to compute , the following lemma will be useful.

### Lemma 4.1

*Let γ be a smooth curve from z*_{1} *to z*_{2} *in the z*-*complex plane, which passes through z*_{0}. *Then*,
4.13*where PV denotes the principal-value integral. If the singularity lies off the path γ, then the last term in* (4.13) *is absent*.

### Proof.

Letting *x*=*z*/*z*_{0}, we find
where *γ*_{0} runs from *z*_{1}/*z*_{0} to *z*_{2}/*z*_{0}, avoids *x*=0 and passes through *x*=1. We recall that
Letting *ρx*=*v*, we find
so that
4.14which immediately gives (4.13), the principal part at *z*_{0} being included by subtracting half the residue at this point. □

The hypergeometric functions in (4.13) are related to the Lerch transcendent, a generalization of the Riemann zeta function
4.15the Lerch transcendent is analytic in the *z*-plane for fixed *p*≠1.

### Proposition 4.2

*For* Re *κ*>−1,
4.16

### Proof.

If *κ*=0 (i.e. *k*=0), then *F*_{1}=*F*_{2}=−*π*, which agrees with (4.16) in the limit *k*→0. Thus, we consider the case of *κ*≠0. Using integration by parts and a partial fraction decomposition, we find
4.17where the contour *γ*_{1} is depicted in figure 2 and we have used
Using the change of variables *u*→1/*u* for the term involving *u*^{−κ}, the above principal-value integral becomes
4.18The integral in (4.18) can be split into two parts, and each resulting integral can be computed by using lemma 4.1. Noting that −*α*<*φ*<*α*, the definition of *v* (see (4.1)) implies that *v* and *v*^{−1} lie on the contour *γ*_{1}. The relevant residue contributions yield the term
4.19Hence, we find
4.20where
4.21

Regarding *h*_{3}(*κ*,*v*), using integration by parts and a partial fraction decomposition, we find
4.22The integral in (4.22) can be split into two parts, and each resulting integral can be computed using lemma 4.1. The definition of *v* implies that the integrand in (4.22) does not have any singularities on [0,1]. Hence, we find
4.23

Substituting (4.20) into (4.17) and using (4.23), we find that the terms involving the hypergeometric functions and the logarithmic functions cancel. Hence, recalling *κ*=−(2*α*/*π*)*k*, we obtain (4.16) for *F*_{1}.

Regarding *F*_{2}, we first deform the contour *γ*_{1} to *γ*_{2}, which is the segment (−i,i) of the imaginary axis (figure 3). Then, using integration by parts and a partial fraction decomposition, we find
4.24Using the fact that the integrand of the above principal-value integral has poles at *x*=±i*y* and employing lemma 4.1, we find that the principal-value integral of the r.h.s. of (4.24) is given by
4.25For *h*_{4}(*κ*,*y*) and *h*_{5}(*κ*,*y*), using the fact that the integrands have a pole at *x*=*y*, we find
4.26and
4.27

Combining (4.24), (4.26) and (4.27) with (4.25), we find that the terms involving the hypergeometric functions and the logarithmic functions cancel and hence we find (4.16) for *F*_{2}. □

### Proposition 4.3

*For* Re *κ*>−1,
4.28

### Proof.

If *κ*=0, it is obvious that *F*_{3}=*F*_{4}=0. Thus, we consider the case *κ*≠0. Using integration by parts and a partial fraction decomposition, *h*_{2}(*κ*,*v*) can be written as
4.29where we have used
The principal-value integral in (4.29) can be evaluated by lemma 4.1,
4.30Substituting (4.30) into (4.29) and combining the resulting expression with (4.23), we find *F*_{3}=0.

Regarding *F*_{4}, deforming the contour *γ*_{1} to *γ*_{2} and then using integration by parts, we find
4.31Using the fact that the integrand of the above principal-value integral has poles at *x*=±i*y* and employing lemma 4.1, we find that the principal-value integral of the r.h.s. of (4.31) is given by
4.32Using (4.26) and (4.27), together with (4.32), we find *F*_{4}=0. □

## 5. Functional identities

The functions *h*_{3}, *h*_{5}, *h*_{6} and *h*_{7} do not involve principal-value integrals. We note that principal-value integrals always yield a term involving i in accordance with the Plemelj formula. Thus, for those real values of *k* for which *h*_{3}, *h*_{5}, *h*_{6} and *h*_{7} are well defined, it should be possible to express these functions in terms of real functions. We have succeeded in doing this for the last three functions, but not for *h*_{3}. Indeed, the following expressions are valid for Re *k*>−1. Although the formulae in this section are valid for complex *y*, we shall restrict attention to 0<*y*<1,
5.1
5.2
and
5.3In addition, for *k* real, the functions *h*_{1} and *h*_{2} possess the following more complicated real forms:
5.4and
5.5where denotes the Meijer G-function.

The basic identities (3.11) and (3.12) yield a plethora of novel identities involving the generalized hypergeometric and related functions.

### Example 5.1

Replacing in the definition (4.5) of *F*_{4}, *h*_{7} by (5.3), *h*_{4} by (4.26) and *h*_{5} by (5.1), the identity *F*_{4}=0 yields, apart from *k*=1,3,5,…, where both sides are singular,
5.6The term in (5.6) represents all the terms within the curly brackets where *y* is replaced by *y*^{−1}; the i appears as a consequence of the fact that the analytic continuation of Gauss’ hypergeometric function acquires an imaginary part for real arguments exceeding unity. Note that the limit of both sides of (5.6) exist.

Letting *k*=2 with *z*=2/(*y*−*y*^{−1}), equation (5.6) yields the following novel identity:
5.7which, by analytic continuation, is valid for all *z*.

For , equation (5.6) yields the curious identity
5.8Letting
it follows that equation (5.8) can be rewritten in terms of the variable *z*.

Similarly, for *k*=4, we find the novel identity for *y*^{2}≤1,
5.9

### Example 5.2

Replacing in the definition (4.3) of *F*_{2}, *h*_{6} by (5.2), *h*_{4} by (4.26) and *h*_{5} by (5.1), the identity *F*_{2}=*f*(*k*) yields the following identity:
5.10In this case, the term represents the expression in curly brackets with *y* replaced by *y*^{−1}.

Replacing in the definition (4.4) of *F*_{4}, *h*_{2} by (5.5), we find the novel identity, valid for Re *k*>−1 and all real *v*,
5.11In particular, for *k*=0 and all real *b*,
5.12

The above identities appear to be new and different in character from any in the standard literature. For example, consider equations (5.7) and (5.9) that are two particular cases stemming from the identity (5.6). These two equations present two hypergeometric identifications that are *not* contained in the most extensive compilation available, that of Prudnikov *et al.* [33]. It is true that Mathematica is able to reduce the l.h.s. of (5.7) to an expression containing a _{2}*F*_{1} whose indefinite integral is listed in [33], but some algebraic manipulation is required to produce the r.h.s. of (5.7) (for the casual user, Mathematica provides no description as to the derivation of its results). Furthermore, Mathematica is silent on (5.9). All the hypergeometric identities required for the manipulations in this section are contained in table 7.3 of [33]. It should be noted that *k is by no means restricted to integer values*. It is restricted solely by the requirement Re *k*>−1. Thus, our results encompass functions such as , for which Mathematica and the literature are silent. We maintain that identities such as (5.6) and (5.11), which are only illustrations of what our methods are capable of providing, are *completely new* to hypergeometric theory. Thus, it seems that by employing the global relation to the solution of certain boundary-value problems, it is possible to construct new formulae in the area of special functions, although it is not yet clear how the form of these identities can be predicted in advance.

## Funding

M.L.G. has been partially supported by the Spanish Ministry of Education and Science (Project MTM 2005-09183)

## Acknowledgements

A.S.F. acknowledges partial support from the Guggenheim Memorial foundation, USA. M.L.G. thanks the University of Valladolid for continuing hospitality. The authors are grateful to Eugene Shargorodsky for several useful suggestions and in particular for his observation expressed in remark 3.2.

- Received February 6, 2013.
- Accepted May 29, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.