# Eigenfunction expansions for a fundamental solution of Laplace’s equation on
${\mathbf{R}}^{3}$ in parabolic and elliptic cylinder coordinates

H S Cohl1 and H Volkmer2 1Information Technology Laboratory, National
Institute of Standards and Technology, Gaithersburg, MD, USA  2Department of
Mathematical Sciences, University of Wisconsin–Milwaukee, P. O. Box 413,
Milwaukee, WI 53201, U.S.A<EMAIL_ADDRESS>
###### Abstract

A fundamental solution of Laplace’s equation in three dimensions is expanded
in harmonic functions that are separated in parabolic or elliptic cylinder
coordinates. There are two expansions in each case which reduce to expansions
of the Bessel functions $J_{0}(kr)$ or $K_{0}(kr)$,
$r^{2}=(x-x_{0})^{2}+(y-y_{0})^{2}$, in parabolic and elliptic cylinder
harmonics. Advantage is taken of the fact that $K_{0}(kr)$ is a fundamental
solution and $J_{0}(kr)$ is the Riemann function of partial differential
equations on the Euclidean plane.

###### pacs:

02.30.Em, 02.30.Gp, 02.30.Hq, 02.30.Jr, 02.30.Mv, 02.40.Dr

###### ams:

35A08, 35J05, 42C15, 33E10, 33C15

##  1 Introduction

A fundamental solution of Laplace’s equation

$\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{\partial
y^{2}}+\frac{\partial^{2}U}{\partial z^{2}}=0$ (1)

is given by (apart from a multiplicative factor of $4\pi$)

$U({\bf x},{\mathbf{x}}_{0})=\frac{1}{\|{\bf
x}-{\mathbf{x}}_{0}\|},\quad\textup{where ${\bf
x}=(x,y,z)\neq{\mathbf{x}}_{0}=(x_{0},y_{0},z_{0})$},$ (2)

and $\|{\bf x}-{\bf x}_{0}\|$ denotes the Euclidean distance between ${\bf x}$
and ${\bf x}_{0}$. In many applications it is required to expand a fundamental
solution in the form of a series or an integral, in terms of solutions of (1)
that are separated in suitable curvilinear coordinates. Examples of such
applications include electrostatics, magnetostatics, quantum direct and
exchange Coulomb interactions, Newtonian gravity, potential flow, and steady
state heat transfer. Morse & Feshbach (1953) [16] (see also Hobson (1955)
[10]; MacRobert (1947) [12]; Heine (1881) [9]) provide a list of such
expansions for various coordinate systems but the formulas for several
coordinate systems are missing. It is the goal of this paper to provide these
expansions for parabolic and elliptic cylinder coordinates. Although these
expansions are partially known from Buchholz (1953) [2] and Lebedev (1972)
[11] for parabolic cylinder coordinates, and from Meixner & Schäfke (1954)
[14] for elliptic cylinder coordinates, we found it desirable to investigate
these expansions in a systematic fashion and provide direct proofs for them
based on eigenfunction expansions.

There will be two expansions for both of these coordinate systems. The first
expansion for a cylindrical coordinate system on ${\mathbf{R}}^{3}$ starts
from the known formula in terms of the integral of Lipschitz (see Watson
(1944) [22, section 13.2]; Cohl et al. (2000) [5, (8)])

$\frac{1}{\|\mathbf{x}-\mathbf{x_{0}}\|}=\int_{0}^{\infty}J_{0}(k\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}})e^{-k|z-z_{0}|}\,dk.$
(3)

The Bessel function $J_{\nu}$ can be defined by (see for instance (10.2.2) in
Olver et al. (2010) [17])

$J_{\nu}(z):=\biggl{(}\frac{z}{2}\biggr{)}^{\nu}\sum_{n=0}^{\infty}\frac{(-z^{2}/4)^{n}}{n!\Gamma(\nu+n+1)},$
(4)

Note that for $W_{k}:{\mathbf{R}}^{2}\times{\mathbf{R}}^{2}\to{\mathbf{R}}$

$W_{k}(x,y,x_{0},y_{0}):=J_{0}(kr),$ (5)

where $r^{2}:=(x-x_{0})^{2}+(y-y_{0})^{2},$ solves the partial differential
equation

$\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{\partial
y^{2}}+k^{2}U=0.$ (6)

Therefore, in a cylindrical coordinate system on ${\mathbf{R}}^{3}$ involving
the Cartesian coordinate $z$, the first expansion (3) reduces to expanding
$W_{k}$ from (5) in terms of solutions of (6) that are separated in a
curvilinear coordinate system on the plane.

The second expansion for a cylindrical coordinate system on ${\mathbf{R}}^{3}$
is based on the known formula given in terms of the Lipschitz-Hankel integral
(see Watson (1944) [22, section 13.21]; Cohl et al. (2000) [5, (9)])

$\frac{1}{\|\mathbf{x}-\mathbf{x_{0}}\|}=\frac{2}{\pi}\int_{0}^{\infty}K_{0}(k\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}})\cos
k(z-z_{0})\,dk,$ (7)

where $K_{\nu}:(0,\infty)\to{\mathbf{R}}$ (cf. (10.32.9) in Olver et al.
(2010) [17]), the modified Bessel function of the second kind (Macdonald’s
function), of order $\nu\in{\mathbf{R}}$, is defined by

$K_{\nu}(z):=\int_{0}^{\infty}e^{-z\cosh t}\cosh(\nu t)\,dt.$ (8)

Now $V_{k}:{\mathbf{R}}^{2}\times{\mathbf{R}}^{2}\setminus\\{({\bf x},{\bf
x}):{\bf x}\in{\mathbf{R}}^{2}\\}\to(0,\infty),$ defined by

$V_{k}(x,y,x_{0},y_{0}):=K_{0}(kr)$ (9)

solves the partial differential equation

$\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{\partial
y^{2}}-k^{2}U=0.$ (10)

In a cylindrical coordinate system on ${\mathbf{R}}^{3}$ involving the
Cartesian coordinate $z$, the second expansion (7) reduces to expanding
$V_{k}$ in terms of solutions of (10) that are separated in curvilinear
coordinates on the plane.

The paper is organized as follows. In section 2 we derive the desired
expansion of $K_{0}(kr)$ in parabolic cylinder coordinates. This expansion is
given in terms of series over Hermite functions. In section 3 we obtain an
integral representation for $J_{0}(kr)$ in terms of separated solutions of
(10) in terms of (modified) parabolic cylinder functions. We show how these
results are based on a general expansion theorem in terms of solutions of the
differential equation

$-u^{\prime\prime}-\frac{1}{4}\xi^{2}u=\lambda u.$ (11)

In sections 4 and 5 we derive the fundamental solution expansions in elliptic
cylinder coordinates for $J_{0}(kr)$ and $K_{0}(kr)$, respectively.

Throughout this paper we rely on the following definitions. The set of natural
numbers is given by ${\mathbf{N}}:=\\{1,2,3,\ldots\\}$, the set
${\mathbf{N}}_{0}:=\\{0,1,2,\ldots\\}={\mathbf{N}}\cup\\{0\\}$, and the set
${\mathbf{Z}}:=\\{0,\pm 1,\pm 2,\ldots\\}.$ The set ${\mathbf{R}}$ represents
the real numbers and the set ${\mathbf{C}}$ represents the complex numbers.

##  2 Expansion of $K_{0}(kr)$ for parabolic cylinder coordinates

Parabolic coordinates on the plane $(\xi,\eta)$ (see for instance Chapter 10
in Lebedev (1972) [11]) are connected to Cartesian coordinates $(x,y)$ by

$x=\frac{1}{2}(\xi^{2}-\eta^{2}),\quad y=\xi\eta,$ (12)

where $\xi\in{\mathbf{R}}$ and $\eta\in[0,\infty)$. To simplify notation we
will first set $k=1$ in (9). Then $V_{1}$ satisfies

$\frac{\partial^{2}U}{\partial x^{2}}+\frac{\partial^{2}U}{\partial
y^{2}}-U=0\quad\textup{if $(x,y)\neq(x_{0},y_{0})$.}$ (13)

Let $(\xi,\eta)$, $(\xi_{0},\eta_{0})$ be parabolic coordinates on
${\mathbf{R}}^{2}$ for $(x,y)$ and $(x_{0},y_{0})$, respectively. Then $V_{1}$
transforms to

$v(\xi,\eta,\xi_{0},\eta_{0}):=K_{0}(r(\xi,\eta,\xi_{0},\eta_{0})),\quad(\xi,\eta)\neq\pm(\xi_{0},\eta_{0})$

and $r(\xi,\eta,\xi_{0},\eta_{0})$ is defined by

$r^{2}=\frac{1}{4}\left[(\xi+\xi_{0})^{2}+(\eta+\eta_{0})^{2}\right]\left[(\xi-\xi_{0})^{2}+(\eta-\eta_{0})^{2}\right],\quad
r>0.$ (14)

Here and in the following we allow all
$\xi,\eta,\xi_{0},\eta_{0}\in{\mathbf{R}}$ such that
$(\xi,\eta)\neq\pm(\xi_{0},\eta_{0})$. From (13) or by direct computation we
obtain that $v$ solves the equation

$\frac{\partial^{2}u}{\partial\xi^{2}}+\frac{\partial^{2}u}{\partial\eta^{2}}-(\xi^{2}+\eta^{2})u=0.$
(15)

Separating variables $u(\xi,\eta)=u_{1}(\xi)u_{2}(\eta)$ in (15), we obtain
the ordinary differential equations

$\displaystyle u_{1}^{\prime\prime}+(2n+1-\xi^{2})u_{1}$ $\displaystyle=$
$\displaystyle 0,$ (16) $\displaystyle
u_{2}^{\prime\prime}-(2n+1+\eta^{2})u_{2}$ $\displaystyle=$ $\displaystyle 0,$
(17)

where we will use only $n\in{\mathbf{N}}_{0}$. Equations (16), (17) have the
general solutions

$\displaystyle u_{1}(\xi)$ $\displaystyle=$ $\displaystyle
c_{1}e^{-\xi^{2}/2}H_{n}(\xi)+c_{2}e^{\xi^{2}/2}H_{-n-1}(i\xi),$
$\displaystyle u_{2}(\eta)$ $\displaystyle=$ $\displaystyle
c_{3}e^{\eta^{2}/2}H_{n}(i\eta)+c_{4}e^{-\eta^{2}/2}H_{-n-1}(\eta),$

where $H_{\nu}:{\mathbf{C}}\to{\mathbf{C}}$ is the Hermite function which can
be defined in terms of Kummer’s function of the first kind $M$ as (cf.
(10.2.8) in Lebedev (1972) [11])

$H_{\nu}(z):=\frac{2^{\nu}\sqrt{\pi}}{\Gamma\left(\frac{1-\nu}{2}\right)}M\left(-\frac{\nu}{2},\frac{1}{2},z^{2}\right)-\frac{2^{\nu+1}\sqrt{\pi}}{\Gamma\left(-\frac{\nu}{2}\right)}zM\left(\frac{1-\nu}{2},\frac{3}{2},z^{2}\right),$

and

$M(a,b,z):=\sum_{n=0}^{\infty}\frac{(a)_{n}z^{n}}{(b)_{n}n!}$ (18)

(see for instance (13.2.2) in Olver et al. (2010) [17]). Note that the Kummer
function of the first kind is entire in $z$ and $a,$ and is a meromorphic
function of $b.$ The Hermite function is an entire function of both $z$ and
$\nu$. If $\nu=n\in{\mathbf{N}}_{0}$ then $H_{\nu}(z)$ reduces to the Hermite
polynomial of degree $n$.

We will expand $v$ as a function of $\xi$ in an orthogonal series of functions
$e^{-\xi^{2}/2}H_{n}(\xi)$, $n\in{\mathbf{N}}_{0}$, so that the coefficients

$f_{n}(\eta,\xi_{0},\eta_{0}):=\int_{-\infty}^{\infty}v(\xi,\eta,\xi_{0},\eta_{0})e^{-\xi^{2}/2}H_{n}(\xi)\,d\xi,$

are to be evaluated. We do this based on the observation that the function
$(\xi,\eta)\mapsto v(\xi,\eta,\xi_{0},\eta_{0})$ is a fundamental solution of
equation (15). It has logarithmic singularities at the points
$\pm(\xi_{0},\eta_{0})$. Arguing as in Volkmer (1984) [21, Theorem 1.11], we
obtain the following integral representation for solutions of (15).

######  Theorem 2.1.

Let $u\in C^{2}({\mathbf{R}}^{2})$ be a solution of (15). Let
$(\xi_{0},\eta_{0})\in{\mathbf{R}}^{2}$, and let $C$ be a a closed rectifiable
curve on ${\mathbf{R}}^{2}$ which does not pass through
$\pm(\xi_{0},\eta_{0})$, and let $n^{\pm}$ be the winding number of $C$ with
respect to $\pm(\xi_{0},\eta_{0})$. Then we have

$2\pi\left[n^{+}u(\xi_{0},\eta_{0})+n^{-}u(-\xi_{0},-\eta_{0})\right]=\int_{C}(u\partial_{2}v-v\partial_{2}u)\,d\xi+(v\partial_{1}u-u\partial_{1}v)\,d\eta,$

where $\partial_{1},\partial_{2}$ denote partial derivatives with respect to
$\xi$, $\eta$, respectively.

We apply Theorem 2.1 to the solution

$u(\xi,\eta)=e^{-\xi^{2}/2}H_{n}(\xi)e^{\eta^{2}/2}H_{n}(i\eta),$

of (15), and for $C$ we take the positively oriented boundary of the rectangle
$|\xi|\leq\xi_{1}$, $|\eta|\leq\eta_{1}$, where $|\xi_{0}|<\xi_{1}$,
$|\eta_{0}|<\eta_{1}$, so $n^{+}=n^{-}=1$. Then let $\xi_{1}\to\infty$ and
note that the integrals over the vertical sides converge to $0$. The integrals
over the horizontal sides of the rectangle give the same contribution because
the integrand changes sign when $(\xi,\eta)$ is replaced by $(-\xi,-\eta)$.
Therefore, we obtain, for $\eta=\eta_{1}>|\eta_{0}|$,

$2\pi
u(\xi_{0},\eta_{0})=\int_{-\infty}^{\infty}(v(\xi,\eta,\xi_{0},\eta_{0})\partial_{2}u(\xi,\eta)-u(\xi,\eta)\partial_{2}v(\xi,\eta,\xi_{0},\eta_{0}))d\xi.$
(19)

Set $f(\eta)=f_{n}(\eta,\xi_{0},\eta_{0})$ and
$g(\eta)=e^{\eta^{2}/2}H_{n}(i\eta)$. Then (19) can be written as

$2\pi u(\xi_{0},\eta_{0})=g^{\prime}(\eta)f(\eta)-g(\eta)f^{\prime}(\eta).$
(20)

By differentiating both sides of (20) with respect to $\eta$ we see that $f$
satisfies (17). Since $f(\eta)$ goes to $0$ as $\eta\to\infty$, it follows
that

$f(\eta)=ce^{-\eta^{2}/2}H_{-n-1}(\eta),$

where $c$ is a constant. Going back to (20), we find that

$2\pi u(\xi_{0},\eta_{0})=cW,$

where $W=i^{n}$ is the (constant) Wronskian of $e^{-\eta^{2}/2}H_{-n-1}(\eta)$
and $g(\eta)$. Therefore, if $\eta>|\eta_{0}|$, we obtain

$f_{n}(\eta,\xi_{0},\eta_{0})=2\pi(-i)^{n}e^{-\xi_{0}^{2}/2}H_{n}(\xi_{0})e^{\eta_{0}^{2}/2}H_{n}(i\eta_{0})e^{-\eta^{2}/2}H_{-n-1}(\eta).$

Using the well-known series expansion in terms of Hermite functions (Lebedev
(1972) [11, Theorem 2, page 71]), we obtain the following result.

######  Theorem 2.2.

For $\xi,\eta,\xi_{0},\eta_{0}\in{\mathbf{R}}$ with $\eta>|\eta_{0}|$,

$K_{0}(r(\xi,\eta,\xi_{0},\eta_{0}))=\displaystyle{\sqrt{\pi}}e^{(\eta_{0}^{2}-\xi_{0}^{2}-\eta^{2}-\xi^{2})/2}\sum_{n=0}^{\infty}\frac{(-i)^{n}}{2^{n-1}n!}H_{n}(\xi)H_{-n-1}(\eta)H_{n}(\xi_{0})H_{n}(i\eta_{0}),$

where $r$ is defined by (14).

The special case $\xi_{0}=\eta_{0}=0$ of Theorem 2.2 can be found in [11,
Problem 7, page 298]. If we multiply each $\xi,\eta,\xi_{0},\eta_{0}$ by
$\sqrt{k}$ then we obtain an expansion for $K_{0}(kr)$. Inserting this
expansion into (7) yields our final result.

######  Theorem 2.3.

Let ${\bf x}$, ${\bf x}_{0}$ be points on ${\mathbf{R}}^{3}$ with parabolic
cylinder coordinates $(\xi,\eta,z)$ and $(\xi_{0},\eta_{0},z_{0})$,
respectively. If $\eta_{\lessgtr}:={\min\atop\max}\\{\eta,\eta_{0}\\}$ then

$\displaystyle\frac{1}{\|{\bf x}-{\bf
x}_{0}\|}=\frac{2}{\sqrt{\pi}}\int_{0}^{\infty}e^{-k/2(\xi^{2}+\eta^{2}+\xi_{0}^{2}-\eta_{0}^{2})}\cos
k(z-z_{0})$ $\displaystyle\hskip
56.9055pt\times\sum_{n=0}^{\infty}\frac{(-i)^{n}}{2^{n-1}n!}H_{n}(\sqrt{k}\xi)H_{-n-1}(\sqrt{k}\eta_{>})H_{n}(\sqrt{k}\xi_{0})H_{n}(i\sqrt{k}\eta_{<})\,dk,$

where ${\bf x}\neq{\bf x}_{0}$.

If we reverse the order of the infinite series and the definite integral in
the above expression we obtain

$\displaystyle\frac{1}{\|{\bf x}-{\bf
x}_{0}\|}=\frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-i)^{n}}{2^{n-1}n!}\int_{0}^{\infty}e^{-k/2(\xi^{2}+\eta^{2}+\xi_{0}^{2}-\eta_{0}^{2})}$
$\displaystyle\hskip 56.9055pt\times
H_{n}(\sqrt{k}\xi)H_{-n-1}(\sqrt{k}\eta)H_{n}(\sqrt{k}\xi_{0})H_{n}(i\sqrt{k}\eta_{0})\cos
k(z-z_{0})\,dk.$

It would be interesting to know the value of the above definite integral.

##  3 Expansion of $J_{0}(kr)$ for parabolic cylinder coordinates

Transforming equation (6) to parabolic coordinates (12) we obtain

$\frac{\partial^{2}u}{\partial\xi^{2}}+\frac{\partial^{2}u}{\partial\eta^{2}}+k^{2}(\xi^{2}+\eta^{2})u=0.$

In order to simplify notation we will (temporarily) set $k=\frac{1}{2}$ and
$\zeta=i\eta$ with $\zeta$ real. Thus we consider

$\frac{\partial^{2}u}{\partial\xi^{2}}-\frac{\partial^{2}u}{\partial\zeta^{2}}+\frac{1}{4}(\xi^{2}-\zeta^{2})u=0,\quad\xi,\zeta\in{\mathbf{R}}.$
(21)

If $u_{1}(\xi)$ and $u_{2}(\zeta)$ are solutions of the ordinary differential
equation (11) for some $\lambda$ then $u(\xi,\zeta)=u_{1}(\xi)u_{2}(\zeta)$
solves (21).

The function $W_{1/2}$ from (5) transformed to $(\xi,\zeta)$ becomes

$w(\xi,\zeta,\xi_{0},\zeta_{0})=J_{0}\left(\frac{1}{2}\tilde{r}(\xi,\zeta,\xi_{0},\zeta_{0})\right),$
(22)

where $\tilde{r}^{2}$ is a symmetric polynomial defined by

$\displaystyle
4\tilde{r}^{2}:=\left[(\xi-\xi_{0})^{2}-(\zeta-\zeta_{0})^{2}\right]\left[(\xi+\xi_{0})^{2}-(\zeta+\zeta_{0})^{2}\right]$
$\displaystyle=8\xi\xi_{0}\zeta\zeta_{0}+\xi^{4}+\xi_{0}^{4}+\zeta^{4}+\zeta_{0}^{4}-2\xi_{0}^{2}\zeta^{2}-2\xi_{0}^{2}\zeta_{0}^{2}-2\zeta^{2}\zeta_{0}^{2}-2\xi^{2}\xi_{0}^{2}-2\xi^{2}\zeta^{2}-2\xi^{2}\zeta_{0}^{2},$

and $J_{0}$ is the order zero Bessel function of the first kind (see (4)). For
fixed $\zeta,\xi_{0},\zeta_{0}\in{\mathbf{R}}$ consider the function
$f:{\mathbf{R}}\to{\mathbf{R}}$ defined by

$f(\xi):=w(\xi,\zeta,\xi_{0},\zeta_{0}).$ (23)

We wish to expand this function in terms of (modified) parabolic cylinder
harmonics according to a general expansion theorem that is derived in the
following subsection.

###  3.1 Spectral theory of (modified) parabolic cylinder harmonics – A
singular Sturm-Liouville problem

We discuss the Sturm-Liouville problem

$-u^{\prime\prime}-{\textstyle\frac{1}{4}}x^{2}u=\lambda
u,\quad-\infty<x<\infty,$ (24)

involving the spectral parameter $\lambda$ subject to

$u\in L^{2}(-\infty,\infty).$

By replacing $\frac{1}{4}x^{2}$ by $-\frac{1}{4}x^{2}$, we obtain the equation
describing the harmonic oscillator whose eigenfunctions (Hermite functions)
and the corresponding spectral theory is well-known. The spectral problem
associated with (24) is far less known.

A discussion of differential equation (24) and its solutions can be found in
[15] and in section 8.2 of (24) Erdélyi et al. (1982) [7] (see also Magnus
(1941) [13]; Wells & Spence (1945) [23]; Cherry (1948) [3]; Darwin (1949)
[6]).

We will follow Chapter 9 in Coddington & Levinson (1955) [4]. First note that,
by [4, Corollary 2, page 231], equation (24) is in the limit-point case at
$x=\pm\infty$. Therefore, one can apply section 5 of Chapter 9 in [4].

For $\lambda,x\in{\mathbf{C}}$ we define the functions $u_{1}(\lambda,x)$,
$u_{2}(\lambda,x)$ as the solutions of (24) uniquely determined by the initial
conditions

$u_{1}(\lambda,0)=u_{2}^{\prime}(\lambda,0)=1,\quad
u_{1}^{\prime}(\lambda,0)=u_{2}(\lambda,0)=0.$

These functions may be expressed in terms of Kummer’s function of the first
kind (18)

$\displaystyle u_{1}(\lambda,x)$ $\displaystyle=$ $\displaystyle
e^{-\frac{i}{4}x^{2}}M\left({\textstyle\frac{1}{4}}+{\textstyle\frac{i}{2}}\lambda,{\textstyle\frac{1}{2}},{\textstyle\frac{i}{2}}x^{2}\right),$
(25) $\displaystyle u_{2}(\lambda,x)$ $\displaystyle=$ $\displaystyle
e^{-{\textstyle\frac{i}{4}}x^{2}}xM\left({\textstyle\frac{3}{4}}+{\textstyle\frac{i}{2}}\lambda,{\textstyle\frac{3}{2}},{\textstyle\frac{i}{2}}x^{2}\right).$
(26)

For $x>0$, the function

$\displaystyle u_{3}(\lambda,x)$ $\displaystyle=$ $\displaystyle
e^{-{\textstyle\frac{i}{4}}x^{2}}U\left({\textstyle\frac{1}{4}}+{\textstyle\frac{i}{2}}\lambda,{\textstyle\frac{1}{2}},{\textstyle\frac{i}{2}}x^{2}\right)$
$\displaystyle=$
$\displaystyle\frac{\sqrt{\pi}}{\Gamma\bigl{(}{\textstyle\frac{3}{4}}+{\textstyle\frac{i}{2}}\lambda\bigr{)}}u_{1}(\lambda,x)-(1+i)\frac{\sqrt{\pi}}{\Gamma\bigl{(}{\textstyle\frac{1}{4}}+{\textstyle\frac{i}{2}}\lambda\bigr{)}}u_{2}(\lambda,x)$

is another solution of (24). Here the Kummer function of the second kind
$U:{\mathbf{C}}\times{\mathbf{C}}\times({\mathbf{C}}\setminus(-\infty,0])\to{\mathbf{C}}$
can be defined as (see Olver et al. (2010) [17, (13.2.42)])

$U(a,b,z):=\frac{\Gamma(1-b)}{\Gamma(a-b+1)}M(a,b,z)+\frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b}M(a-b+1,2-b,z).$

Except when $z=0$, each branch of $U$ is entire in $a$ and $b.$ We assume that
$U(a,b,z)$ has its principal value. The asymptotic behavior of the Kummer
function of the second kind [17, (13.2.6)] shows that $u_{3}(\lambda,\cdot)\in
L^{2}(0,\infty)$ provided that $\Im\lambda<0$. Since (24) is in the limit-
point case at $+\infty$, $u_{3}$ is the only solution with this property
except for a constant factor. The Titchmarsh-Weyl functions
$m_{\pm\infty}(\lambda)$, $\Im\lambda\neq 0$, are defined by the property that
$u_{1}(\lambda,x)+m_{\pm\infty}(\lambda)u_{2}(\lambda,x)$ is square-integrable
at $x=\pm\infty$. Therefore,

$m_{\infty}(\lambda)=\left\\{\begin{array}[]{ll}\displaystyle\displaystyle{-(1+i)\frac{\Gamma(\frac{3}{4}+\frac{i}{2}\lambda)}{\Gamma(\frac{1}{4}+\frac{i}{2}\lambda)}}&\quad\mathrm{if}\
\Im\lambda<0,\\\\[5.0pt]
\displaystyle\displaystyle{-(1-i)\frac{\Gamma(\frac{3}{4}-\frac{i}{2}\lambda)}{\Gamma(\frac{1}{4}-\frac{i}{2}\lambda)}}&\quad\mathrm{if}\
\Im\lambda>0.\end{array}\right.$

By symmetry, we have

$m_{-\infty}(\lambda)=-m_{\infty}(\lambda).$

Using the notation of [4, Theorem 5.1, page 251], on obtains

$M_{11}(\lambda)=\frac{-1}{2m_{\infty}(\lambda)},\quad M_{12}=M_{21}=0,\quad
M_{22}(\lambda)=\frac{m_{\infty}(\lambda)}{2}.$

Now using [4, page 250, last line], we find that, for
$\lambda\in{\mathbf{R}}$,

$\rho^{\prime}_{1}(\lambda):=\rho^{\prime}_{11}(\lambda)=\frac{1}{\pi}\lim_{\epsilon\to
0+}\Im\left(M_{11}(\lambda+i\epsilon)\right)=\frac{e^{\frac{\pi}{2}\lambda}}{4\sqrt{2}\pi^{2}}\left|\Gamma({\textstyle\frac{1}{4}}+{\textstyle\frac{i}{2}}\lambda)\right|^{2},$

since [17, (5.4.5)]

$\Gamma({\textstyle\frac{1}{4}}+iy)\Gamma({\textstyle\frac{3}{4}}-iy)=\frac{\pi\sqrt{2}}{\cosh(\pi
y)+i\sinh(\pi y)}.$

Moreover, $\rho_{12}(\lambda)=\rho_{21}(\lambda)=0$ and

$\rho^{\prime}_{2}(\lambda):=\rho^{\prime}_{22}(\lambda)=\frac{1}{\pi}\lim_{\epsilon\to
0+}\Im\left(M_{22}(\lambda+i\epsilon)\right)=\frac{e^{\frac{\pi}{2}\lambda}}{2\sqrt{2}\pi^{2}}\left|\Gamma({\textstyle\frac{3}{4}}+{\textstyle\frac{i}{2}}\lambda)\right|^{2}.$

Since the $\rho$-functions are real-analytic functions (with no jumps), we see
that the spectrum of (24) is the whole real line ${\mathbf{R}}$ and there are
no eigenvalues. The latter also follows from the known asymptotic behavior of
the solutions of (24) (see [17, Chapter 12]).

Using a variant of Stirling’s formula (see (5.11.9) in Olver et al. (2010)
[17])

$|\Gamma(x+iy)|\sim\sqrt{2\pi}|y|^{x-1/2}e^{-\pi|y|/2}\quad\textup{as
$x,y\in{\mathbf{R}}$, $|y|\to\infty$},$ (28)

we can determine the asymptotic behavior of $\rho^{\prime}_{j}$, namely

$\displaystyle\rho_{1}^{\prime}(\lambda)$ $\displaystyle\sim$
$\displaystyle\frac{1}{2\pi}|\lambda|^{-1/2}e^{\pi(\lambda-|\lambda|)/2}\quad\textup{as
$|\lambda|\to\infty$,}$ (29) $\displaystyle\rho_{2}^{\prime}(\lambda)$
$\displaystyle\sim$
$\displaystyle\frac{1}{2\pi}|\lambda|^{1/2}e^{\pi(\lambda-|\lambda|)/2}\quad\textup{as
$|\lambda|\to\infty$.}$ (30)

Applying [4, Theorem 5.2, page 251] to the analysis above, we obtain the
following result on the spectral resolution associated with equation (24).

######  Theorem 3.1.

For a given function $f\in L^{2}({\mathbf{R}}),$ form the functions

$g_{j}(\lambda)=\int_{-\infty}^{\infty}u_{j}(\lambda,x)f(x)\,dx,\quad j=1,2,\
\lambda\in{\mathbf{R}}.$ (31)

Then

$g_{j}\in L^{2}({\mathbf{R}},\rho_{j}),\quad j=1,2,$

or, equivalently,

$\int_{-\infty}^{\infty}|g_{j}(\lambda)|^{2}\rho^{\prime}_{j}(\lambda)\,d\lambda<\infty,\quad
j=1,2.$

The function $f$ can be represented in the form

$f(x)=\sum_{j=1}^{2}\int_{-\infty}^{\infty}u_{j}(\lambda,x)g_{j}(\lambda)\rho_{j}^{\prime}(\lambda)\,d\lambda.$
(32)

Moreover, we have Parseval’s equation

$\int_{-\infty}^{\infty}|f(x)|^{2}\,dx=\sum_{j=1}^{2}\int_{-\infty}^{\infty}|g_{j}(\lambda)|^{2}\rho^{\prime}_{j}(\lambda)\,d\lambda.$

Equations (31), (32) establish a one-to-one correspondence between $f$ and
$(g_{1},g_{2})$. The integrals appearing in (31), (32) have to be interpreted
in the $L^{2}$-sense. For instance, (31) means that

$\int_{-n}^{n}u_{j}(\lambda,x)f(x)dx$

converges to $g_{j}(\lambda)$ in $L^{2}({\mathbf{R}},\rho_{j})$ as
$n\to\infty$. Of course, if

$\int_{-\infty}^{\infty}\left|u_{j}(\lambda,x)f(x)\right|dx<\infty\quad\textup{for
every $\lambda\in{\mathbf{R}}$,}$

then (31) is also true pointwise.

###  3.2 Applying the spectral theory to the expansion of $J_{0}(kr)$ in
parabolic cylinder coordinates

Since $f(\xi)=O(|\xi|^{-1})$ in (23) as $|\xi|\to\infty$, we have that $f\in
L^{2}({\mathbf{R}})$. Therefore, we can expand $f$ using Theorem 3.1,
according to (32). For $\lambda\in{\mathbf{R}}$, we form the integrals

$g_{j}(\lambda,\zeta,\xi_{0},\zeta_{0})=\int_{-\infty}^{\infty}w(\xi,\zeta,\xi_{0},\zeta_{0})u_{j}(\lambda,\xi)\,d\xi,\quad
j=1,2.$ (33)

These are absolutely convergent integrals because $f(\xi)=O(|\xi|^{-1})$ and
$u_{j}(\lambda,\xi)=O(|\xi|^{-1/2})$ as $|\xi|\to\infty$.

Since $w(\xi,\zeta,\xi_{0},\zeta_{0})$ from (22) solves (21) for fixed
$(\xi_{0},\zeta_{0}),$ and $u_{j}(\lambda,\xi)$ from (25), (26) solves (24),
it follows from differentiation under the integral sign followed by
integration by parts (see for instance Schäfke (1963) [18, Satz 8, page 26])
that $g_{j}$ solves (24) as a function of $\zeta$. Since the function $w$ is
symmetric in its four variables, it is also true that $g_{j}$ solves (24) as
function of $\xi_{0}$ for fixed $\zeta_{0},\zeta$ and as function of
$\zeta_{0}$ for fixed $\xi_{0},\zeta$. From these properties of $g_{j}$ it
follows easily that there are functions $c_{jk\ell
m}:{\mathbf{R}}\to{\mathbf{R}}$ with $j,k,\ell,m=1,2$, depending on $\lambda$
but not on $\zeta,\xi_{0},\zeta_{0}$ such that

$g_{j}(\lambda,\zeta,\xi_{0},\zeta_{0})=\sum_{k,\ell,m=1}^{2}c_{jk\ell
m}(\lambda)u_{k}(\lambda,\zeta)u_{\ell}(\lambda,\xi_{0})u_{m}(\lambda,\zeta_{0}).$
(34)

This formula holds for all $\lambda,\zeta,\xi_{0},\zeta_{0}\in{\mathbf{R}}$.
Substituting $\zeta=\xi_{0}=\zeta_{0}=0$ in (33), (34), we obtain

$c_{j111}(\lambda)=\int_{-\infty}^{\infty}J_{0}({\textstyle\frac{1}{4}}\xi^{2})u_{j}(\lambda,\xi)\,d\xi.$

If $j=2,$ we integrate over an odd function, so $c_{2111}(\lambda)=0$. By
differentiating (34) with respect to $\zeta$ and/or $\xi_{0}$ and/or
$\zeta_{0}$ and then substituting $\zeta=\xi_{0}=\zeta_{0}=0$ we find (after
some calculations)

$c_{jk\ell m}(\lambda)=\left\\{\begin{array}[]{ll}\displaystyle
c_{1}(\lambda)&\quad\mathrm{if}\ j=k=\ell=m=1,\\\\[2.0pt] \displaystyle
c_{2}(\lambda)&\quad\mathrm{if}\ j=k=\ell=m=2,\\\\[2.0pt] \displaystyle
0&\quad\textup{otherwise}.\end{array}\right.$

Here $c_{j}:{\mathbf{R}}\to{\mathbf{R}}$ for $j=1,2$ is given by (see A)

$\displaystyle
c_{1}(\lambda):=\int_{-\infty}^{\infty}J_{0}({\textstyle\frac{1}{4}}\xi^{2})u_{1}(\lambda,\xi)\,d\xi=\frac{2\sqrt{2}\pi
e^{-\pi\lambda/2}}{\cosh(\pi\lambda)|\Gamma\left(\frac{3}{4}+\frac{i\lambda}{2}\right)|^{2}},$
(35) $\displaystyle
c_{2}(\lambda):=-\int_{-\infty}^{\infty}\xi^{-1}J_{1}({\textstyle\frac{1}{4}}\xi^{2})u_{2}(\lambda,\xi)\,d\xi=\frac{-4\sqrt{2}\pi
e^{-\pi\lambda/2}}{\cosh(\pi\lambda)|\Gamma\left(\frac{1}{4}+\frac{i\lambda}{2}\right)|^{2}}.$
(36)

According to (32),

$\displaystyle w(\xi,\zeta,\xi_{0},\zeta_{0})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2}\int_{-\infty}^{\infty}\\!\\!\\!c_{j}(\lambda)\rho^{\prime}_{j}(\lambda)u_{j}(\lambda,\xi)u_{j}(\lambda,\zeta)u_{j}(\lambda,\xi_{0})u_{j}(\lambda,\zeta_{0})d\lambda.$
(37)

Note that

$\displaystyle c_{1}(\lambda)\rho_{1}^{\prime}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi\cosh(\pi\lambda)}\left|\frac{\Gamma\left(\frac{1}{4}+i\frac{\lambda}{2}\right)}{\Gamma\left(\frac{3}{4}+i\frac{\lambda}{2}\right)}\right|^{2}=\frac{1}{4\pi^{3}}\left|\Gamma\left({\textstyle\frac{1}{4}}+{\textstyle\frac{i\lambda}{2}}\right)\right|^{4},$
(38) $\displaystyle c_{2}(\lambda)\rho_{2}^{\prime}(\lambda)$ $\displaystyle=$
$\displaystyle-\frac{2}{\pi\cosh(\pi\lambda)}\left|\frac{\Gamma\left(\frac{3}{4}+i\frac{\lambda}{2}\right)}{\Gamma\left(\frac{1}{4}+i\frac{\lambda}{2}\right)}\right|^{2}=\frac{-1}{\pi^{3}}\left|\Gamma\left({\textstyle\frac{3}{4}}+{\textstyle\frac{i\lambda}{2}}\right)\right|^{4},$
(39)

where we used [17, (5.4.4), (5.5.5)]. It follows from (28) that

$\displaystyle c_{1}(\lambda)\rho_{1}^{\prime}(\lambda)$ $\displaystyle\sim$
$\displaystyle\frac{1}{\pi|\lambda|\cosh(\pi\lambda)}\quad\textup{as
$|\lambda|\to\infty$,}$ (40) $\displaystyle
c_{2}(\lambda)\rho_{2}^{\prime}(\lambda)$ $\displaystyle\sim$
$\displaystyle-\frac{|\lambda|}{\pi\cosh(\pi\lambda)}\quad\textup{as
$|\lambda|\to\infty$}.$ (41)

It is known (see Atkinson (1964), [1, (8.2.5)]) that, for fixed
$x\in{\mathbf{C}}$, there is a constant $C$ such that
$u_{j}(\lambda,x)=O(e^{C|\lambda|^{1/2}})$. It follows from (40), (41), that
the integrands in (37) decay exponentially and therefore the corresponding
integrals are absolutely convergent. By the identity theorem for analytic
functions we see that equation (37) is true for all
$\xi,\zeta,\xi_{0},\zeta_{0}\in{\mathbf{C}}$.

After setting $\zeta=i\eta$ and $\zeta_{0}=i\eta_{0}$ in (37), one obtains the
following result.

######  Theorem 3.2.

Let $\xi,\eta,\xi_{0},\eta_{0}\in{\mathbf{R}}$. Then

$\displaystyle J_{0}\left(\frac{1}{2}r(\xi,\eta,\xi_{0},\eta_{0})\right)$
$\displaystyle\hskip
19.91684pt=\sum_{j=1}^{2}\int_{-\infty}^{\infty}c_{j}(\lambda)\rho^{\prime}_{j}(\lambda)u_{j}(\lambda,\xi)u_{j}(\lambda,i\eta)u_{j}(\lambda,\xi_{0})u_{j}(\lambda,i\eta_{0})\,d\lambda,$

where $r$ is given by (14) and $c_{j}(\lambda)\rho_{j}^{\prime}(\lambda)$ is
given by (38), (39).

In the special case $\xi_{0}=\eta_{0}=0$ (or correspondingly $\xi=\eta=0$),
Theorem 3.2 can be found in Buchholz (1953) [2, (16), page 175]. Of course, if
we multiply each $\xi,\eta,\xi_{0},\eta_{0}$ by $\sqrt{2k}$ we get the
expansion of $J_{0}(k\,r(\xi,\eta,\xi_{0},\eta_{0}))$. This leads to the
$J_{0}(kr)$ expansion of a fundamental solution for the three-dimensional
Laplace equation in parabolic cylindrical coordinates.

######  Theorem 3.3.

Let ${\bf x}$, ${\bf x}_{0}$ be points on ${\mathbf{R}}^{3}$ with parabolic
coordinates $(\xi,\eta,z)$ and $(\xi_{0},\eta_{0},z_{0})$, respectively, Then

$\displaystyle\frac{1}{\|{\bf x}-{\bf
x}_{0}\|}=\sum_{j=1}^{2}\int_{0}^{\infty}\int_{-\infty}^{\infty}c_{j}(\lambda)\rho^{\prime}_{j}(\lambda)$
$\displaystyle\hskip 14.22636pt\times
u_{j}(\lambda,2\sqrt{k}\xi)u_{j}(\lambda,2i\sqrt{k}\eta)u_{j}(\lambda,2\sqrt{k}\xi_{0})u_{j}(\lambda,2i\sqrt{k}\eta_{0})e^{-k|z-z_{0}|}\,d\lambda\,dk.$

###  3.3 The Riemann method of integration

The function $w(\xi,\zeta,\xi_{0},\zeta_{0})$ (as a function of $(\xi,\zeta)$)
is a solution of the partial differential equation (21) and it satisfies the
condition $w(\xi,\zeta,\xi_{0},\zeta_{0})=1$ if
$\xi-\xi_{0}=\pm(\zeta-\zeta_{0})$. This shows that $w$ is the Riemann
function of (21); for instance, see Garabedian (1986) [8].

The Riemann method of integration applied to the partial differential equation
(21) as in Volkmer (1980) [19] gives, for all
$\zeta,\xi_{0},\zeta_{0}\in{\mathbf{R}}$,

$\displaystyle
2u(\xi_{0},\zeta_{0})=u(\zeta-\zeta_{0}+\xi_{0},\zeta)+u(-\zeta+\zeta_{0}+\xi_{0},\zeta)$
(42)
$\displaystyle\int_{-\zeta+\zeta_{0}+\xi_{0}}^{\zeta-\zeta_{0}+\xi_{0}}\left[w(\xi,\zeta,\xi_{0},\zeta_{0})\partial_{2}u(\xi,\zeta)-\partial_{2}w(\xi,\zeta,\xi_{0},\zeta_{0})u(\xi,\zeta)\right]\,d\xi,$

where $u\in C^{2}({\mathbf{R}}^{2})$ is a solution of (21). This formula for
$\xi_{0}=\zeta=0$ and $u(\xi,\zeta)=u_{1}(\lambda,\xi)u_{2}(\lambda,\zeta)$
with $u_{1},u_{2}$ from (25), (26) (after replacing $\zeta_{0}$ by $\zeta$),
implies that

$\int_{-\zeta}^{\zeta}J_{0}({\textstyle\frac{1}{4}}(\xi^{2}-\zeta^{2}))u_{1}(\lambda,\xi)\,d\xi=2u_{2}(\lambda,\zeta).$
(43)

This equation allows us to transform the even solution $u_{1}$ into the odd
solution $u_{2}$ of equation (24). One can prove (43) directly by denoting the
left-hand side of (43) by $f(\zeta)$ and then showing that $f$ is an odd
solution of (24) with $f^{\prime}(0)=2$.

By differentiating (42) first with respect to $\xi_{0},\zeta_{0}$, and using
$w_{1}=u_{2}$, $w_{2}=u_{1}$, we obtain (after a lengthy calculation) that

$\int_{-\zeta}^{\zeta}\frac{\xi\zeta}{\xi^{2}-\zeta^{2}}J_{1}({\textstyle\frac{1}{4}}(\xi^{2}-\zeta^{2}))u_{2}(\lambda,\xi)\,d\xi=2u_{1}(\lambda,\zeta)-2u_{2}^{\prime}(\lambda,\zeta).$

This formula can also be proved directly.

Let $h:{\mathbf{R}}^{2}\to{\mathbf{R}}$ be the function defined by

$h(\xi,\zeta):=\left\\{\begin{array}[]{ll}\displaystyle
J_{0}\left({\textstyle\frac{1}{4}}(\xi^{2}-\zeta^{2})\right)&\quad\mathrm{if}\
|\xi|<|\zeta|,\\\\[5.0pt] \displaystyle
0&\quad\textup{otherwise.}\end{array}\right.$

For fixed $\zeta$ this is an even function in $L^{2}({\mathbf{R}})$ which can
be expanded according to Theorem 3.1 (without knowing $c_{j}(\lambda)$), so
that

${\rm
sign}(\zeta)h(\xi,\zeta)=2\int_{-\infty}^{\infty}u_{1}(\lambda,\xi)u_{2}(\lambda,\zeta)\rho^{\prime}_{1}(\lambda)\,d\lambda.$

If $\xi=0$, we obtain

${\rm
sign}(\zeta)J_{0}({\textstyle\frac{1}{4}}\zeta^{2})=2\int_{-\infty}^{\infty}u_{2}(\lambda,\zeta)\rho_{1}^{\prime}(\lambda)\,d\lambda.$

By Theorem 3.1, this formulas allows us to conclude

$\int_{0}^{\infty}J_{0}({\textstyle\frac{1}{4}}\zeta^{2})u_{2}(\lambda,\zeta)\,d\zeta=\frac{\rho_{1}^{\prime}(\lambda)}{\rho_{2}^{\prime}(\lambda)},$

and this is in agreement with (62).

##  4 Expansion of $J_{0}(kr)$ for elliptic cylinder coordinates

Consider equation (6) for $k>0$ and elliptic coordinates on the plane

$x=c\cosh\xi\cos\eta,\quad y=c\sinh\xi\sin\eta,$ (44)

where $\xi\in[0,\infty)$, $\eta\in{\mathbf{R}},$ and $c>0$. Transforming
$u(\xi,\eta)=U(x,y)$ we obtain

$\frac{\partial^{2}u}{\partial\xi^{2}}+\frac{\partial^{2}u}{\partial\eta^{2}}+k^{2}c^{2}(\cosh^{2}\xi-\cos^{2}\eta)u=0.$
(45)

Separating variables $u(\xi,\eta)=u_{1}(\xi)u_{2}(\eta),$ leads to

$\displaystyle-u_{1}^{\prime\prime}(\xi)+(\lambda-2q\cosh 2\xi)u_{1}(\xi)$
$\displaystyle=$ $\displaystyle 0,$ (46) $\displaystyle\hskip
14.22636ptu_{2}^{\prime\prime}(\eta)+(\lambda-2q\cos 2\eta)u_{2}(\eta)$
$\displaystyle=$ $\displaystyle 0,$ (47)

where $q=\frac{1}{4}c^{2}k^{2}>0$.

For $q\in{\mathbf{R}}$, Mathieu’s equation (47) is a Hill’s differential
equation with period $\pi$; see [17] or [14]. Here $q$ is positive but in the
next section $q$ will be negative. As a Hill’s equation, Mathieu’s equation
(47) admits nontrivial $2\pi$-periodic solutions if and only if $\lambda$ is
equal to one of its eigenvalues $a_{n}(q)$, $n\in{\mathbf{N}}_{0}$ or
$b_{n}(q)$, $n\in{\mathbf{N}}$. If $\lambda=a_{n}(q),$ then Mathieu’s equation
has an even $2\pi$-periodic solution ${\rm ce}_{n}(\eta,q)$, and if
$\lambda=b_{n}(q)$ then Mathieu’s equation has an odd $2\pi$-periodic solution
${\rm se}_{n}(\eta,q)$. These functions are normalized according to

$\int_{0}^{2\pi}{\rm ce}_{n}^{2}(\eta,q)\,d\eta=\int_{0}^{2\pi}{\rm
se}_{n}^{2}(\eta,q)\,d\eta=\pi.$

Moreover,

${\rm ce}_{n}(\eta+\pi,q)=(-1)^{n}{\rm ce}(\eta,q),\quad{\rm
se}_{n}(\eta+\pi,q)=(-1)^{n}{\rm se}(\eta,q).$

Note that all solutions of Mathieu’s equation are entire functions of $\eta$.

For these $2\pi$-periodic solutions of the Mathieu equation, we have the
following expansion theorem.

######  Theorem 4.1.

Let $f(z)$ be a $2\pi$-periodic function that is analytic in an open doubly-
infinite strip $S$ that contains the real axis. Then

$f(z)=\alpha_{0}{\rm ce}_{0}(z,q)+\sum_{n=1}^{\infty}(\alpha_{n}{\rm
ce}_{n}(z,q)+\beta_{n}{\rm se}_{n}(z,q)),$ (48)

where

$\alpha_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x){\rm
ce}_{n}(x,q)\,dx,\quad\beta_{n}=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x){\rm
se}_{n}(x,q)\,dx.$

The series (48) converges absolutely and uniformly on any compact subset of
the strip $S$.

Let $(x_{0},y_{0})$ and $(x,y)$ be points on ${\mathbf{R}}^{2}$ with distance
$r$. Let $(x_{0},y_{0})$, $(x,y)$ have elliptic coordinates
$(\xi_{0},\eta_{0})$ and $(\xi,\eta)$, respectively. Then

$\displaystyle
r^{2}=c^{2}\left[(\cosh\xi\cos\eta-\cosh\xi_{0}\cos\eta_{0})^{2}+(\sinh\xi\sin\eta-\sinh\xi_{0}\sin\eta_{0})^{2}\right].$
(49)

Clearly, $J_{0}(kr)$ as a function of $(\xi,\eta)$ solves (45). We substitute
$\zeta=i\xi$ and $\zeta_{0}=i\xi_{0}$. Then (45) changes to

$\frac{\partial^{2}u}{\partial\zeta^{2}}-\frac{\partial^{2}u}{\partial\eta^{2}}+k^{2}c^{2}(\cos^{2}\eta-\cos^{2}\zeta)u=0,$
(50)

and $J_{0}(kr)$ is transformed to

$w(\zeta,\eta,\zeta_{0},\eta_{0})=J_{0}(k\tilde{r}),$

where

$\displaystyle\tilde{r}^{2}$ $\displaystyle=$ $\displaystyle
c^{2}[(\cos\zeta\cos\eta-\cos\zeta_{0}\cos\eta_{0})^{2}-(\sin\zeta\sin\eta-\sin\zeta_{0}\sin\eta_{0})^{2}]$
$\displaystyle=$ $\displaystyle
c^{2}(\cos(\zeta-\eta)-\cos(\zeta_{0}-\eta_{0}))(\cos(\zeta+\eta)-\cos(\zeta_{0}+\eta_{0})).$

The function $w(\zeta,\eta,\zeta_{0},\eta_{0})$ is an analytic function on
${\mathbf{C}}^{4}$ and it solves equation (50) as a function of
$(\zeta,\eta)$. It is the Riemann function of this differential equation
because $w(\zeta,\eta,\zeta_{0},\eta_{0})=1$ if
$\zeta-\zeta_{0}=\pm(\eta-\eta_{0})$. For fixed $\eta,\zeta_{0},\eta_{0}$ we
wish to expand the function $\zeta\mapsto w(\zeta,\eta,\zeta_{0},\eta_{0})$ in
a series of Mathieu functions according to Theorem 4.1. To this end we have to
evaluate the integral

$\int_{-\pi}^{\pi}w(\zeta,\eta,\zeta_{0},\eta_{0}){\rm
ce}_{n}(\zeta,q)\,d\zeta,$

and a similar integral with ${\rm ce}_{n}$ replaced by ${\rm se}_{n}$. Using
Riemann’s method of integration applied to a pentagonal curve, it has been
shown in [20] that

$\displaystyle\int_{-\pi}^{\pi}w(\zeta,\eta,\zeta_{0},\eta_{0}){\rm
ce}_{n}(\zeta,q)\,d\zeta$ $\displaystyle=$ $\displaystyle\mu_{n}(q){\rm
ce}_{n}(\eta,q){\rm ce}_{n}(\zeta_{0},q){\rm ce}_{n}(\eta_{0},q),$ (51)
$\displaystyle\int_{-\pi}^{\pi}w(\zeta,\eta,\zeta_{0},\eta_{0}){\rm
se}_{n}(\zeta,q)\,d\zeta$ $\displaystyle=$ $\displaystyle\nu_{n}(q){\rm
se}_{n}(\eta,q){\rm se}_{n}(\zeta_{0},q){\rm se}_{n}(\eta_{0},q).$ (52)

There do not exist explicit formulas for the quantities $\mu_{n}(q)$ and
$\nu_{n}(q)$ but they can be determined as follows. Mathieu’s equation (47)
with $\lambda=a_{n}(q)$, $n\in{\mathbf{N}}_{0}$, has the solution
$u_{1}(\eta)={\rm ce}_{n}(\eta,q)$. We choose a second linear independent
solution $u_{2}(\eta)$. Then there is $\sigma$ such that

$u_{2}(\eta+\pi)=\sigma u_{1}(\eta)+(-1)^{n}u_{2}(\eta)$

and

$\mu_{n}(q)=\frac{2(-1)^{n}\sigma}{W[u_{1},u_{2}]},$ (53)

where $W[u_{1},u_{2}]$ denotes the Wronskian of $u_{1}$ and $u_{2}$.
Similarly, Mathieu’s equation (47) with $\lambda=b_{n}(q)$,
$n\in{\mathbf{N}}$, has the solution $u_{3}(\eta)={\rm se}_{n}(\eta,q)$. We
choose a second linear independent solution $u_{4}(\eta)$. Then there is
$\tau$ such that

$u_{4}(\eta+\pi)=\tau u_{3}(\eta)+(-1)^{n}u_{4}(\eta)$

and

$\nu_{n}(q)=\frac{2(-1)^{n}\tau}{W[u_{3},u_{4}]}.$ (54)

Now applying Theorem 4.1 and substituting $\zeta=i\xi$, $\zeta_{0}=i\xi_{0},$
we obtain the following result.

######  Theorem 4.2.

Let $\xi,\eta,\xi_{0},\eta_{0}\in{\mathbf{C}}$, and let $k>0$, $c>0$,
$q=\frac{1}{4}c^{2}k^{2}$. Then

$\displaystyle J_{0}(kr)=\frac{1}{\pi}\sum_{n=0}^{\infty}\mu_{n}(q){\rm
ce}_{n}(i\xi,q){\rm ce}_{n}(\eta,q){\rm ce}_{n}(i\xi_{0},q){\rm
ce}_{n}(\eta_{0},q)$ $\displaystyle\hskip
34.14322pt+\frac{1}{\pi}\sum_{n=1}^{\infty}\nu_{n}(q){\rm se}_{n}(i\xi,q){\rm
se}_{n}(\eta,q){\rm se}_{n}(i\xi_{0},q){\rm se}_{n}(\eta_{0},q),$

where $r$ is given by (49).

Theorem 4.2 agrees with expansion (23) (for $j=1$ and $\nu=0$), section 2.66
in Meixner & Schäfke [14], who have a slightly different notation. They use
${\rm me}_{n}(z,q)$, $n\in{\mathbf{Z}}$, where

$\displaystyle{\rm me}_{n}(z,q)$ $\displaystyle:=$ $\displaystyle\sqrt{2}{\rm
ce}_{n}(z,q)\quad\textup{if $n\in{\mathbf{N}}_{0},$}$ $\displaystyle{\rm
me}_{-n}(z,q)$ $\displaystyle:=$ $\displaystyle-\sqrt{2}i{\rm
se}_{n}(z,q)\quad\textup{if $n\in{\mathbf{N}}.$}$

Moreover, the coefficients $\mu_{n}(q)$ and $\nu_{n}(q)$ are represented in a
different form. The proof of Theorem 4.2 based on Riemann’s method of
integration appears to be new.

We now use (3) to obtain our final result in this section.

######  Theorem 4.3.

Let ${\bf x}$, ${\bf x}_{0}$ be points on ${\mathbf{R}}^{3}$ with elliptic
cylinder coordinates $(\xi,\eta,z)$ and $(\xi_{0},\eta_{0},z_{0})$,
respectively. Then

$\displaystyle\frac{1}{\|{\bf x}-{\bf
x}_{0}\|}=\frac{1}{\pi}\sum_{n=0}^{\infty}\int_{0}^{\infty}\mu_{n}(q){\rm
ce}_{n}(i\xi,q){\rm ce}_{n}(\eta,q){\rm ce}_{n}(i\xi_{0},q){\rm
ce}_{n}(\eta_{0},q)e^{-k|z-z_{0}|}\,dk$
$\displaystyle+\frac{1}{\pi}\sum_{n=1}^{\infty}\int_{0}^{\infty}\nu_{n}(q){\rm
se}_{n}(i\xi,q){\rm se}_{n}(\eta,q){\rm se}_{n}(i\xi_{0},q){\rm
se}_{n}(\eta_{0},q)e^{-k|z-z_{0}|}\,dk,$

where $q=\frac{1}{4}c^{2}k^{2}$.

##  5 Expansion of $K_{0}(kr)$ for elliptic cylinder coordinates

Consider equation (10) for $k>0$. Transforming to elliptic coordinates (44),
we obtain

$\frac{\partial^{2}u}{\partial\xi^{2}}+\frac{\partial^{2}u}{\partial\eta^{2}}-k^{2}c^{2}(\cosh^{2}\xi-\cos^{2}\eta)u=0.$
(55)

Separating variables $u(\xi,\eta)=u_{1}(\xi)u_{2}(\eta),$ leads again to (46),
(47) but now $q=-\frac{1}{4}c^{2}k^{2}$ is negative.

We will need the following solutions of the modified Mathieu equation (46)
when $q<0$; see [17, §28.20]. Set $q=-h^{2}$ with $h>0$. For
$n\in{\mathbf{N}}_{0}$, ${\rm Ie}_{n}(\xi,h)$ is the even solution of (46)
with $\lambda=a_{n}(q)$ with asymptotic behavior

${\rm Ie}_{n}(\xi,h)\sim I_{n}(2h\cosh\xi)\quad\textup{as $\xi\to+\infty$},$

while ${\rm Ke}_{n}(\xi,h)$ is the recessive solution determined by

${\rm Ke}_{n}(\xi,h)\sim K_{n}(2h\cosh\xi)\quad\textup{as $\xi\to+\infty$},$

where $I_{n}(z):=i^{-n}J_{n}(iz)$ and $K_{n}(z)$ are the modified Bessel
functions of the first [17, (10.27.6)] and second kinds respectively, with
integer order $n$ (see (4), (8)). Similarly, for $n\in{\mathbf{N}}$, ${\rm
Io}_{n}(\xi,h)$ is the odd solution of (46) with $\lambda=b_{n}(q)$ with
asymptotic behavior

${\rm Io}_{n}(\xi,h)\sim I_{n}(2h\cosh\xi)\quad\textup{as $\xi\to+\infty$},$

while ${\rm Ko}_{n}(\xi,h)$ is the recessive solution determined by

${\rm Ko}_{n}(\xi,h)\sim K_{n}(2h\cosh\xi)\quad\textup{as $\xi\to+\infty$}.$

For fixed $\xi,\xi_{0},\eta_{0}\in{\mathbf{R}}$ we wish to expand the function

$v(\xi,\eta,\xi_{0},\eta_{0}):=K_{0}(kr(\xi,\eta,\xi_{0},\eta_{0}))$

with $r$ given by (49) into a series of periodic Mathieu functions according
to Theorem 4.1. The corresponding integrals appearing in the expansion will be
computed based on the observation that $(\xi,\eta)\mapsto
v(\xi,\eta,\xi_{0},\eta_{0})$ is a fundamental solution of (55). In fact, it
is a solution of (55) and it has logarithmic singularities at the points
$\pm(\xi_{0},\eta_{0}+2m\pi)$, where $m$ is any integer. Arguing as in Volkmer
(1984) [21, Theorem 1.11], we have the following representation theorem for a
solution of (55).

######  Theorem 5.1.

Let $u\in C^{2}({\mathbf{R}}^{2})$ be a solution of (55). Let
$(\xi_{0},\eta_{0})\in{\mathbf{R}}^{2}$, and let $C$ be a a closed rectifiable
curve on ${\mathbf{R}}^{2}$ which does not pass through any of the points
$\pm(\xi_{0},\eta_{0}+2m\pi)$, $m\in{\mathbf{Z}}$. Let $n^{\pm}_{m}$ be the
winding number of $C$ with respect to $\pm(\xi_{0},\eta_{0}+2m\pi)$. Then we
have

$\displaystyle
2\pi\sum_{m}\left[n^{+}_{m}u(\xi_{0},\eta_{0}+2m\pi)+n^{-}_{m}u(-\xi_{0},-\eta_{0}-2m\pi))\right]$
$\displaystyle\hskip
22.76219pt=\int_{C}(u\partial_{2}v-v\partial_{2}u)\,d\xi+(v\partial_{1}u-u\partial_{1}v)\,d\eta,$

where $\partial_{1},\partial_{2}$ denote partial derivatives with respect to
$\xi$, $\eta$, respectively.

In Theorem 5.1 we choose

$u(\xi,\eta)=u_{1}(\xi)u_{2}(\eta),$

where

$u_{1}(\xi)={\rm Ke}_{n}(\xi,h),\quad u_{2}(\eta)={\rm ce}_{n}(\eta,q).$

Let $\xi_{0}>0$ and $\eta_{0}\in{\mathbf{R}}$. We take the curve $C$ to be the
positively oriented boundary of the rectangle $\xi_{1}\leq\xi\leq\xi_{2}$,
$\eta_{0}-\pi\leq\eta\leq\eta_{0}+\pi$, where $|\xi_{1}|<\xi_{0}<\xi_{2}$.
Consider the line integral $\int_{C}$ in Theorem 5.1. Since $u_{2}$ has period
$2\pi,$ the line integrals along the horizontal segments of $C$ cancel each
other. When $\xi_{2}\to+\infty$ the asymptotic behavior of $u_{1}(\xi)$ shows
that the integral along the right-hand vertical segment of $C$ tends to $0$ as
$\xi_{2}\to+\infty$. Therefore, setting

$f(\xi)=\int_{-\pi}^{\pi}v(\xi,\eta,\xi_{0},\eta_{0})u_{2}(\eta)\,d\eta=\int_{\eta_{0}-\pi}^{\eta_{0}+\pi}v(\xi,\eta,\xi_{0},\eta_{0})u_{2}(\eta)\,d\eta$

for $|\xi|<\xi_{0}$, one obtains

$2\pi
u_{1}(\xi_{0})u_{2}(\eta_{0})=u_{1}(\xi)f^{\prime}(\xi)-u_{1}^{\prime}(\xi)f(\xi).$
(56)

We now argue as in section 2. By differentiating (56) with respect to $\xi$,
we find that $f(\xi)$ satisfies the modified Mathieu equation (46). It is easy
to see that $f(\xi)$ is an even function, so $f(\xi)=cu_{3}(\xi)$, where
$u_{3}(\xi)={\rm Ie}_{n}(\xi,h)$ and $c$ is a constant. Then (56) implies that

$2\pi u_{1}(\xi_{0})u_{2}(\eta_{0})=cW[u_{1},u_{3}].$

Since $W[u_{1},u_{3}]=1$,

$\frac{1}{\pi}\int_{-\pi}^{\pi}v(\xi,\eta,\xi_{0},\eta_{0})u_{2}(\eta)\,d\eta=2u_{1}(\xi_{0})u_{2}(\eta_{0})u_{3}(\xi)\quad\textup{if
$|\xi|<\xi_{0}$}.$ (57)

By the same reasoning, we see that (57) is also true when $u_{1}(\xi)={\rm
Ko}_{n}(\xi,h)$, $u_{2}(\eta)={\rm se}_{n}(\eta,q)$, $u_{3}(\xi)={\rm
Io}_{n}(\xi,h)$.

Expanding the function $\eta\mapsto v(\xi,\eta,\xi_{0},\eta_{0})$ according to
Theorem 4.1, the following result is obtained.

######  Theorem 5.2.

Let $\xi,\eta,\xi_{0},\eta_{0}\in{\mathbf{R}}$ such that $|\xi|<\xi_{0}$, and
let $k>0$, $c>0,q=-\frac{1}{4}c^{2}k^{2}$, $h=\frac{1}{2}ck$. Then

$\displaystyle K_{0}(kr)$ $\displaystyle=$ $\displaystyle
2\sum_{n=0}^{\infty}{\rm Ie}_{n}(\xi,h){\rm ce}_{n}(\eta,q){\rm
Ke}_{n}(\xi_{0},h){\rm ce}_{n}(\eta_{0},q)$ $\displaystyle+$ $\displaystyle
2\sum_{n=1}^{\infty}{\rm Io}_{n}(\xi,h){\rm se}_{n}(\eta,q){\rm
Ko}_{n}(\xi_{0},h){\rm se}_{n}(\eta_{0},q),$

where $r$ is given by (49).

Theorem 5.2 agrees with Meixner & Schäfke (1954) [14, section 2.66] although
our notation and proof are different.

Inserting this result in (7), we obtain our final result.

######  Theorem 5.3.

Let ${\bf x}$, ${\bf x}_{0}$ be points on ${\mathbf{R}}^{3}$ with elliptic
cylinder coordinates $(\xi,\eta,z)$ and $(\xi_{0},\eta_{0},z_{0})$,
respectively. If $\xi_{\lessgtr}:={\min\atop\max}\\{\xi,\xi_{0}\\}$ then

$\displaystyle\frac{1}{\|{\bf x}-{\bf
x}_{0}\|}=\frac{4}{\pi}\sum_{n=0}^{\infty}\int_{0}^{\infty}{\rm
Ie}_{n}(\xi_{<},h){\rm ce}_{n}(\eta,q){\rm Ke}_{n}(\xi_{>},h){\rm
ce}_{n}(\eta_{0},q)\cos k(z-z_{0})\,dk$
$\displaystyle+\frac{4}{\pi}\sum_{n=1}^{\infty}\int_{0}^{\infty}{\rm
Io}_{n}(\xi_{<},h){\rm se}_{n}(\eta,q){\rm Ko}_{n}(\xi_{>},h){\rm
se}_{n}(\eta_{0},q)\cos k(z-z_{0})\,dk,$

where $q=-\frac{1}{4}c^{2}k^{2}$, $h=\frac{1}{2}ck$, and ${\bf x}\neq{\bf
x}_{0}$.

##  Appendix A Integrals for the (modified) parabolic cylinder harmonics
expansion of $J_{0}(kr)$

The following formulas are valid for $\Im\lambda<0$ :

$\displaystyle
I_{1}:=\int_{0}^{\infty}J_{0}\left({\textstyle\frac{1}{4}}\xi^{2}\right)u_{3}(\lambda,\xi)\,d\xi$
$\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\pi}(1-i)\frac{G_{1}}{G_{2}^{2}},$ (58)
$\displaystyle
I_{2}:=\int_{0}^{\infty}\xi^{-1}J_{1}\left({\textstyle\frac{1}{4}}\xi^{2}\right)u_{3}(\lambda,\xi)\,d\xi$
$\displaystyle=$
$\displaystyle-\sqrt{\pi}\left(\lambda\frac{1}{G_{2}}+2i\frac{G_{2}}{G_{1}^{2}}\right),$
(59)

where

$\displaystyle G_{1}$ $\displaystyle=$ $\displaystyle
G_{1}(\lambda)=\Gamma\left({\textstyle\frac{1}{4}}+{\textstyle\frac{i}{2}}\lambda\right),$
$\displaystyle G_{2}$ $\displaystyle=$ $\displaystyle
G_{2}(\lambda)=\Gamma\left({\textstyle\frac{3}{4}}+{\textstyle\frac{i}{2}}\lambda\right).$

We believe that these integrals may be known but do not have a reference. We
will derive (59). In (3.1) we use the integral representation [17, (13.4.4)]

$\Gamma(a)U(a,b,z)=\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt,\quad\Re
z,\Re a>0.$

Substituting $4s=\xi^{2}$ and changing the order of integration, one obtains

$I_{2}=\frac{1}{2G_{1}}\int_{0}^{\infty}t^{-\frac{3}{4}+\frac{i}{2}\lambda}(1+t)^{-\frac{3}{4}-\frac{i}{2}\lambda}\int_{0}^{\infty}s^{-1}J_{1}(s)e^{-is(2t+1)}\,ds\,dt.$
(60)

From [22, page 405], we have, for $t>0$,

$\displaystyle\int_{0}^{\infty}s^{-1}J_{1}(s)\cos(s(2t+1))\,ds$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\int_{0}^{\infty}s^{-1}J_{1}(s)\sin(s(2t+1))\,ds$
$\displaystyle=$ $\displaystyle t+(t+1)-2\sqrt{t}\sqrt{t+1}.$

Substituting these formulas in (60), we can evaluate $I_{2}$ using three times
the formula for the beta function [17, (5.12.3)]

$B(z,w)=\frac{\Gamma(z)\Gamma(w)}{\Gamma(z+w)}=\int_{0}^{\infty}t^{z-1}(1+t)^{-z-w}\,dt,\quad\Re
z,\Re w>0.$

This gives (59).

The proof of (58) is similar, but in (3.1) one should first use [17,
(13.2.40)]

$U(a,b,z)=z^{1-b}U(1+a-b,2-b,z).$

The formulas (58), (59) remain valid for real $\lambda$. By separating real
and imaginary parts, we obtain for $\lambda\in{\mathbf{R}}$,

$\displaystyle\int_{0}^{\infty}J_{0}({\textstyle\frac{1}{4}}\xi^{2})u_{1}(\lambda,\xi)\,d\xi$
$\displaystyle=$
$\displaystyle\frac{\Re(G_{1}\overline{G}_{2})+\Im(G_{1}\overline{G_{2}})}{|G_{2}|^{2}},$
(61)
$\displaystyle\int_{0}^{\infty}J_{0}({\textstyle\frac{1}{4}}\xi^{2})u_{2}(\lambda,\xi)\,d\xi$
$\displaystyle=$
$\displaystyle{\textstyle\frac{1}{2}}\left|\frac{G_{1}}{G_{2}}\right|^{2},$
(62)
$\displaystyle\int_{0}^{\infty}\xi^{-1}J_{1}({\textstyle\frac{1}{4}}\xi^{2})u_{1}(\lambda,\xi)\,d\xi$
$\displaystyle=$ $\displaystyle
2\left|\frac{G_{2}}{G_{1}}\right|^{2}-\lambda,$ (63)
$\displaystyle\int_{0}^{\infty}\xi^{-1}J_{1}({\textstyle\frac{1}{4}}\xi^{2})u_{2}(\lambda,\xi)\,d\xi$
$\displaystyle=$ $\displaystyle
2\frac{\Re(G_{1}\overline{G}_{2})+\Im(G_{1}\overline{G_{2}})}{|G_{1}|^{2}},.$
(64)

We may use

$\Re(G_{1}\overline{G}_{2})+\Im(G_{1}\overline{G_{2}})=\frac{\pi\sqrt{2}e^{-\frac{1}{2}\pi\lambda}}{\cosh(\pi\lambda)}.$

Formulas (61), (64) give us the integrals (35), (36) noting that we integrate
even functions in (35), (36).

## Acknowledgements

Part of this work was conducted while H. S. Cohl was a National Research
Council Research Postdoctoral Associate in the Information Technology
Laboratory at the National Institute of Standards and Technology,
Gaithersburg, Maryland, U.S.A.

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