Consider the following harmonic function satisfying the Laplace equation in cylindrical coordinates, i.e., $\Delta f = 0$. $$ f(r,z) = \frac{z-h}{\left( r^2 + (z-h)^2 \right)^{\frac{3}{2}}} \, , $$ where $r$ and $z$ are the radial and axial coordinates in the system of cylindrical coordinates $\left(r,\phi,z\right)$, with $\phi$ denoting the azimuthal angle.
The goal is to express the harmonic function $f(r,z)$ as infinite series in terms of the eigensolutions for bipolar coordinates of the form $$ f(r,z) \stackrel{?}{=} \left( \cosh\xi - \cos\eta \right)^\frac{1}{2} \sum_{n \ge 0} \left( \alpha_n \sinh \left( \left( n+\frac{1}{2}\right) \xi \right) +\beta_n \cosh \left( \left( n+\frac{1}{2}\right) \xi \right) \right) P_n(\cos\eta) \, , $$ wherein the system of toroidal coordinates $\left(\xi, \eta, \phi\right)$, where $\xi \in [0, \infty)$ and $\eta, \phi \in [0, 2\pi)$. Note that $\phi$ is common in both systems of coordinates. In addition, $\alpha_n$ and $\beta_n$ are unknown series coefficients to be determined, and $P_n$ denotes Legendre polynomial of degree $n$.
It is worthwhile noting that the series representation above also satisfies the Laplace equation in bipolar coordinates.
Toroidal coordinates are related to cylindrical coordinates by the transformations $(r, z) = \left( c \sinh \xi/\Lambda, c \sin\eta /\Lambda\right)$ where $\Lambda = \cosh \xi - \cos\eta$ and $c=1$ is a geometric constant. The azimuthal angle $\phi$ is common to both systems of coordinates.
Any help of hint are highly appreciated.
Thank you!
Daddy
While identifying those coefficients might be a challenging task as the expansion does not involve orthogonal functions, transforming the hyperbolic functions into trigonometric functions using the substitution $\xi = i \zeta$ might perhaps help to solve your problem.