Relation between formulations in Laplace's equation and Helmholtz's equation

We can consider Laplace's equation to be the special case Helmholtz's equation as k tends to 0. Therefore one expects that the FMM formulation for Laplace's equation is obtained as the limit of $k \rightarrow 0$ in the FMM formulation for Helmholtz's equation. However, the formulation obtained above does not allow such operation in the form as it is. It is because the behaviours of $j_n(k\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert)$ and $h_n(k\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert)$ near k=0 are O(k^-n-1) and O(kⁿ) and, hence,

$$\begin{eqnarray*}\lim_{k\rightarrow0}{\cal I}_n^m(k,\overrightarrow{O\mbox{\bold... ...\cal O}_n^m(k,\overrightarrow{O\mbox{\boldmath$ x $ }}) = \infty \end{eqnarray*}$$

In order to obtain a formulation allowing an obvious transition between these governing equations we need some modifications to cancel these behaviours.

Noting the power series of $j_n(k\vert\overrightarrow{O\mbox{\boldmath$\space y $ }}\vert)$ and $h_n(k\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert)$, we obtain the following formulation

 

$$\frac{e^{ik R}}{4 \pi R} =\frac{1}{4 \pi}\sum_{n=0}^{\infty}\sum_... ...boldmath$ y $ }})\hat{{\cal O}}_n^m(k,\overrightarrow{O\mbox{\boldmath$ x $ }})$$ (3.134)

where $\hat{{\cal I}}_n^m(k,\overrightarrow{O\mbox{\boldmath$\space y $ }})$ and $\hat{{\cal O}}_n^m(k,\overrightarrow{O\mbox{\boldmath$\space x $ }})$ are defined as

$$\begin{eqnarray*}\hat{{\cal I}}_n^m(k,\overrightarrow{O\mbox{\boldmath$ y $ }})=... ... x $ }}\vert) S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}). \end{eqnarray*}$$

Since

$$\begin{eqnarray*}\lim_{k\rightarrow0} \hat{{\cal I}}_n^m(k,\overrightarrow{O\mbo... ...math$ x $ }})=S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}), \end{eqnarray*}$$

the expression in (3.135) is seen to approach (3.120) as $k \rightarrow 0$.