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Series expansion of |x-y|

We introduce two vectors $\overrightarrow{O\mbox{\boldmath$\space y $ }}$ and $\overrightarrow{O\mbox{\boldmath$\space x $ }}$ which satisfy the inequality: $\vert\overrightarrow{O\mbox{\boldmath$\space y $ }}\vert
< \vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert$. An expansion for the fundamental solution of three-dimensional Helmholtz's Equation[17] is given by
 
$\displaystyle \frac{e^{ik{\scriptstyle \vert\mbox{\boldmath$ x $ }-\mbox{\boldm...
...\boldmath$ x $ }}\vert) \overline{R_{n,m}}(\widehat{Oy}) S_{n,m}(\widehat{Ox}),$     (F.1)

where $\widehat{Ox}$ and $\widehat{Oy}$ are unit vectors defined as

\begin{eqnarray*}\quad \widehat{Ox} = \frac{\overrightarrow{O\mbox{\boldmath$ x ...
...th$ y $ }}}{\vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert},
\end{eqnarray*}


jn is the spherical Bessel function, hn(1) is defined with the spherical Bessel functions jn and yn as

\begin{eqnarray*}h_n^{(1)}(k\vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert) ...
...t) + i y_n(k\vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert),
\end{eqnarray*}


and jn and yn are expressed in Taylor (Laurent) series as
  
jn(kz) = $\displaystyle (2 k z)^n \sum_{p=0}^{\infty} \frac{(-1)^p (n+p)!}{p! (2n+2p+1)!}(k z)^{2 p},$ (F.2)
yn(kz) = $\displaystyle -\frac{1}{2^n( k z)^{n+1}} \sum_{p=0}^{\infty} \frac{\Gamma(2 n - 2 p + 1)}{p! \Gamma(n - p + 1)}(k z)^{2 p}.$ (F.3)

We next compare the coefficient of k in the left-hand side of (F.1) with that in the right-hand side of (F.1) to obtain a series expansion of $\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$. The left-hand side of (F.1) is expanded into a power series of wave number as
$\displaystyle \frac{e^{ik{\scriptstyle \vert\mbox{\boldmath$ x $ }-\mbox{\boldm...
... + \frac{ik}{2}\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert + \ldots$     (F.4)

and therefore the coefficient of k in the left-hand side of (F.1) is
 
$\displaystyle \frac{i\vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert}{2}.$     (F.5)

To begin with, we note that the coefficient of kin $j_n(k\vert\overrightarrow{O\mbox{\boldmath$\space y $ }}\vert)j_n(k\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert)$ is zero. The coefficient of k in $j_n(k\vert\overrightarrow{O\mbox{\boldmath$\space y $ }}\vert)y_n(k\vert\overrightarrow{O\mbox{\boldmath$\space x $ }}\vert)$ is

\begin{eqnarray*}&&-\frac{\vert\overrightarrow{O\mbox{\boldmath$ y $ }}\vert^n}{...
...vert\overrightarrow{O\mbox{\boldmath$ x $ }}\vert^{n-1}}\right).
\end{eqnarray*}


Hence the coefficient of k in the right-hand side of (F.1) is
 
    $\displaystyle \sum_{n=0}^{\infty}\sum_{m=-n}^{n} \frac{i}{2}\left( \frac{1}{2n+...
...$ }}\vert^{n-1}}\right) \overline{R_{n,m}}(\widehat{Oy}) S_{n,m}(\widehat{O x})$  
    $\displaystyle = \frac{i}{2}\sum_{n=0}^{\infty}\sum_{m=-n}^{n}\left(
\frac{S_{n,...
...}})\overline{R_{n,m}}(\overrightarrow{O\mbox{\boldmath$ y $ }})}{2n-1} \right).$ (F.6)

Comparing (F.5) with (F.6), we obtain a series expansion of $\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$ given as
 
$\displaystyle \vert\mbox{\boldmath$ x $ }-\mbox{\boldmath$ y $ }\vert$ = $\displaystyle \sum_{n=0}^{\infty}\sum_{m=-n}^{n}\left(
\frac{S_{n,m}(\overright...
... }})\overline{R_{n,m}}(\overrightarrow{O\mbox{\boldmath$ y $ }})}{2n-1} \right)$ (F.7)
  = $\displaystyle \sum_{n=0}^{\infty}\sum_{m=-n}^{n}\left(
\frac{\overline{S_{n,m}}...
...dmath$ x $ }})R_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }})}{2n-1} \right),$ (F.8)

where $U_{n,m}(\overrightarrow{O\mbox{\boldmath$\space y $ }})$ and $T_{n,m}(\overrightarrow{O\mbox{\boldmath$\space x $ }})$ are functions defined as
 
$\displaystyle U_{n,m}(\overrightarrow{O\mbox{\boldmath$ y $ }}) = \vert\overrig...
...box{\boldmath$ x $ }}\vert^2 S_{n,m}(\overrightarrow{O\mbox{\boldmath$ x $ }}).$     (F.9)

Hayami and Sauter [41] obtained the same series expansion of $\vert\mbox{\boldmath$\space x $ }-\mbox{\boldmath$\space y $ }\vert$ in another way.
next up previous contents
Next: Series expansion of the Up: Applications of Fast Multipole Previous: L2L translation formula
Ken-ichi Yoshida
2001-07-28