Next: Series expansion of the Up: Applications of Fast Multipole Previous: L2L translation formula

# Series expansion of |x-y|

We introduce two vectors and which satisfy the inequality: . An expansion for the fundamental solution of three-dimensional Helmholtz's Equation[17] is given by

 (F.1)

where and are unit vectors defined as

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

and jn and yn are expressed in Taylor (Laurent) series as

 jn(kz) = (F.2) yn(kz) = (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 .
• the coefficient of k in the left-hand side of (F.1)
The left-hand side of (F.1) is expanded into a power series of wave number as
 (F.4)

and therefore the coefficient of k in the left-hand side of (F.1) is

 (F.5)

• the coefficient of k in the right-hand side of (F.1)
To begin with, we note that the coefficient of kin is zero. The coefficient of k in is

Hence the coefficient of k in the right-hand side of (F.1) is

 (F.6)

Comparing (F.5) with (F.6), we obtain a series expansion of given as

 = (F.7) = (F.8)

where and are functions defined as

 (F.9)

Hayami and Sauter [41] obtained the same series expansion of in another way.

Next: Series expansion of the Up: Applications of Fast Multipole Previous: L2L translation formula
Ken-ichi Yoshida
2001-07-28