Poisson's Equation in Spherical Coordinates

in spherical coordinates. According to Section 2.3, the general three-dimensional Green's function for Poisson's equation is

(329) |

When expressed in terms of spherical coordinates, this becomes

(330) |

where

(331) |

is the angle subtended between and . According to Equation (298), we can write

(332) |

for , and

(333) |

for . Thus, it follows from the spherical harmonic addition theorem, (322), that

where represents the lesser of and , whereas represents the greater of and .

According to Section 2.3, the general solution to Poisson's equation, (329), is

(335) |

Thus, Equation (335) yields

where

(337) | ||

(338) |