Laplace's Equation in Spherical Coordinates

(322) |

in spherical coordinates. Let us write

(323) |

It follows from Equation (308) that

(324) |

However, given that the spherical harmonics are mutually orthogonal [in the sense that they satisfy Equation (311)], we can separately equate the coefficients of each in the above equation, to give

(325) |

for all and . It follows that

(326) |

where the and are arbitrary constants. Hence, the general solution to Laplace's equation in spherical coordinates is written

(327) |

If the domain of solution includes the origin then all of the must be zero, in order to ensure that the potential remains finite at . On the other hand, if the domain of solution extends to infinity then all of the (except ) must be zero, otherwise the potential would be infinite at .