Any real-valued function with continuous second Partial Derivatives which satisfies
Laplace's Equation

(1) |

To find a class of such functions in the Plane, write the Laplace's Equation in Polar Coordinates

(2) |

(3) |

(4) |

(5) |

(6) |

(7) |

(8) |

(9) |

(10) |

(11) |

Other solutions may be obtained by differentiation, such as

(12) | |||

(13) |

(14) | |||

(15) |

and

(16) |

(17) | |||

(18) |

The Poisson Kernel

(19) |

**References**

Ash, J. M. (Ed.) *Studies in Harmonic Analysis.* Washington, DC: Math. Assoc. Amer., 1976.

Axler, S.; Pourdon, P.; and Ramey, W. *Harmonic Function Theory.* Springer-Verlag, 1992.

Benedetto, J. J. *Harmonic Analysis and Applications.* Boca Raton, FL: CRC Press, 1996.

Cohn, H. *Conformal Mapping on Riemann Surfaces.* New York: Dover, 1980.

© 1996-9

1999-05-25