By letting , the Real and Imaginary Parts of must
satisfy the Cauchy-Riemann Equations and Laplace's Equation, so they automatically provide a scalar
Potential and a so-called stream function. If a
physical problem can be found for which the solution is valid, we obtain a solution--which may have been very difficult
to obtain directly--by working backwards. Let

(1) |

(2) | |||

(3) |

For ,

(4) | |||

(5) |

which is a double system of Lemniscates (Lamb 1945, p. 69). For ,

(6) | |||

(7) |

This solution consists of two systems of Circles, and is the Potential Function for two Parallel opposite charged line charges (Feynman

(8) | |||

(9) |

gives the field near the edge of a thin plate (Feynman

(10) | |||

(11) |

This is two straight lines (Lamb 1945, p. 68). For ,

(12) |

(13) |

(14) | |||

(15) |

These are two Perpendicular Hyperbolas, and is the Potential Function near the middle of two point charges or the field on the opening side of a charged Right Angle conductor (Feynman 1989, §7-3).

**References**

Feynman, R. P.; Leighton, R. B.; and Sands, M. *The Feynman Lectures on Physics, Vol. 1.*
Redwood City, CA: Addison-Wesley, 1989.

Lamb, H. *Hydrodynamics, 6th ed.* New York: Dover, 1945.

© 1996-9

1999-05-26