## Conformal Solution

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)

the Real and Imaginary Parts then give
 (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 et al. 1989, §7-5; Lamb 1945, p. 69). For ,
 (8) (9)

gives the field near the edge of a thin plate (Feynman et al. 1989, §7-5). For ,
 (10) (11)

This is two straight lines (Lamb 1945, p. 68). For ,
 (12)

gives the field near the outside of a rectangular corner (Feynman et al. 1989, §7-5). For ,
 (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).

See also Cauchy-Riemann Equations, Conformal Map, Laplace's Equation

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 Eric W. Weisstein
1999-05-26