## 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).