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) |
(6) | |||
(7) |
(8) | |||
(9) |
(10) | |||
(11) |
(12) |
(13) |
(14) | |||
(15) |
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