VI. Two-dimensional potential problems concerning a single closed boundary

Solutions of Laplace’s equation, ∂2 V/∂x 2 + ∂2 V/∂y 2 + ∂2 V/∂z 2 = 0 . . . . . (1. 11) are required in many branches of Applied Mathematics, such as hydrodynamics, electro-and magneto-statics, steady flow of heat or electricity, etc. The two-dimensional form of the equation, ∂2 V/∂x 2 + ∂2 V/∂y 2 = 0, . . . . (1. 12) has a general solution V =f (x + ɩy ) + F (x – ɩy ), . . . (1. 21)f and F being arbitrary functions of their complex arguments. In the applications, one function alone is usually sufficient, and it is customary to writew =ϕ + ɩψ =f (z ). . . . . (1. 22) withz =x + ɩy , whenϕ and ψ usually have each some physical significance. Moreover, in most cases, the boundary conditions which have to be satisfied either are, or can be reduced to, the prescription of the boundary values ofϕ or ψ, of their derivatives.