ABSTRACT
This section provides some physical motivation for the study of conformal mapping that follows. Its purpose is to establish the connection between a conformal mapping and the solution of what is called a two-dimensional boundary value problem for the Laplace equation, which is studied later in some detail. We start by recalling that a two-dimensional harmonic function f(x, y) in a domain D of the (x, y)-plane satisfies the Laplace equation
(5.1)
Let w f(z) u iv be an analytic function that maps a domain D in the z-plane one-one and conformally onto a domain D∗ in the w-plane. Let us examine what happens to the Laplace Equation (5.1) if the Cartesian coordinates (x, y) are changed to the differentiable curvilinear coordinates variables u u(x, y) and v v(x, y) in the function f(z) u iv, when f(x, y) becomes the function (u, v) f(u(x, y), v(x, y)). From the chain rule for differentiation, we have
(5.2)
and a further differentiation with respect to x gives
(5.3)
∂ ∂
∂ ∂
∂ ∂
∂ ∂
f
x x u u x
∂ ∂
∂ ∂
u u
x
2 ∂
∂ ∂ ∂
∂ ∂
x v v x
∂ ∂
∂ ∂
v v
x
∂ ∂
∂ ∂
∂ ∂
∂ ∂
∂ ∂
f
x u u x v
v x
,
f f f
( , ) .x y x y
∂ ∂
∂ ∂
Examination of Equation (5.2) shows that the operation of differentiation of f with respect to x is related to the operation of differentiation of (u, v) with respect to u and v by the linear operator
(5.4)
In terms of this operator in Equation (5.3) becomes
(5.5)
The corresponding expression for ∂2f/∂y2 follows directly from Equation (5.5) by replacing x by y whenever it occurs in a partial derivative.
