Introduction

In order to proceed further, we must review the rotational effect of a conformal mapping w = f(z) at a point . If the contour C has the parameterization z(t) = x(t) + i y(t), then a vector tangent to C at the point is

The image of C is a contour K given by

and a vector tangent to K at the point is

If the angle of inclination of is , then the angle of inclination of is

Hence the angle of inclination of the tangent to C at is rotated through the angle to obtain the angle of inclination of the tangent to K at the point .

Theorem 10.6. (Schwarz-Christoffel Formula) Let P be a polygon in the w plane with vertices and exterior angles , where , for . There exists a one-to-one conformal mapping w=f(z) from the upper half plane onto G that satisfies the boundary conditions

The derivative is
and the function f(z) can be expressed as an indefinite integral,

where A and B are suitably chosen constants. Two of the points may be chosen arbitrarily, and the constants A and B determine the size and position of P.

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(c) John Mathews, 1998, 2006