In Chapter 17 we developed the geometric approach of conformal mapping. This meant that, for a complex analytic function w = f(z) defined in a domain D of the z-plane, we associated with each point in D a corresponding point in the w-plane. This gave us a conformal mapping (angle-preserving), except at critical points where f′(z) = 0.
Now, in this chapter, we shall apply conformal mappings to potential problems. This will lead to boundary value problems and many engineering applications in electrostatics, heat flow, and fluid flow. More details are as follows.
Recall that Laplace's equation ∇2Φ = 0 is one of the most important PDEs in engineering mathematics because it occurs in gravitation (Secs. 9.7, 12.11), electrostatics (Sec. 9.7), steady-state heat conduction (Sec. 12.5), incompressible fluid flow, and other areas. The theory of this equation is called potential theory (although “potential” is also used in a more general sense in connection with gradients (see Sec. 9.7)). Because we want to treat this equation with complex analytic methods, we restrict our discussion to the “two-dimensional case.” Then Φ depends only on two Cartesian coordinates x and y, and Laplace's equation becomes
An important idea then is that its solutions ...