Complex Analysis - Maths KU

7.5 Applications
7.5 Applications 429
In this section 7.5 we revisit the method introduced in Section 4.5 for solving Dirichlet problems;
here we incorporate the new mappings defined in this chapter or in Appendix III. We
also describe a similar process for solving a new type of boundary-value problem that relies
on finding a conformal mapping between two domains. This allows us to investigate more
complicated boundary-value problems arising in the two-dimensional modeling of electrostatics
and heat flow. We conclude this section with an application of conformal mapping
to the problem of finding an irrotational flow of an incompressible fluid, that is, the flow of
an ideal fluid, in a region of the plane.
7.5.1 Boundary-Value Problems
Dirichlet Problems Revisited Suppose that D is a domain in the
z-plane and that g is a function defined on the boundary C of D. The problem
of finding a function φ(x, y) that satisfies Laplace’s equation ∇ 2 φ =0,or
∂2φ ∂x2 + ∂2φ =0, (1)
∂y2 in D and that equals g on the boundaryof D, is called a Dirichlet problem.
In Section 4.5, we saw that analytic functions can be used to solve certain
Dirichlet problems. We obtained a solution of a Dirichlet problem in a domain
D byfinding an analytic mapping of D onto a domain D ′ in which the
associated, or transformed, Dirichlet problem can be solved. That is, we
found a mapping w = f(z) ofD onto D ′ such that f(z) =u(x, y)+iv(x, y)
is analytic in D. ByTheorem 4.5, if Φ(u, v) is a solution of the transformed
Dirichlet problem in D ′ , then φ(x, y) =Φ(u(x, y), v(x, y)) is a solution of
the Dirichlet problem in D. Thus, our method presented in Section 4.5 for
solving Dirichlet problems consisted of the following four steps:
• Find an analytic mapping w = f(z) =u(x, y)+iv(x, y) of the domain
D onto a domain D ′ ,
• transform the boundaryconditions from D to D ′ ,
• solve the transformed Dirichlet problem in D ′ , and
• set φ(x, y) =Φ(u(x, y), v(x, y)).
For a more detailed discussion of these steps refer to Section 4.5 and Figure
4.19.
In this chapter we investigate a number of topics that can help complete
these four steps. The table of conformal mappings discussed in Section 7.1,
the linear fractional transformations studied in Section 7.2, and the Schwarz-
Christoffel transformation of Section 7.3 provide a valuable source of mappings
to use in Step 1. In addition, if D ′ is taken to be either the upper half-plane
y>0 or the open unit disk |z| < 1, then the Poisson integral formulas of
Section 7.4 provide a means to determine a solution of the associated Dirichlet
problem in D ′ .