Complex Analysis - Maths KU

168 Chapter 3 Analytic Functions
y
C
D
Find φ so that
∇ 2 φ = 0 in D
Figure 3.9 Dirichlet problem
φ = k 0
y
∇ 2 φ = 0
x
and φ=
g
on C
–1 1
φ = k 1
Figure 3.10 Figure for Example 2
x
Table 3.1 summarizes some of the applications of the complex potential
function Ω(z) and the names given to the level curves
φ(x, y) =c1 and ψ(x, y) =c2.
Application Level curves φ (x, y) =c1 Level curves ψ (x, y) =c2
electrostatics equipotential curves lines of force
fluid flow equipotential curves streamlines of flow
gravitation equipotential curves lines of force
heat flow isotherms lines of heat flux
Table 3.1 Complex potential function Ω(z) =φ(x, y)+iψ(x, y)
Dirichlet Problems A classical and important problem in applied
mathematics involving Laplace’s equation is illustrated in Figure 3.9 and put
into words next.
Dirichlet Problem
Suppose that D is a domain in the plane and that g is a function defined
on the boundary C of D. The problem of finding a function φ(x, y), which
satisfies Laplace’s equation in D and which equals g on the boundary C
of D is called a Dirichlet problem.
Such problems arise frequently in the two-dimensional modeling of electrostatics,
fluid flow, gravitation, and heat flow.
In the next example we solve a Dirichlet problem. Although the problem
is quite simple, its solution will aid us in the solution of another problem in
Section 4.5.
EXAMPLE 2 A Simple Dirichlet Problem
Solve the Dirichlet problem illustrated in Figure 3.10. The domain D is a
vertical infinite strip defined by −1