Complex Analysis - Maths KU

434 Chapter 7 Conformal Mappings
y
1
0.5
0
–0.5
–1
–1 –0.5 0 0.5 1
x
Figure 7.51 Equipotential curves and
lines of force for Example 2
the equipotential curves and lines of force, respectively. The Mathematicagenerated
plot in Figure 7.51 shows the equipotential curves in color and the
lines of force in black.
Neumann Problems Theorem 7.1 states that an analytic mapping is
conformal at a point where the derivative is nonzero. This fact did not appear
to be of immediate importance in previous examples when we solved Dirichlet
problems, but it is extremelyimportant in another class of boundary-value
problems associated with Laplace’s equation called Neumann problems.
Neumann Problem
Suppose that D is a domain in the plane and that h is a function defined on
the boundary C of D. The problem of finding a function φ(x, y) that satisfies
Laplace’s equation in D and whose normal derivative dφ/dn equals
h on the boundary C of D is called a Neumann problem.
Certain types of Neumann problems occur naturally in the study of electrostatics,
fluid flow, and heat flow. For example, consider the problem of
determining the steady-state temperature φ in a domain D with boundary
C. If the temperatures on the boundary C of D are specified, then we have
a Dirichlet problem. However, it mayalso be the case that all or part of
the boundaryis insulated. This means that there is no heat flow across the
boundary, and, it can be shown that this implies that the directional derivative
of φ in the direction of the normal vector n to C is 0. We call this derivative
the normal derivative and denote it by dφ/dn. In summary, an insulated
boundarycurve in a heat flow problem corresponds to a boundarycondition of
the form dφ/dn = 0, and, thus, is an example of a Neumann problem. As the
following theorem asserts, conformal mappings preserve boundaryconditions
of the form dφ/dn =0.
Theorem 7.8 Preservation of Boundary Conditions
Suppose that the function f(z) =u(x, y)+iv(x, y) is conformal at every
point of a smooth curve C. Let C ′ be the image of C under w = f(z). If
the normal derivative dΦ/dN of the function Φ(u, v) satisfies
dΦ
dN =0
at everypoint on C ′ in the w-plane, then the normal derivative dφ/dn of
the function φ(x, y) =Φ(u(x, y), v(x, y)) satisfies
at everypoint of C in the z-plane.
dφ
dn =0