Complex Analysis - Maths KU

164 Chapter 3 Analytic Functions
L 1
L 2
(x 0 , y 0 )
3.4 Applications
22. In this problem you are guided through the start ofthe proofofthe proposition:
If u(x, y) is a harmonic function and v(x, y) is its harmonic conjugate,
then the function φ(x, y) =u(x, y)v(x, y) is harmonic.
Proof Suppose f(z) =u(x, y)+iv(x, y) is analytic in a domain D. We saw
in Section 3.1 that the product oftwo analytic functions is analytic. Hence
[f(z)] 2 is analytic. Now examine [f(z)] 2 and finish the proof.
In Section 3.3 we 3.4 saw that if the function f(z) =u(x, y)+iv(x, y) is analytic in a domain D,
then the real and imaginary parts of f are harmonic; that is, both u and v have continuous
second-partial derivatives and satisfy Laplace’s equation in D:
∂2u ∂x2 + ∂2u = 0 and
∂y2 ∂2v ∂x2 + ∂2v =0. (1)
∂y2 Conversely, if we know that a function u(x, y) is harmonic in D, we can find a unique (up
to an additive constant) harmonic conjugate v(x, y) and construct a function f(z) that is
analytic in D.
In the physical sciences and engineering, Laplace’s partial differential equation is often
encountered as a mathematical model of some time-independent phenomenon, and in that
context the problem we face is to solve the equation subject to certain physical side conditions
called boundary conditions. See Problems 11–14 in Exercises 3.4. Because of the
link displayed in (1), analytic functions are the source of an unlimited number of solutions
of Laplace’s equation, and we may be able to find one that fits the problem at hand. See
Sections 4.5 and 7.5. This is just one reason why the theory of complex variables is so
essential in the serious study of applied mathematics.
We begin this section by showing that the level curves of the real and imaginary parts
of an analytic function f(z) =u(x, y)+iv(x, y) are two orthogonal families of curves.
u(x, y) = u 0
v(x, y) = v 0
Figure 3.3 Tangents L1 and L2 at
point of intersection z0 are
perpendicular.
Orthogonal Families Suppose the function f(z) =u(x, y)+iv(x, y)
is analytic in some domain D. Then the real and imaginary parts of f can be
used to define two families of curves in D. The equations
u(x, y) =c1 and v(x, y) =c2, (2)
where c1 and c2 are arbitrary real constants, are called level curves of u and
v, respectively. The level curves (2) are orthogonal families. Roughly, this
means that each curve in one family is orthogonal to each curve in the other
family. More precisely, at a point of intersection z0 = x0 + iy0, where we shall
assume that f ′ (z0) �= 0, the tangent line L1 to the level curve u(x, y) =u0
and the tangent line L2 to the level curve v(x, y) =v0 are perpendicular.
See Figure 3.3. The numbers u0 and v0 are defined by evaluating u and v
at z0, that is, c1 = u(x0, y0) = u0 and c2 = v(x0, y0) = v0. To prove
that L1 and L2 are perpendicular at z0 we demonstrate that the slope of one