1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

116 CHAP'l'ER 3 • ANALYTIC AND HARMONIC FUNCTI ONS

Technically we should always specify the domain of function when defining
it. When no such specification is given, it is assumed that the domain is the
entire complex plane, or the largest set for which the expression defining the
function makes sense.

• EXAMPLE 3.13 Show that u (x, y) = xy^3 - x^3 y is a harmonic function, and
  find a conjugate harmonic function v (x, y).

Solution We follow the construction process of Theorem 3.9. The first partial

derivatives are

(3-28)

To verify that u is harmonic, we compute the second partial derivatives and note

that Uxx (x, y) + Uyy (x, y) = -6xy + 6xy = 0, sou satisfies Laplace's Equation

(3-26). To construct v, we start with Equation (3-27) and the first of Equations

(3-28) to get

J

1 3

v(x,y) = (y^3 - 3x^2 y) dy+C(x) =

4

y^4 -

2

x^2 y^2 + C(x).

Differentiating the left and right sides of this equation with respect to x and

using - Uy (x, y) = v., (x,y) and Equations (3-28) on the left side yield

-3xy^2 +x^3 = 0-3xy^2 +C' (x),

which implies that

Finally, if we integrate this equation, we get

Harmonic functions arise as solutions to many physical problems. Applica-

tions include two-dimensional models of heat flow, electrostatics, and fluid flow.

We now give an example of the latter.

We assume that an incompressible and frictionless fluid flows over the com-

plex plane and that all cross sections in planes parallel to the complex plane are