1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

114 CHAPTER 3 • ANALYTIC AND HARMONIC FUNCTIONS

• EXAMPLE 3.11 If u (x, y) = x^2 - y^2 , then u.,,, (x,y) +Uvv (x,y) = 2-2 = O;
  hence u is a harmonic function for all z. We find t hat v ( x, y) = 2xy is also a
  harmonic function and that

u,, =Vy= 2x and Uy= -v., = -2y.

Therefore, v is a harmonic conjugate of u, and the function J given by

J(z) = x^2 - y^2 +i2xy = z^2

is an analytic function.

Theorem (3.8) makes the constrnction of harmonic functions from known
analytic functions an easy task.

•EXAMPLE 3.12 The function f (z) = z^3 = x^3 - 3xy^2 + i (3x^2 y - y^3 ) is
analytic for all values of z; hence it follows that

u (x,y) =Re(! (z)] = x^3 - 3xy^2

is harmonic for all z and that

v (x, y) =Im [f (z)) = 3x^2 y - y^3

is a harmonic conjugate of u (x, y).

Figures 3.2 and 3.3 show the graphs of these two functions. The partial
derivatives are u., (x,y) = 3x^2 - 3y^2 , u 11 (x,y) = -6xy, v., (x,y) = 6xy, and
Vy ( x, y) = 3x^2 - 3y^2. They satisfy the Cauchy- Riemann equations because they

Figure 3.2 u (x, y) = x^3 - 3xy^2 • Figure 3.3 v (x,y) = 3x^2 y-y^3.