3.3 • HARMONIC FUNCTIONS 115are the real and imaginary parts of an analytic function. At the point ( x, y) =
(2, -1), we have u"' (2,-1) = v 11 (2, -1) = 9, and these partial derivatives appear
along the edges of the surfaces for u and v where x = 2 and y = - 1. Similarly,
1.1 11 (2,-1) = 12 and v"' (2, -1) = -12 also appear along the edges of the surfaces
for u and v where x = 2 iμld y = -1.
We can use complex analysis to show easily that certain combinations of
harmonic funct ions a.re harmonic. For example, if v is a harmonic conjugate of
u, then their product If> (x,y) = u (x,y) v (x, y) is a harmonic function. This con-
dition can be verified directly by computing the partial derivatives and showing
that Equation (3-26) holds, but the details are tedious. If we use complex vari-
able techniques instead, we can start with the fact that f (z) = u (x, y) +iv (x, y)
is an analytic function. Then we observe that the square of f is also an analytic
function, which is
[/ (z)j^2 = [u(x,y)j^2 - (v(x,y)]^2 +i211(x,y)v(x,y).We then know immediately that the imaginary part, 2u(x,y)v(x, y) , is a
harmonic function by Theorem 3.8. A constant multiple of a harmonic function
is harmonic, so it follows that <P is harmonic. We leave as an exercise to show
that if 111 and u2 are two harmonic functions that are not related in the preceding
fashion, then their product need not be harmonic.