1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

3.3 • HARMONIC FUNCTIONS 113

is known as Laplace's equation (sometimes referred to as the pot ential equa-
tion). If</>, </>x, </>in </>xx, </>xy, </>yx. a.nd w a.re a.II continuous, a.nd if</> (x, y) satis-
fies Laplace's equation, then (x,y) is harmonic on D. Harmonic functions are
important in applied mathematics, engineering, and mathematical physics. They
are used to solve problems involving steady state temperatures, two-dimensional
electrostatics, and ideal fluid flow. In Chapter 11 we describe how complex anal-
ysis techniques can be used to solve some problems involving harmonic functions.
We begin with an important theorem relating analytic and harmonic functions.

If we have a function tt (x, y) that is harmonic on the domain D and if we

can find another harmonic function v (x, y) such that the partial derivatives for

tt and v satisfy the Cauchy- Riemann equations throughout D, then we say that

v(x,y) is a harmonic conjugate ofu(x,y). It then follows t hat the function

f (z) = u(x,y) +iv(x,y) is analytic on D.