322 Chapter6
every harmonic complex function f is analytic, since not any two solutions
of the Laplace equation can be taken as the components u and v of an
analytic function in A, for they must be related by the Cauchy-Riemann
equations u., =Vy and Uy = -v.,. Whenever v is related to u by the pre-
ceding equations it is said that vis a harmonic conjugate of u. However, if
v is a harmonic conjugate of u, in general u is not a harmonic conjugate of
v. In fact, a harmonic conjugate of vis -u, as follows from -if= v - iu
(Exercises 6.1, problem 16a).
Given one of the components u, v (necessarily a harmonic function) on a
simply connected region R, the harmonic conjugate can be determined by
solving the system u., = vy, Uy = -v.,. If, for instance, u is given, we have
dv = v.,dx + vydy = -uydx + u.,dy (6.6-4)
the last expression being an exact differential since (-uy)y ( u.,).,, or
Uxx + Uyy = 0. Hence, v can be found by standard methods of integration.
An explicit formula is -
l
(x,y)
v(x, y) = (-uy dx + u., dy)
(xo,yo)(6.6-5)where the line integral is to be evaluated along any rectifiable arc contained
in Rand joining the fixed point (x 0 , Yo) to the variable point (x, y). Since
the choice of the fixed point ( x 0 , y 0 ) is arbitrary, it is clear that v is deter-
mined up to an arbitrary real constant. It follows that given a harmonic
function u(x,y) in a simply coi'.inected region R, it can always be regarded
as the real part of a function f = u +iv analytic in R, which is determined
up to an arbitrary purely imaginary constant.
If the region R is multiply connected, the line integral in (6.6-5) in
general takes different values for different arcs connecting the points ( x 0 , y 0 )
and (x,y), and so it leads to a multiple-valued conjugate v(x,y), as well
as to a multiple-valued function f = u + iv, except when the path of
integration is restricted to a simply connected subregion of R. Hence if
a determination of v is chosen then the function f = u + iv is locally
analytic in R, i.e., analytic in some neighborhood of each point of R (see
Example 2 belmy). Two different determinations or branches of f differ
by a pure imaginary constant, so all branches have the same derivative,
namely, f' = u., +iv., at every point of R, this derivative being single-valued
in R.
Examples 1. Let u = x^2 + x - y^2 • We have
u., = 2x + 1, Uy= -2y, Uxx = 2, Uyy = -2