324 Chapter6
so that v is multiple-valued. If we choose the principal value of the
argument in the subregion Ri = {z: z + Jzl =f; O}, we have
v=Argz+C
and we obtain the analytic function
f(z) = %ln(x^2 + y^2 ) + i Argz + iC
= ln JzJ + iArgz + iC= Logz+iC
in R 1. This example shows that given a harmonic function in a certain
region R, it is not always possible to find a single-valued harmonic conjugate
in the same set. However, f(z) can be made locally analytic in R by using
a suitable determination of v. For instance, in the example above suppose
that a point z 0 is given on the negative real axis. Let Bo be an angle other
than (2k+ l)?T and define the single-valued function arg 0 z to be argz with
the restriction B 0 :::; arg z < B 0 + 271". Then
f(z) = ln JzJ + i arg 0 z + iC
is analytic at zo.
Another method. Given the function u( x, y) harmonic in a neighborhood
of some point (x 0 , y 0 ), it is possible to compute by a simple formula, and
without integration, the function f(z), analytic in some neighborhood of
zo = xo + iy 0 , which has u(x,y) as its real part. For a justification of
the method we need to anticipate a result to be established in Chapter 8,
namely, that an analytic function at z 0 has a power series representation
of the form z:::=o an(z - zo)n valid in some disk Jz - zoJ < r. This result
implies that both u( x, y) and v( x, y) possess Taylor series representations
in powers of x - xo and y - Yo valid in the same disk. It follows that the
functions u(x,y) and v(x,y) will have a meaning when the real variables
are replaced by complex variables and, in particular, when we let
1 1
x=xo+ 2(z-zo)=
2(z+zo)
1 1Y = Yo + 2 i ( z - zo) = 2 i ( z - zo) (6.6-7)
provided that Jz-z 0 J < r. Since the substitution (6.6-7) implies x+iy = z,
we obtain the identity
(6.6-8)