A function *f* of a complex variable *z* = *x* + *iy* can be factored into real and imaginary parts:

where *x* and *y* are real numbers, and *u* and *v* are real-valued functions of two real values.

Suppose you are given *u*(*x*, *y*) and you want to find *v*(*x*, *y*). The function *v* is called a harmonic conjugate of *u*.

## Finding *v*

If *u* and *v* are the real and imaginary parts of an analytic function, then *u* and *v* are related via the Cauchy-Riemann equations. These are first order differential equations that one could solve to find *u* given *v* or *v* given *u*. The approach I’ll present here comes from [1] and relies on algebra rather than differential equations.

The main result from [1] is

So given an expression for *u* (or *v*) we evaluate this expression at *z*/2 and *z*/2*i* to get an expression for *f*, and from there we can find an expression for *v* (or *u*).

This method is simpler in practice than in theory. In practice we’re just plugging (complex) numbers into equations. In theory we’d need to be a little more careful in describing what we’re doing, because *u* and *v* are not functions of a complex variable. Strictly speaking the right hand side above applies to the extensions of *u* and *v* to the complex plane.

## Example 1

Shaw gives three exercises for the reader in [1]. The first is

We find that

We know that the constant term is purely imaginary because *u*(0, 0) = 0.

Then

and so

is a harmonic conjugate for *u* for any real number β.

The image above is a plot of the function *u* on the left and its harmonic conjugate *v* on the right.

## Example 2

Shaw’s second example is

We begin with

and so

From there we find

## Example 3

Shaw’s last exercise is

Then

This leads to

from which we read off

## Related posts

[1] William T. Shaw. Recovering Holomorphic Functions from Their Real or Imaginary Parts without the Cauchy-Riemann Equations. SIAM Review, Dec., 2004, Vol. 46, No. 4, pp. 717–728.