# Harmonic functions.

Definition 3.4.1   Let be a function of two real variables and , defined over a domain . Suppose that has second order partial derivatives on . The function is called an harmonic function if it verifies the equation:

Example 3.4.2   Take . Then over we have:

It follows that over , i.e. is harmonic .

Now suppose that is analytic in a neighborhood of . Moreover suppose that the partial derivatives of and are differentiable and that the second partial derivatives are continuous functions on . From Cauchy-Riemann equations follows:

As the second partial derivatives are continuous on , we have and . It follows that:

 and

The computations that we performed before Definition 4.1 can be summarized in a theorem:

Theorem 3.4.3   If is a function of a complex variable, analytic over a domain , then and are harmonic over .

Example 3.4.4   Let . The function is an entire function, as we proved previously. With and , we have:

On the one hand, e have:

On the other hand, we have:

The functions and are both harmonic.

Definition 3.4.5   Let be a function of two real variables, harmonic over a domain . Let be a function of two real variables, defined over , and such that is analytic over . Then is called an harmonic conjugate of .

Example 3.4.6   Let . This is a polynomial function, thus it has partial derivatives of any order.
(i)
The function is harmonic:

(ii)
If there exists such that is analytic over , then and verify the Cauchy-Riemann equations (v.s. section 3):

where and are independent of and respectively. As these two formulas define the same function, and must be constant, i.e. , where .
(iii)
Finally, we have:

Please compare this with example 2.1.

We can now discover another important property of the analytic conjugates.

Theorem 3.4.7   Let , as usual. Suppose that is analytic over some domain . Then the level curves of are orthogonal to the level curves of .

Proof. Use Cauchy-Riemann equations and show that the gradients of and are orthogonal, whence the result.

Example 3.4.8   Take . With the usual algebraic notation, we have and .

For general , then equation defines an equilateral hyperbola, whose symmetry axes are the coordinate axes (red curves in Figure 2), and the equation defines an equilateral hyperbola whose asymptotes are the coordinates axes (blue curves in Figure 2). For , we have the union of the angle bisectors of the coordinate axes and the union of the coordinate axes respectively.

Proof.

Noah Dana-Picard 2007-12-24