Question

There are some theorems in harmonic function theory that resemble results in complex analysis, like:

• Holomorphic functions and complex functions are analytic;
• Cauchy's integral formula in complex analysis and the mean value theorem in harmonic function theory;
• The principle of maximum and minimum that works for harmonic and holomophic functions.
• The real and imaginary parts of a holomorphic function are harmonic;

These results suggest that there are connections between these two areas and I would like to ask: how can each of these theories be used to develop the other?

PS: I'm really sorry for my really bad English.

Answer 1

The Cauchy Reimann Equations imply that every holomorphic function satisfies Laplace's Equation and is therefore its real and imaginary components are harmonic.

You can also take a harmonic function u and construct, up to a constant, its harmonic conjugate v so that u and v satisfy the Cauchy Reimann Equations. Thus u+iv is a holomorphic function.

These proofs can be found in just about any Complex Analysis book. My favorite is Complex Analysis by Lars Ahlfors.

So in a sense a harmonic function is just the real component of a holomorphic function.