Harmonic Functions and the Schwarz Problem on ⅅ

Abstract:

Let us recall that a harmonic function u on a domain D of the complex plane ℂ is a real-valued function in C²(D) such that

for all z = (x, y) in D. From Exercise (9) in Chapter 10, we see that the real and imaginary parts of a holomorphic function on D are harmonic on D. This connection provides great insight into the study of harmonic functions by means of holomorphic functions. Some features of this connection are given in this chapter in the context of the unit disk ⅅ…