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5 Introduction to harmonic functions

Topic 5 NotesJeremy Orloff5 Introduction to harmonic IntroductionHarmonic functions appear regularly and play a fundamental role in math, physics andengineering. In this topic we ll learn the definition, some key properties and their tightconnection to complex analysis. The key connection to is that both the real andimaginary parts of analytic functions are harmonic . We will see that this is a simpleconsequence of the cauchy -Riemann equations. In the next topic we will look at someapplications to harmonic functionsWe start by defining harmonic functions and looking at some of their functionu(x,y) is called harmonic if it is twice continuously differen-tiable and satisfies the following partial differential equation: 2u=uxx+uyy= 0.

5.4 A second proof that u and v are harmonic This fact is important enough that we will give a second proof using Cauchy’s integral formula. One bene t of this proof is that it reminds us that Cauchy’s integral formula can transfer a general question on analytic functions to a question about the function 1=z. We start with an easy to derive ...

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