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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.

Assuming the curves are smooth the proof of the theorem is trivial: We know from 18.02 that the gradient ruis orthogonal to the level curves of uand the same is true for rvand the level curves of v. Since, by Lemma 5.4, the gradients are orthogonal this implies the curves are orthogonal. Finally, we show that f0(z) 6= 0 means the curves are ...

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