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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.(1)Equation 1 is called Laplace s equation.

connection to complex analysis. The key connection to 18.04 is that both the real and imaginary parts of analytic functions are harmonic. We will see that this is a simple consequence of the Cauchy-Riemann equations. In the next topic we will look at some applications to hydrodynamics. 5.2 Harmonic functions

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