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6: Harmonic Functions

Last updated

Sep 5, 2021
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- 5.5: Amazing consequence of Cauchy’s integral formula
- 6.2: Harmonic Functions

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Harmonic functions appear regularly and play a fundamental role in math, physics and engineering. In this topic we’ll learn the definition, some key properties and their tight 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.

6.2: Harmonic Functions
We start by defining harmonic functions and looking at some of their properties.
6.3: Del notation
Here’s a quick reminder on the use of the notation ∇ .
6.4: A second Proof that u and v are Harmonic
This fact that u and v are harmonic is important enough that we will give a second proof using Cauchy’s integral formula. One benefit 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 fact.
6.5: Maximum Principle and Mean Value Property
6.6: Orthogonality of Curves

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