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Mathematics LibreTexts

6.2: Harmonic Functions

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We start by defining harmonic functions and looking at some of their properties.

Definition: Harmonic Functions

A function u(x,y) is called harmonic if it is twice continuously differentiable and satisfies the following partial differential equation:

2u=uxx+uyy=0.

Equation ??? is called Laplace’s equation. So a function is harmonic if it satisfies Laplace’s equation. The operator 2 is called the Laplacian and 2u is called the Laplacian of u.


This page titled 6.2: Harmonic Functions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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