What is motivation to study Harmonic functions? Recall that a function $f:U (\subset \mathbb R^n) \to \mathbb R$ is called harmonic if $\Delta f=0$ where $\Delta$ is Laplacian operator.

What is a good motivation to study harmonic functions?


The harmonic functions are important for various reasons:
1) They are locally the real part of holomorphic functions, hence providing useful examples in complex analysis
2)They can be decomposed into series involving sines and cosines (Fourier Series), thus providing food for thought for fourier analysis.
3) They enjoy many properties of Anaytic Functions-Maximum Principke, Liouville's Theorem, Harnack inequality, Mean value property, Regularity property etc.
4) They have useful applications in Mathematical Physics, Stochastic Processes and Electrical Engineering as they can be used to model acoustic phenomena or similar phenomena involving waves or Harmonics
