Question

Define Harmonic function with an example.

Answer 1

$<font color="red"><h1>HELLO FRIEND</h1></font> <b><font color="green"><marquee>Here is your answer.</marquee></font></b><hr size=3 color="blue"> <b>Harmonic function </b> In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R where U is an open subset of R^ⁿthat satisfies Laplace's equation, i.e. everywhere on U.  <hr size=3 color="green"><font color=#800080> <b>HOPE IT HELPS<BR>PLEASE MARK IT AS A BRAINLIEST ANSWER<BR>PLEASE FOLLOW ME</b><BR><BR></font><font color="red"><b>About me: </b><I>Shivam shaurya$

Answer 2

A Harmonic function is basically a function that is continuous and also it’s first and second derivatives are continuous over a certain domain (eg. −π−π to ππ), or in other words that he function can satisfy Laplace's equation:

∇2f=0∇2f=0

It’s called harmonic because in general periodic functions (which are related to waves… hence describe harmonic motion) do satisfy the Laplace Equation.

There are many applications for harmonic functions.

For example:

They can satisfy the Wave equation, hence describe waves.

The Schrödinger equation being also a wave equation in essence, also requires harmonic functions, since to have an eigenfunction of the system it must me twice differentiable and continuous within the boundaries.

In electromagnetism the charge density as described by the Poisson's equation also must in have an harmonic function.

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