# What is the importance of harmonic functions

## harmonious function

*harmonic mapping,* one in an open crowd *D.* ⊂ & Copf; defined function *u* : D → & reals; which satisfies the Laplace equation. The following must apply more precisely:

*u*is in*D.*twice continuously real differentiable, d. H. all second partial derivatives of*u*exist in*D.*and are there steadily.- In
*G*applies.

This equation is also called Laplace's equation or potential equation, and harmonic functions are also called potential functions.

In addition to the complex formulation given here, there is also a real definition of the harmonic function (i.e., & Copf; is replaced by & reals;^{2} replaced), as well as obvious generalizations in the & reals;* ^{n}* For

*n*> 2. However, in the latter case, no more functional theory methods can be used.

The amount of all in *D.* harmonic functions forms a complex vector space with the pointwise scalar multiplication and addition of functions, which contains the constant functions. The definition can be extended to open sets \ (D \ subset \ hat {\ mathbb {C}} \). Is ∞ ∈ *D.*, Is called *u* harmoniously with ∞, if there is one in an environment *U* of 0 harmonic function *u*^{*} gives such that *u**(*z*) = *u*(1/*z*) for all *z* ∈ *U* \ {0}.

Is *f* one in *D.*holomorphic function, so are *u* : = Re *f* and *v* : = Im *f* harmonic functions in *D.* This immediately gives you many examples of harmonic functions. For example, the functions are all harmonious in & Copf ;.

Is *u* harmonious in *D.*, *f* one in an open set D * ⊂ & Copf; holomorphic function with *f*(*D.**) ⊂ *D.* and *u**(*z*) := *u*(*f*(*z*)) For *z* ∈ *D.**, so is *u** harmonious in *D.*^{*}.

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