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:

  1. u is in D. twice continuously real differentiable, d. H. all second partial derivatives of u exist in D. and are there steadily.
  2. 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 zU \ {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 zD.*, so is u* harmonious in D.*.