Harmonic Functions in Complex Analysis
If Ω denotes a set that is open in Ɍ3, then the real valued function u x, y, z on Ω characterized by a continuous second partials is declared harmonic if, and only if the Laplace ∆u=0 is identical on Ω. This is when the Laplace u is shown as
∆u= ∂2u ∂x2+ ∂2u∂y2+ ∂2u∂z2In the case of an open set Ω in Ɍ3, then a similar definition can be made. u will therefore, be harmonic only if;
∆u= ∂2u ∂x2+ ∂2u∂y2=0 on Ω.
The following are basic examples of harmonic functions;
u= x2+ y2- 2z2, Ω = Ɍ3u= 1r Ω = Ɍ3- (0, 0, 0) where r= x2+ y2+z2 .
More to this, considering Ω, an open set in Ɍ3, the real part of the analytic function is always harmonic. Therefore, a function like u= rncosnθ is harmonic because u is real in zn.Applications of harmonic functions
Harmonic equations are very important and widely used by physicists and mathematicians in the analysis of various physical situations. Examples of them include;
The following are some of the properties characterized with harmonic functions;
The Maximum Principle
This describes the basic result that characterize different forms of harmonic functions. It says that if u is harmonic function on Ω, and B is a region bounded and closed in Ω, then the maximum, and also minimum, of u on B is assumed at the boundary of B. From the Weak Maximum Principle, suppose the harmonic function u x, y, z is contained in Ω, then an isolated critical point of u in Ω cannot give a relative maxima or minima of u. More specifically, if a harmonic function on Ω only contains isolated critical points, then E is a closed solid region in Ω, the absolute maxima or minima of u on E occur on the boundary of E.
The Heat Equation
This is an interesting application of the maximum principle. It explains that if T (x, y, z,t) is a function of temperature on the domain Ω, then a constant c exists such that;
∂T∂t=c∆x, y,zT where ∆x, y,zT= ∂2T ∂2x+ ∂2T∂2y+ ∂2T∂2zThis is a derivation from the Gauss’s theorem. Suppose a function of temperature T (x, y, z,t) is at ∂T∂t=0 i.e. at steady state. This occurs at the point where temperature, T is independent of time, t. The heat equation, at the steady state, implies that the Laplace of T satisfies ∆T = 0. The result of the maximum principle for the steady state temperature function T, on Ω is, therefore, that the coldest and hottest points on the subset B of Ω which is closed and bounded, will occur on the boundary of B. this is a practical application of the Maximum Principle. A pan, for instant, that has been sitting on fire for some time is definitely hotter at the skillet compared to the handle which is at the furthest point.
The Average Value Property
This property states that assuming u is a harmonic function which is defined on an open set Ω, then the value of u at the center of p of a ball or disk in Ω is the average vale of u on the boundary of the ball or disk. More specific, from the average value property it is seen that if u is a harmonic on Ω С Ɍ2, p ϵ Ω and Br is a closed disk in Ω centered at p.
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Sibelska, Agnieszka. “On The Geometrical Properties of Some Classes of Complex Harmonic Functions Defined by Analytic or Coefficient Conditions with Complex Parameter”. Journal of Applied Analysis 20.1 (2014): Web resource.