Harmonic Functions in Complex Analysis

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Harmonic Functions
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;
Heat transfer
Electrostatic potential
Gravitational potential
Fluid flow
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 Princip…