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        general >> complex analysis >>  Harmonic Functions
        
(Message started by: Icarus on Nov 18^th, 2002, 5:54pm)

Title: Harmonic Functions
Post by Icarus on Nov 18^th, 2002, 5:54pm

Finally, William has posted something in this forum! :)

As to why analytic functions are trippy in general, they are all harmonic functions: In order for a complex function to have a derivative, its real and imaginary parts must be harmonic: if f(z) = u(x,y) + iv(x,y), where z = x + iy, then

[partial]_x²u + [partial]_y²u = 0

and

[partial]_x²v + [partial]_y²v = 0

If f(z) = r(x, y)e^{i[theta](x,y)}, then the same equations are true of r and [theta]:

[partial]_x²r + [partial]_y²r = 0
[partial]_x²[theta] + [partial]_y²[theta] = 0

The thing is, if you have a harmonic function u, then you can find the appropriate harmonic function v so that u + iv is analytic. And given r or [theta], you can find the other so that re^i[theta] is analytic.

So really your trippy complex analysis graphs are just trippy harmonic function graphs. Of course this begs the question as to why harmonic functions are so trippy. But I'm going to leave my thoughts on this until some other people have had their say.

[e]Edited to replace the "@" symbols I used originally with the math symbolry William later added, and to remove my gripe about the lack of such symbolry! ;)[/e]

Title: Re: Harmonic Functions
Post by pjay on Dec 6^th, 2003, 11:54am

everyone has a different take on this but i would say that harmonic functions give rise to wonderful graphs since they are exactly the set of functions which satisfy the mean-value property: the property that f(z) is the average value of
f(z+rw) where w lies on the unit circle and r>0 (we have to make sure this circle lies in the domain of f). Notice that on R^1 these are exactly the set of all lines. So we can see immediately that this property causes the functions to behave in a very special way. One application of this is that suppose we fix the value of all temperatures on the boundary of a nice domain so that the fixed temperature is a continuous function on the boundary. then the equilibrium temperature of points inside the domain (the temperature of a given point after waiting a long time) will be exactly the harmonic function that extends continuously to the fixed boundary temperature function (imagine a heater controlled by a thermostat at each point of the boundary). Anyways, that example was meant to give the reader an intuitive reason as to why harmonic functions give rise to nice graphs. BTW, the above is a desrciption of the heat equation when we set the time derivative to 0 (this happens in equilibrium).