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February 3rd, 2007, 07:38 PM   #1
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Partial Differential Equation

Verify that this function is everywhere harmonic.
(e)^x Cos(y)
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February 3rd, 2007, 09:44 PM   #2
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We need to differentiate the function twice with respect to x (treating y as though it's a constant), then start again and differentiate twice with respect to y (treating x as though it's a constant).

In the process, we note that our results are valid everywhere (the functions used are differentiable everywhere).

The resulting second derivatives are easily found and are (e^x) cos(y) and -(e^x) cos(y).

Since they are defined for all values of x and y and their sum is zero, the original function is everywhere harmonic.

That was easy, wasn't it?
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