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February 3rd, 2007, 07:38 PM  #1 
Newbie Joined: Nov 2006 From: Cubs Town Posts: 5 Thanks: 0  Partial Differential Equation
Verify that this function is everywhere harmonic. (e)^x Cos(y) 
February 3rd, 2007, 09:44 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 21,124 Thanks: 2332 
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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differential, equation, partial 
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