SOLUTION: show that the following function ( the real part of f(z))are harmonic and find its corresponding v( x,y ) :

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: show that the following function ( the real part of f(z))are harmonic and find its corresponding v( x,y ) :       Log On


   

Question 351313: show that the following function ( the real part of f(z))are harmonic and find its corresponding v( x,y ) :


e^x *( cos y ) ,


ln (squar x + squar y ) ,


y /squar ( 1 - x ) squar y


what hamonic is mean mathematiclly ?
plz explain>>>
thanks
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Harmonic means that a function satisfies the Laplace equation, fxx%2Bfyy=0.
(fx is the partial derivative of f with respect to x, fxx is partial derivative of fx with respect to x)
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f%28x%2Cy%29=e%5Ex%2Acos%28y%29
fx=e%5Ex%2Acos%28y%29
fxx=e%5E%28x%29%2Acos%28y%29
fy=e%5Ex%28-sin%28y%29%29
fyy=-e%5E%28x%29%2Asin%28y%29
fxx%2Bfyy=e%5Ex%2Acos%28y%29-e%5E%28x%29%2Acos%28y%29=0, so the function is harmonic.
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f%28x%2Cy%29=ln%28x%5E2%2By%5E2%29
fx=2x%2Flog%28x%5E2%2By%5E2%29
fxx=2%2F%28x%5E2%2By%5E2%29-4x%5E2%2F%28x%5E2%2By%5E2%29%5E2
fy=2y%2Flog%28x%5E2%2By%5E2%29
fyy=2%2F%28x%5E2%2By%5E2%29-4y%5E2%2F%28x%5E2%2By%5E2%29%5E2
fxx%2Bfyy=4%2F%28x%5E2%2By%5E2%29-4%28x%5E2%2By%5E2%29%2F%28x%5E2%2By%5E2%29%5E2
fxx%2Bfyy=4%2F%28x%5E2%2By%5E2%29-4%2F%28x%5E2%2By%5E2%29
fxx%2Bfyy=0
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I leave the last one for you to finish.
Follow the same procedure.