Finding Harmonic function
Find harmonic function in cartesian coordinates u(x,y) which satisfies in polar coordinates the next condition:
u(rcos(0),rsin(0))=u(rcos(pi/4),rsin(pi/4))=a
given that:
u(x0,y0)=2a for 0<y0<x0
Answer
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Can you maybe let me in on how you came up with this function?
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We need a harmonic function that is zero on the x-axis and on {y=x}. When harmonic functions are zero on two lines, there is some "reflection principle" that implies that they must be zero also on further lines that are found by reflecting one line on the other one (and this can be iterated). So in our case it must also be zero on the y axis and on {y=-x}. So I tried x y (x-y)(x+y) = x y (x²-y²) = x³ y - x y³ which is easily seen to be harmonic (because of the "antisymmetry" in (x,y)).
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("We need" => rather: "I wanted"...)
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( I started from the idea to write the function as a sum u1 + u2 where u1 = a everywhere and u2 = 0 on the axes, and equal to a (so that u1+u2 = 2a) in (x0,y0). Sorry if I was a bit brief in the solution.)
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I edited the solution to add these explanations. I can try to find a reference for the "reflection principle" if you are interested (but you can probably find it through Google.)
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