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Math Help - (Easy?) Boundary Value Problem

  1. #1
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    (Easy?) Boundary Value Problem

    Hi,

    I have to solve a boundary value problem that involves finding function \varphi that is harmonic on an annulus centered at 1+i in the plane and takes the value zero on the inner boundary (radius 1) and ten on the outer (radius 2.) Once in possession of this function, I have to evaluate it at (0,0). The solutions I was given don't give the function but just the value of phi (1/2.) I'm not getting this answer; I tried setting up a system of equations with the solution form \varphi =A+\ln |z-(1+i)| +B but kept getting A=\frac{10}{\ln 2}, B=0 I'd appreciate at least an understanding of why their answer is right. Also, the solution should involve elementary methods (no crazy transforms please.) Thanks!
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  2. #2
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    Quote Originally Posted by mstrfrdmx View Post
    Hi,

    I have to solve a boundary value problem that involves finding function \varphi that is harmonic on an annulus centered at 1+i in the plane and takes the value zero on the inner boundary (radius 1) and ten on the outer (radius 2.) Once in possession of this function, I have to evaluate it at (0,0). The solutions I was given don't give the function but just the value of phi (1/2.) I'm not getting this answer; I tried setting up a system of equations with the solution form \varphi =A+\ln |z-(1+i)| +B but kept getting A=\frac{10}{\ln 2}, B=0 I'd appreciate at least an understanding of why their answer is right. Also, the solution should involve elementary methods (no crazy transforms please.) Thanks!
    Presumably you mean \varphi(z) =A\ln |z-(1+i)| +B. Your answer looks correct to me, giving \varphi(0) = 5. Are you sure that the book says 10, not 1, for the value on the outer boundary? If it really says 10, then my guess is that there's a misprint in the book.
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