1. (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!

2. Originally Posted by mstrfrdmx
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.