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Find a function that is harmonic on the washer-shaped region between the circles with radii 1 and 2 and center (1,i). It should be 0 and 10 on the inner and outer circle.

Guess the answer \phi (x,y)=A{\mathrm  {Log}}|z-(1+i)|+B\,.

At the point i\,, A{\mathrm  {Log}}|-1|+B=0\implies B=0\,

At the point 1-i\,, A{\mathrm  {Log}}|-2i|=10\implies A={\frac  {10}{{\mathrm  {Log}}2}}\,

So the answer is

\phi (z)={\frac  {10}{{\mathrm  {Log}}2}}{\mathrm  {Log}}|z-(1+i)|\,.

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