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Find a function that is harmonic on the washer-shaped region between the circles |z|=1,|z|=2\, and takes the values 20 and 30 on the inner and outer circle.

For washer-shaped regions, guess \phi (x,y)=A{\mathrm  {Log}}|z|+B\, where {\mathrm  {Log}}\, is the principal value.

Then these equations must hold:

\phi (1,0)=A{\mathrm  {Log}}1+B=20\implies B=20\,

\phi (2,0)=A{\mathrm  {Log}}2+B=30\implies A={\frac  {10}{{\mathrm  {Log}}2}}\,


\phi (x,y)={\frac  {10{\mathrm  {Log}}|z|}{{\mathrm  {Log}}2}}+20\,

For a washer centered at z_{0}\,, guess the function \phi (x,y)=A{\mathrm  {Log}}|z-z_{0}|+B\,.

Main Page : Complex Variables : Exponential and Log