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Find a function that is harmonic on the wedge-shaped region between the rays in the complex plane with principal argument 3\pi /4,5\pi /4\,.

Guess \phi (z)=A{\mathrm  {arg}}_{0}z+B\, where {\mathrm  {arg}}_{0}\, is a branch measured from 0 to 2\pi so that there are no discontinuities in the region of interest.

Then these equations must hold:

A{\frac  {3\pi }{4}}+B=20\,

A{\frac  {5\pi }{4}}+B=30\,

Solve to get A=20/\pi ,B=5\,

The solution is

\phi (z)={\frac  {20}{\pi }}{\mathrm  {arg}}_{0}z+5\,, 0\leq {\mathrm  {arg}}_{0}z<2\pi \,

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