CVFHF2

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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}_0z + B\, where \mathrm{arg}_0\, is a branch measured from 0 to 2π 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}_0z + 5\,, 0\le \mathrm{arg}_0z < 2\pi\,


Main Page : Complex Variables : Exponential and Log

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