CVFHF2

From Exampleproblems

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

CVFHF2

From Exampleproblems

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