CVFHF1

From Exampleproblems

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:

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

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

So

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

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

CVFHF1

From Exampleproblems

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