CVFHF4

From Exampleproblems

Find a function that is harmonic on the washer-shaped region between the circles with radii 1 and 2 and center $(1, i)$ . It should be 0 and 10 on the inner and outer circle.

Guess the answer $\phi(x,y) = A \mathrm{Log}|z-(1+i)|+B\,$ .

At the point $i\,$ , $A \mathrm{Log}|-1|+B=0 \implies B=0\,$

At the point $1-i\,$ , $A \mathrm{Log}|-2i| = 10 \implies A=\frac{10}{\mathrm{Log}2}\,$

So the answer is

$\phi(z) = \frac{10}{\mathrm{Log}2} \mathrm{Log}|z-(1+i)| \,$ .

Main Page : Complex Variables : Exponential and Log

CVFHF4

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