Solving Harmonic Function: Find v(x,y) for u + iv Analytic on C

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Homework Statement


I have shown that u(x,y) = e-xsin(y) is harmonic. That is uxx+ uyy = 0. How do I find a harmonic function v(x,y) such that u + iv is analytic on C.


Homework Equations





The Attempt at a Solution


I tried to find v(x,y) in the same fashion as you find a scalar potential, given a gradient but that was no go.
Would I find the Cauchy-Riemann equations & go from there?
 
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Never mind. Got it.
It was v(x,y) = e-xcos(y) + K
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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