# harmonic conjugates and cauchy riemann eqns

I'm trying to find function $v(x,y)$ such that the pair $(u,v)$ satisfies the Cauchy-Riemann equations for the following functions $u(x,y)$:

a) $u = \log(x^2+y^2)$ $$u_x = v_y \Rightarrow \frac{2x}{x^2+y^2} = v_y \Rightarrow v = \frac{2xy}{x^2+y^2}?$$

b) $u = \sin x \cosh y$ $$u_x = \cos x \cosh y = v_y \Rightarrow v = \sinh y \cos x + C$$

c) $u = \frac{x}{x^2+y^2}$

$u_x = v_y$, but I am getting a mess with integration. The reason is that is there a way to do this by integration, or is the way I have started this seem correct? Thanks!

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Please use TeX. –  Paul Oct 22 '12 at 9:28
Sure, still learning –  mary Oct 22 '12 at 9:29

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