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!