The following exercise asks me to find $a$ and $b$ so that $v(x,y)$ is harmonic:
$$ v(x,y) = y(bx^{2}-y^{2}) + \frac{y-a}{y^{2} + (x-a)^{2}} $$
So far I've tried both using Laplace (resulting expression is pretty ugly) and Milne-Thomson to try and construct an analytic function with v(x,y) as its imaginary part and see if I got any ideas where to go from there, but what resulted also seems hard to analyze:
$$ f(z) = -\frac{1}{z-a} + \frac b{3}{z^{3} - i\frac{a}{(z-a)^2} + C} $$
Any help or push in the right direction would be appreciated.