Let $\displaystyle f=u + iv$ be analytic. In each of the following, find $\displaystyle v$ given $\displaystyle u$.

(1) $\displaystyle u= x^2 - y^2$

(2) $\displaystyle u = \frac{x}{x^2 + y^2}$ (Figured this one out)

(3) $\displaystyle u = 2x^2 +2x +1 -2y^2$

(4) $\displaystyle u = cosh y sin x$

(5) $\displaystyle u = cosh x cos y$

If someone could show me step by step how to do one or two of these problems I would appreciate it. I'm not sure where I am going with these since I don't have the answers. Thanks!

These are Cauchy-Riemann equation problems.