Chapter 13.3, Problem 104ES

In each part, show that $u\left(x,y\right)and\upsilon \left(x,y\right)$ satisfy the Cauchy-Riemann equations

$\frac{\partial u}{\partial x}=\frac{\partial \upsilon }{\partial y}\text{and}\frac{\partial u}{\partial y}=-\frac{\partial \upsilon }{\partial x}$

$\begin{array}{l}\left(\text{a}\right)u=x^2-y^2,\upsilon =2xy\\ \left(\text{b}\right)u=e^x\cos y,\upsilon =e^x\sin y\\ \left(\text{c}\right)u=\ln \left(x^2+y^2\right),\upsilon =2{\tan }^{-1}\left(y/x\right)\end{array}$