Chapter 14.10, Problem 1P

Prove the theorem stated just after (10.2) as follows. Let $\varphi (u,v)$ be a harmonic function (that is, $\varphi$ satisfies ${\partial }^2\varphi /\partial u^2+{\partial }^2\varphi /\partial v^2=0$ ). Show that there is then an analytic function $g(w)=\varphi (u,v)+i\psi (u,v)$ (see Section 2). Let $w=f(z)=u+iv$ be another analytic function (this is the mapping function). Show that the function $h(z)=g(f(z))$ is analytic. Hint: Show that $h(z)$ has a derivative. (How do you find the derivative of a function of a function, for example, $\ln \sin z?$ ) Then (by Section 2 ) the real part of $h(z)$ is harmonic. Show that this real part is $\varphi (u(x,y),v(x,y).$