Let $R \subset C$ be a connected domain, and let $u$: $R \rightarrow R$ be a harmonic function, i.e., $u_{xx} + u_{yy} = 0$.
(a) Does there exist a holomorphic function $f : R \rightarrow C$ such that $Re f(x + iy) = u(x; y)?$
(b) If a function $f$ as in (a) does exist, how many such functions are there?
(c) Same questions assuming that $R$ is simply connected.
(d) Find all harmonic functions of the form $u(x; y) = h(x^2 + y^2)$. What is their natural domain?
(e) Take a function $u(x, u) \not= 0$ from (d), restrict it to the right half-plane $R = ${$z | Re z > 0$} , and find all holomorphic functions $f : R \rightarrow C$ such that $Re f(x + iy) = u(x; y).$
Now I found parts a and b when $R$ is simply connected but couldn't figure out when it is just connected.
Also I guess there is infinetly many functions such that it satisfies part (a). But how can I show it? I am having some problems in part (d) and (e) as well. Any help is greatly appreciated.
It doesn't always exist when $R$ is not simply connected, and (d) will provide some examples of that where $R = \mathbb C \backslash \{0\}$. If it does exist, you can always add an imaginary constant, so in that sense there are infinitely many.
For (d), you can directly compute $u_{xx} + u_{yy}$ for $u(x,y) = h(x^2 + y^2)$, obtaining a differential equation for $h$.