Specifically, the question is as follows:
Let $\mathbb{D}$ be the open unit disk in $\mathbb{C}$ and suppose that $f:\bar{\mathbb{D}}\rightarrow\mathbb{C}$ is a continuous function such that both $\Re f$ and $\Im f$ are harmonic.
(a) Show that
$$f(re^{i\theta})=\frac{1}{2\pi}\int_{-\pi}^\pi f(e^{it})P_r(\theta-t)dt$$
for all $re^{i\theta}$ in $\mathbb{D}$.(b) Show that $f$ is holomorphic in $\mathbb{D}$ if and only if
$$\int_{-\pi}^\pi f(e^{it})e^{int}dt=0$$
for all $n\geq1$.
I believe (a) is a direct consequence of Poisson's representation theorem for harmonic functions, stated as follows:
Let $\mathbb{D}$ be the open unit disk and $f:\partial\mathbb{D}\rightarrow\mathbb{R}$ be continuous. Then there exists a continuous function $u:\bar{\mathbb{D}}\rightarrow\mathbb{R}$ such that $u(z)=f(z)$ if $z\in\partial\mathbb{D}$, $u$ is harmonic on $\mathbb{D}$, and $u(re^{i\theta})=\frac{1}{2\pi}\int_{-\pi}^\pi P_r(\theta-t)f(e^{it})dt$ and $u$ is unique.
It seems to me that this can just be applied to the real and imaginary parts of the function $f$ from the question to derive the desired result, but maybe I'm making a careless mistake?
For (b), it's not obvious to me how part (a) assists in the solution. This is a homework question, so I am looking for assistance with understanding it, but not a full solution. Thanks.
Best Answer
Poisson's representation formula here assumes only a function on the boundary. Also, one does not know the relationship between $u:\bar{\mathbb{D}}\rightarrow\mathbb{R}$ from the theorem and $f:\bar{\mathbb{D}}\rightarrow\mathbb{R}$ from the question (unless the proof of the theorem is constructive, then maybe you can deduce it), so you cannot use it like this here.
Hint for (a):
Hint for (b):