I have been trieng this exercises during many days and I did not can deal with it:
Show that harmonic functions are $C^{\infty}$
I have the following lemmas from the text:
Lemma 1: If $f(z)$ is analytic on a domain $D$, then $f(z)$ is infinitely differentiable, and the successive complex derivatives $f'(z), f"(z), ... $, are all analytic on $D$.
Lemma 2: Let $D$ be a bounded domain with piecewise smooth boundary. If $f(z)$ is an analytic function on $D$ that extends smoothly to the boundary of $D$, then f(z) has complex derivatives of all orders on $D$.
And also the Cauchy integral formula
My plan was to prove that harmonic functions were analitic and then use the lemma to have the result. My problem is showing this intermediate part. I found some information on the web, they use the poisson formula, they create a power series and many other strange things for me. Is ther any way to show the stament with out dying on the try.
Let $u(x,y)$ by harmonic so that $\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0$ with the first and second partial derivatives being continuous. Let $v=\int \frac{\partial u}{\partial x} \, dy + \text{const.}$ Then the function $f=u+iv$ is analytic. The function $u$ is the real part of an analytic function, so it is infinitely differentiable with respect to $x$ and $y$. The function $v$ is called the harmonic conjugate of $u$.