Let $f$ be holomorphic on $U=\{2<|z|<\infty\}$ with $\int_{|z|=3|}f(z)dz=0$, show $f$ has a primitive on $U$.

Question

My approach I learnt of constructing primitive is usually in the proof of Morera's theorem. $F(z)=\int_\{[a,z]\}f(w)dw$. However, I found this not that useful because we don't have triangular path control. Could you give me some suggestions?

Firstly, is constructing the Primitive a right way to approach this problem? Secondly, is it the right way to construct "Primitive"? Thirdly, if I have defined in this way, what I should do to show it is well defined? Fourthly, where can I use $\int_{|z|=3|}f(z)dz=0$?

i.e. I am just asking the question stated in the topic.

Answer 1

Hints: 1. For any integer $n\ne -1,$ $z^n$ has an antiderivative on $\mathbb C\setminus \{0\}.$ 2. Could the Laurent expansion of your $f$ have a $z^{-1}$ term?

Answer 2

The approach you suggest does work. Following in the steps of the proof of Morera's theorem (as given in Wiki) we must require that the integral from a point $a\in U$ to another point $z\in U$ is path independent for the primitive to be well-defined.

If we have two such curves $\gamma$ and $\tau$ one can show that $\int_\gamma f = \int_\tau f$ if $\int_{\gamma \tau^{-1}}f = 0$ where $\gamma\tau^{-1}$ is following $\gamma$ from $a$ to $z$ and then following $\tau$ in reverse from $z$ to $a$. This is a closed curve and you need to show that this is zero. There are two kinds of closed curves you can have. The first one is where the interior is in $U$. By Cauchy's theorem the integral over such a curve is $0$. The problematic curves are those that wrap around the region $|z| \leq 2$ (for example $f$ might have a pole in this region like $f(z) = \frac{1}{z}$ for which the theorem at hand does not hold).

The condition $\int_{|z|=3}f = 0$ is what rules out such counter-examples and ensures that the integral is zero. Note that we can replace $|z|=3$ by any other curve that wraps around this region.

To deal with a curve that wraps around $|z| \leq 2$ the idea is to deform such a curve to $|z| = 3$ for which we know the integral (and we are can deform a curve in $U$ without changing the value of the integral since $f$ is holomorphic).