# Holomorphic function and deritives

Is it true that there exists a holomorphic function whose derivative is $1/(z^2-1)$? How to tackle problems like these?

• Write it as a taylor series with help of the geometric series. Then integrate termwise. – Paul K Dec 14 '16 at 9:03
• In what domain? – lhf Dec 14 '16 at 9:04
• In $\Bbb C$ i guess, since it is complex analysis? But it is not specified in this problem. – J.doe Dec 14 '16 at 9:05
• @menag Then what? – J.doe Dec 14 '16 at 9:06
• Alright, I got my answer, which is false. Can someone confirm that? – J.doe Dec 14 '16 at 9:11

## 1 Answer

$\dfrac1{z^2-1}$ is holomorphic in the open unit disk and so has a primitive there.

The primitive is given explicitly by integrating the Taylor series: $$\int \frac1{z^2-1} =-\int \sum_{n=0}^{\infty} z^{2n} =-\sum_{n=0}^{\infty} \frac{z^{2n+1}}{2n+1}$$

Both series converge in the open unit disk.

There is no primitive on the whole complex plane because the function is not even defined on the whole complex plane.

• Wait.. so you are saying that this statement is true? – J.doe Dec 14 '16 at 9:22