Holomorphic function and deritives

Question

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

Answer 1

$\dfrac1{z^2-1}$ 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.