If$f(z)$ is analytic , then what about $f'(z)?? $

Question

If$f(z)$ is analytic , then what about $f'(z)$?

can we conclude that $f^{(k)}(z)$ is analytic for any k$\in $$ \mathbb{N} $

Answer 1

Yeah with cauchy's formula $$f^{(n)}(z_0)=\frac{n!}{2\pi} \int_{\partial B} \frac{f(\xi)}{(\xi - z_0)^{n+1}} \, \mathrm{d}\xi$$ you can calculate all derivates only knowing your function.

Maybe this will look easier, as $f$ is analytic it can be written as a power series $$f(z)= \sum_{k=0}^\infty a_k z^k $$ Using that you can change differention and summing you get that $f'$ is analytic too.