Holomorphic function such that $f(1/k)=k^2/(k^2+1)$

Question

Function $f$ is holomorphic in neighbourhood of $0$ and for every positive integer $k$ $$f\left(\frac{1}{k}\right)=\frac{k^2}{k^2+1}$$ How do I find values of $f^{(n)}(0)$ for $n=1,2,3,...$?

Answer 1

You can show using the Analytic Continuation Principle that the only holomorphic function that satisfy that condition is $f(z) = \frac{1}{1+z^2}$. Knowing that you only need to derivate.