How to find an entire function $f(z)$ such that $f(n) = \sqrt {|n|}$ for every integer $n$?

Question

How to find an entire function $f(z)$ such that $f(n) = \sqrt {|n|}$ for every integer $n$?

Now my thinking is to create a series $\displaystyle\sum_{k=-\infty}^\infty f_{k}(z)$ such that $f_{k}(z)=\sqrt{|k|}$ and the series is convergent for every $z$.

Thanks for any help

Answer 1

the existence of such thing is standard, see Rudin under 'interpolation'. To actually construct it , you are multiplying by $\sum \sqrt{\vert n \vert} \frac 1 {x-n}$ which has the correct poles at n but doesn't converge. Try $\sum \sqrt{\vert n \vert} (\frac 1 {x-n} - \frac 1 n)$ which has the same poles & residues but speeds up the convergence