An example of a function that is analytic only on the upper-half plane

Question

I've been asked to give an example of a complex function that meets the criteria outlined in the question above but I'm not sure if this is something I should know given an example or if its a result that can be derived from the Cauchy-Riemann equations.

I'd appreciate any help.

Answer 1

The function

$$f(z)=\sum_{n=0}^\infty e^{izn^2}$$

converges for $\Im(z)>0$ and diverges otherwise.