Let be open and consider a function so that for all the function is analytic. Prove that is analytic.
Any ideas? Can I use Morera's Theorem? How?
Then is analytic (as a function of z) for each fixed But it is not (Riemann) integrable as a function of t, except at z=0.
If you have some sufficiently strong additional condition controlling the integrability of f as a function of t, then you could hope to prove the result by approximating the integral by Riemann sums, and using the fact (corollary of Morera's theorem) that a uniform limit of analytic functions is analytic.