# Thread: Prove that f is analytic

1. ## Prove that f is analytic

Let $\displaystyle \mathcal U\subset\mathbb C$ be open and consider a function $\displaystyle f:[0,1]\times\mathcal U\longrightarrow\mathbb C$ so that for all $\displaystyle t\in[0,1]$ the function $\displaystyle f(t,\cdot):\mathcal U\longrightarrow\mathbb C$ is analytic. Prove that $\displaystyle F(z)=\int_0^1 f(t,z)\,dt$ is analytic.

Any ideas? Can I use Morera's Theorem? How?

Thanks.

2. ## Re: Prove that f is analytic

Originally Posted by Homing
Let $\displaystyle \mathcal U\subset\mathbb C$ be open and consider a function $\displaystyle f:[0,1]\times\mathcal U\longrightarrow\mathbb C$ so that for all $\displaystyle t\in[0,1]$ the function $\displaystyle f(t,\cdot):\mathcal U\longrightarrow\mathbb C$ is analytic. Prove that $\displaystyle F(z)=\int_0^1 f(t,z)\,dt$ is analytic.

Any ideas? Can I use Morera's Theorem? How?

Thanks.
I think that you need some extra condition for this result to hold. Suppose for example that

$\displaystyle f(t,z) = \begin{cases}z & \text{if }t \text{ is rational,} \\ -z & \text{if }t \text{ is irrational.} \end{cases}$

Then $\displaystyle f(t,z)$ is analytic (as a function of z) for each fixed $\displaystyle t\in[0,1].$ 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.

3. ## Re: Prove that f is analytic

Perhaps the continuity of $\displaystyle f(t,z)$ and the connectedness of $\displaystyle \mathcal U$ ?