Let $U$ be a complex domain and suppose $f:U \times \mathbb{R} \rightarrow \mathbb{C}$ is continuous and, for all $t$, is analytic in $s$.
That is, $f(s, t)$ is continuous and $g_t(s) := f(s, t)$ is analytic for each fixed $t$.
Now define $F(s) := \int_\mathbb{R} f(s, t) \, \text{d}t$.
What conditions do we need on $f$ so that $F$ is analytic on $U$. Specifically,
- If the integral converges for all $s \in U$, is $F$ analytic?
- If not, then what about if the integral converges absolutely for all $s \in U$
- What if the integral converges uniformly?
Basically, what combination of uniform and/or absolute convergence of the integral gives analyticity of $F$?
The basic way we can tell that something is analytic is with Morera's theorem. What we need to show is that for every closed piecewise $C^1$ contour (sufficient to be a triangle) $\Gamma$ such that $\Gamma$ and its interior are contained in $U$, $$ \int_\Gamma F(s)\; ds = 0$$ We know by Cauchy-Goursat that for every $t$, $\int_\Gamma f(s,t) \; ds = 0$. So what we want to do is permute the integrals (Fubini's theorem): we want
$$\int_\Gamma \int_{\mathbb R} f(s,t)\; dt \; ds = \int_{\mathbb R} \int_\Gamma f(s,t)\; ds \; dt = 0$$
The basic requirement for this is that the double integral of the absolute value converges:
$$ \int_\Gamma \int_{\mathbb R} |f(s,t)|\; dt \; ds < \infty$$
In particular this is true if $\int_{\mathbb R} |f(s,t)|\; dt$ is uniformly bounded on every compact set $K \subset U$.