Let $f \in L^1$ with compact support.
I want to show that $\hat f: \mathbb R \to \mathbb C, f(z):=\int_{\mathbb R}f(x)e^{-ixz}d\lambda(x), $ can be extended to a holomorphic function on $\mathbb C$ and I was wondering if I can do it like this:
Let $U_a:=\{z \in \mathbb C: -a<Im(z)<a\}$ and $g(x,z):=f(x)e^{-ixz}$
1) $x\mapsto g(x,z)$ is integrable for all $z \in U_a$: $f$ is integrable and vanishes outside of its support. On the (compact) support $x\mapsto e^{-ixz}$ is integrable because it is continuous. Therefore $x\mapsto g(x,z)$ is integrable for all $z \in U_a$.
2) $z \mapsto g(x,z)$ is holomorphic because for a fixed $x$ $f(x)$ is just a constant and $z \mapsto e^{-ixz}$ is holomorphic.
3) $|g(x,z)|=|f(x)e^{-ixz}|=|f(x)e^{Im(z)}x|<|f(x)|e^{a|x|}$
which is a integrable function $\mathbb R \to \overline {\mathbb R}$
Therefore $\int_{\mathbb R} g(.,z) d\lambda(.)$ is holomorphic on $\mathbb C$ (because $U_a$ was arbitrary) according to the 'holomorphy under the integral' lemma
You can start by arguing that $\hat{f}(\lambda)=\int_{\mathbb{R}}f(x)e^{-i\lambda x}dx$ is continuous everywhere on $\mathbb{C}$, which follows from the dominated convergence theorem. Then you can apply Morera's theorem by showing that $\int_{\Delta}\hat{f}(\lambda)d\lambda=0$ for all triangles in $\mathbb{C}$, which follows by interchanging the order of integration using Fubini's Theorem: $$ \int_{\Delta}\hat{f}(\lambda)d\lambda=\int_{\mathbb{R}}f(x)\int_{\Delta}e^{-ix\lambda} d\lambda dx = 0. $$ The conclusion of Morera's theorem is that $\hat{f}$ is an entire function.