Show that $F(z)=\int_{-\infty}^{\infty} e^{-t^2+4 z t} d t$ is holomorphic in $\mathbb{C}$

Question

I am trying to show that

$$F(z)=\int_{-\infty}^{\infty} e^{-t^2+4 z t} d t$$

is a holomorphic function in $\mathbb{C}$. The idea is to use the following theorem for parametric integrals:

Suppose $I$ is an interval, possibly unbounded, $A \subset \mathbb{C}$ is an open subset and suppose $\phi(t,z)$ is defined on $(I\backslash P) \times A$, where $P=\{t_1,\dots,t_N\}$. Suppose that for each $t \in (I\backslash P)$, $\phi(t,z) \in H(A)$ and $\phi, D_z \phi \in C((I\backslash P) \times A)$. Suppose further that for each $K \subset A$ a compact subset, there exists $F_K$ such that $$|\phi(t,z)| \leq F_K(t), \quad \forall t \in I \backslash P, z \in K,$$ where $$\int_{I} F_K(t) dt < \infty.$$ Then if $f(z)= \int_{I} \phi(t,z) dt$ then $f \in H(A)$ and $f^\prime (z)= \int_{I} D_z \phi(t,z) dt$.

ATTEMPT

Let $\phi(t,z)=e^{-t^2+4zt}$. Obviously $\phi(t,\cdot) \in H(\mathbb{C})$ for $t$ fixed, and $\phi, D_z \phi$ are continuous in $\mathbb{R} \times \mathbb{C}$. Let $K \subset \mathbb{C}$ be a compact subset. We know that $$|\phi(t,z)|=e^{-t^2+4 Re(z)t}.$$

Now we can suppose that $K \subset \{z: \alpha < Re(z) < \beta\}$. The problem in trying to bound $|\phi(t,z)|$ is that $t$ can be both negative and positive, and so can $\alpha$ and $\beta$, depending on where the compact is located in the complex plane. For example:

If $K$ is in the half-plane where $Re(z) >0$, then $\alpha,\beta >0$, and if $t <0$, then $4Re(z)t <0$, and thus $$|\phi(t,z)| \leq e^{-t^2},$$

and it is well known that $\int_{0}^{\infty} e^{-t^2} dt< \infty$. So we want to define $F_K$ as $$F_K=\left\{ \begin{array}{cl} e^{-t^2} & : \ t < 0 \\ \text{something} & : \ t \geq 0 \end{array} \right.$$

The problem is that, for $t \geq 0$ I can't seem to find a good bound to get a convergent integral, and by working like this it seems that I will have to do many cases by exhaustion. Is there a simpler way? I know of the way using Morera and Fubini, but that is not the objective here.

Answer 1

If suffices to consider the case $z \in K \subset \{z: -\alpha < Re(z) < \alpha\}$ for $\alpha > 0$. Then $$ |\phi(t,z)| \leq \begin{cases} e^{-t^2+4 \alpha t} \text{ if } t \ge 0 \\ e^{-t^2-4 \alpha t} \text{ if } t < 0 \end{cases} $$ so that for all $t \in \Bbb R$ $$ |\phi(t,z)| \leq e^{-t^2+4 \alpha t} + e^{-t^2-4 \alpha t} \\ = e^{-(t-2\alpha)^2+ 4 \alpha^2} + e^{-(t+2\alpha)^2+ 4 \alpha^2} =: F_k(t) $$ where $\int_{-\infty}^\infty F_K(t) dt < \infty$.