I'm trying to evaluate, if it is possible, a Gaussian integral over a wedge of the complex plane
$$ I = \int_W d^2z \,e^{-|z|^2 + z^*a + za^*}, $$ ($d^2z = \frac{dz\,dz^*}{2} = dx\,dy = r\,dr\,d\phi$) where $W=\{z\mid -\theta \le \arg(z) \le \theta\}$, with $\theta < \pi/2$. In other words, the integral can also be written
$$ I = \int_{-\theta}^\theta d\phi \int_0^\infty dr \,r e^{-r^2 + e^{-i\phi}r a + e^{i\phi}ra^*}. $$
In the case where $W$ is instead the entire complex plane, the answer is well known and equal to $I=\pi e^{|a|^2}$.
Attempt at solution
I made an attempt using Green's theorem to turn $I$ into a contour integral, and I was hoping to continue with Cauchy's integral formula from there, but got stuck due to a non-analytic function in the contour integral. I'm not sure if this is a good path forward, but I'll show you what I did. My complex integration knowledge is rusty so be aware of mistakes.
First, assume that we have a complex function $f(z)$ that can be written $f(z) = f(z,z^*) = \partial_{z^*} F(z,z^*)$. Then Green's theorem gives
$$ \int_W d^2 z\, f(z) = \int_W \frac{dz \, dz^*}{2} \partial_z^* F(z,z^*) = \int_W dx\,dy\, (\partial_x + i\partial_y)F = \frac{1}{2} \int_{\partial W} F \,dy - iF\,dx \\ = -\frac{i}{2} \int_{\partial W} F(z)\,dz, $$ where the last integral is a counter-clockwise contour integral over the boundary $\partial W$ of $W$. Here I used that $\partial_z^* = (\partial_x + i\partial_y)/2$ and Green's theorem $\int_S dx\,dy\,(\partial_x L + \partial_y M) = \int_{\partial S} M \,dx - L \,dy$.
Returning to the integral $I$ above we have $f(z) = e^{-|z|^2 + z^*a + za^*} = \partial_{z^*} F(z,z^*)$ with
$$ F(z,z^*) = \frac{e^{-|z|^2 + z^*a + za^*}}{a-z} = \frac{f(z)}{a-z}. $$
Now, it is tempting at this stage to use Cauchy's integral formula $\int_{\partial W} dz \,\frac{g(z)}{z-a} = 2\pi i g(a)$ for holomorphic (i.e. analytic) $g(z)$. However this does not work because $f(z)$ above is not holomorphic due to the $|z|^2$ and $z^*$ terms in the exponent.
Writing out the contour integral over the wedge $\partial W$ explicitly we have
$$ \int_{\partial W} F(z) \, dz = \int_{\partial W} dz \,\frac{e^{-|z|^2 + z^*a + za^*}}{a-z} \\ = \int_{0}^\infty dr \,\frac{e^{-r^2 + r \left(e^{-i\theta} a + e^{i\theta} a^*\right)}}{a-e^{-i\theta}r} + \int_\infty^0 dr\, \frac{e^{-r^2 + r \left(e^{i\theta} a + e^{-i\theta} a^*\right)}}{a-e^{i\theta}r} + 0 $$ Here I have divided the contour into three parts. One where $\arg(z) = -\theta$, one where $\arg(z) = \theta$, and the "arc" out at infinity which vanishes due to the $-|z|^2$ term in the exponent.
I wasn't able to make any further progress on the integrals over $r$ above. Perhaps there is a way to evaluate them directly, or perhaps one can come up with an analytic function $g(z)$ that is equal to $f(z)$ on the contour and use Cauchy's integral formula?