I recently ran across a problem where I can reduce an integral to the form $\int_{0}^{(1+i)R} e^{-ax^2} \,dx$, I want to find the solution as $R$ goes to infinity. Since the integral looks to be a Gaussian integral, is the solution $\frac{\sqrt{\pi}}{2\sqrt{a}}$? Or is this incorrect since the Gaussian Integral is only for the real line?
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Maybe this helps: https://math.stackexchange.com/questions/1297096/gaussian-integral-with-a-shift-in-the-complex-plane . If not, can you show how you arrived at the expression $\int_{0}^{(1+i)R} e^{-ax^2} ,dx$ ? It seems ambiguous / not well-defined. – Andreas Tsevas Feb 14 '23 at 09:46
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@Andreas I got the expression from trying to evaluate the Fresnel integral for a real $a$, $\int_{0}^{\infty} e^{ia \frac{x^2}{2}} dx$ from $0$ to $\infty$ by integrating it over 3 contours, one of them being $\gamma=[0, (1+i)R]={(1+i)t, 0\geq t \geq R }$ which got me this integral but with a $(1+i)$ in front. I want to show the integral over this contour evaluates to a finite value as R goes to infinity, so if the Gaussian integral can be applied then I can conclude it does equal a finite value. – Eric L. Feb 14 '23 at 09:57