# Improper Integral of Exponential over Quadratic

Trying to solve a damped wave equation this integral propped up.

$$I_a(t) = \int_{0}^{\infty} \frac{x\,e^{-x^2 + \text{i}xt}}{x^2 - a} dx$$

Where $$t \in \mathbb{R}$$ and $$a \in \mathbb{C}$$ with $$\text{Im}[a] \neq 0$$.

This integral is doable without the imaginary part on the exponential (according to Mathematica). I know Jordan's lemma doesn't apply because the integrand does not go to zero in either side of the contour.

Any help would really be appreciated!

• It's not indefinite. It's definitely definite. But it is improper. Commented Nov 16, 2021 at 6:44
• Oops! Thank you! Commented Nov 16, 2021 at 6:46

Well I have an hint which I think will be sufficient for you proceed. It's a bit long for comment. Let, $$I(s)=\int_{0}^{\infty}\frac{xe^{s(x^{2}-a)+ixt}}{x^{2}-a} dx$$ $$I(-1)e^{-a}$$ is the required integral. $$I'(s)=\int_{0}^{\infty}\frac{x(x^{2}-a)e^{s(x^{2}-a)+ixt}}{x^{2}-a} dx$$ $$I'(s)=\int_{0}^{\infty} xe^{s(x^{2}-a)+ixt} dx$$ In case you are confused still how to go from here. Let me help. First rewrite $$e^{ixt}$$ using Euler formula. Next use the formula mentioned in this video
• Thank you so much! That is a really clever way of looking at it! I just have one question. I managed to get an analytic form of the derivative $I'(s)$, but to go back to $I(s)$ we need to integrate along s right? I don't know what limits to use for that. But honestly thank you so much! Commented Nov 16, 2021 at 18:41
• @panoik, would you mind showing your evaluation of $I'(s)$?? Commented Nov 16, 2021 at 19:03
• And also it was not mentioned in the question about the $\Re(a)$, I mean is it any real number or only positive real or else. According to that one can choose the bounds. Or else find it's anti derivative and find the value of constant of integration. Commented Nov 16, 2021 at 19:12
• $\Re (a)$ can be anything that's why I didn't specify. You can get $I'(s) = e^{- s a + t^2/2s} /s D_{-2}(-\frac{c}{2\sqrt{s}})$ where $D_{-2}(x)$ is the parabolic cyllinder function. Commented Nov 16, 2021 at 23:00