What is the value of the following integral ?: $$ I=\int_0^{\infty}\exp\left(-\,{c \over x^{2} + a^{2}} - x^{2}\right) \,\mathrm{d}x $$

By change of variable $u = x^{2} + a^{2}$: $$I = \mathrm{e}^{a^{2}} \int_{a^{2}}^{\infty}{1 \over 2\,\sqrt{\, u - a^{2}\,}\,}\, \exp\left(-\,{c \over u} - u\right)\,\mathrm{d}u $$

I have worked on it a lot but nothing I found$\ldots$ Any thoughts ?.

  • $\begingroup$ It depends on what you would define as "value". But you could try to express the integral using the Gamma function. $\endgroup$ – MrYouMath Oct 6 '16 at 17:01
  • $\begingroup$ $$dx = \frac1{2 \sqrt{u-a^2}} du$$ $\endgroup$ – arthur Oct 6 '16 at 17:05
  • $\begingroup$ @arthur Thanks. I edited the problem statement. $\endgroup$ – Susan_Math123 Oct 6 '16 at 18:25
  • $\begingroup$ @MrYouMath I tried, but I couldn't ...can you tell me how? $\endgroup$ – Susan_Math123 Oct 6 '16 at 18:25

I will outline a possible approach. For simplicity, I will initially assume $a=c=1$, then provide a generalization for $a,c>0$. We have $$ I(a,c)=\int_{0}^{+\infty}\exp\left(-\frac{1}{x^2+a}-x^2\right)\,dx = \int_{0}^{+\infty}\frac{e^{-x}}{2\sqrt{x}}\cdot\exp\left(-\frac{c}{x+a}\right)\,dx \tag{1}$$ where the Laplace transform of $\frac{e^{-x}}{2\sqrt{x}}$ is given by $$ \mathcal{L}\left(\frac{e^{-x}}{2\sqrt{x}}\right)=\frac{\sqrt{\pi}}{2\sqrt{1+s}}\tag{2} $$ and the inverse Laplace transform of $\exp\left(-\frac{1}{x+1}\right)$ is given by: $$ \mathcal{L}^{-1}\left(\exp\left(-\frac{1}{x+1}\right)\right) = \left(\delta(s)-\frac{J_1(2\sqrt{s})}{\sqrt{s}}\right)e^{-s}\tag{3} $$ so that: $$\begin{eqnarray*}I(1,1)&=&\frac{\sqrt{\pi}}{2}-\sqrt{\pi}\int_{0}^{+\infty}\frac{J_1(2t)}{\sqrt{t^2+1}}e^{-t^2}\,dt\\&=&\frac{\sqrt{\pi}}{2}-\sqrt{\pi}\int_{0}^{+\infty}J_1(2\sinh\theta)\exp\left(-\sinh^2\theta\right)\,d\theta. \tag{4}\end{eqnarray*} $$ where the Bessel function $J_1(t)$ is an odd function, and for every odd $m\in\mathbb{N}$ the integral $$ \int_{0}^{+\infty}\sinh(\theta)^{m}e^{-\sinh^2\theta}\,d\theta $$ is a linear combination with rational coefficients of $1$ and $e\sqrt{\pi}\,\text{Erfc}(1)$, due to integration by parts. Since the coefficient of $t^{2n+1}$ in $J_1(t)$ is given by $\frac{(-1)^n}{2^{2n+1}n!(n+1)!}$, the RHS of $(4)$ can be rearranged as a very fast-convergent series. The same principle holds if $a,c$ are positive but $\neq 1$.

  • $\begingroup$ Many thanks for outlining the possible approach. Can you please convert it to a complete solution with final value as a series or something? $\endgroup$ – Susan_Math123 Oct 18 '16 at 3:01
  • $\begingroup$ And, can you clarify on how you go from step (1) to (4) ? (how you use the Laplace transform and inverse Laplace to achieve (4)? I am in the grace period for the bounty so I appreciate your fast response. $\endgroup$ – Susan_Math123 Oct 18 '16 at 15:56
  • $\begingroup$ @Sus20200: I just exploited the properties of the Laplace transform outlined here: en.wikipedia.org/wiki/… $\endgroup$ – Jack D'Aurizio Oct 18 '16 at 16:11

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