Integral $a\int_{-\infty }^{\infty } \frac{e^{\frac{x^2}{a^2+x^2}}}{a^2+x^2} \,\operatorname dx$

Question

I'm looking to calculate $a\int_{-\infty }^{\infty } \frac{e^{\frac{x^2}{a^2+x^2}}}{a^2+x^2} \,\operatorname dx$.

Mathematica10 can't integrate this, but numerical integration gives an answer of 5.50843 for any $a$ I've tried, so I assume that there is an analytic answer to be found.

Is this a standard integral?

Many thanks,

Have you tried using complex analysis? – UserX Sep 22 '14 at 22:26 — UserX, Sep 22 '14 at 22:26
Perhaps try substituting $x=a \tan{\theta}$. – Ron Gordon Sep 22 '14 at 22:39 — Ron Gordon, Sep 22 '14 at 22:39

score 4 · Accepted Answer · answered Sep 22 '14 at 22:44

4

Sub $x=a \tan{\theta}$ and the integral becomes

$$\sqrt{e} \int_{-\pi/2}^{\pi/2} d\theta \, e^{-\frac12 \cos{2 \theta}}$$

which is indeed a standard integral, equal to $\pi \sqrt{e} I_0(1/2) \approx 5.50842977...$, where $I_0$ is the modified Bessel function of the first kind of zeroth order.

answered Sep 22 '14 at 22:44

Ron Gordon

138,521

Very nice. Can you point me to a resource that lists that standard integral? I also need the result for the more general integral where the exponent is to the power +1/2.b.cos2θ, where b is a positive constant. – Dave Sep 22 '14 at 23:12
1

@Dave See Formula 9.6.16 in Abramowitz and Stegun. – Dilip Sarwate Sep 22 '14 at 23:16

Integral $a\int_{-\infty }^{\infty } \frac{e^{\frac{x^2}{a^2+x^2}}}{a^2+x^2} \,\operatorname dx$

1 Answers1