Does $\int \frac{exp( -b\sqrt{a+x})}{\sqrt{x}} dx$ have a solution?

Question

Is there a solution for the following integral:

$$ \int_0^{\infty} \frac{\exp( -b\sqrt{a+x})}{\sqrt{x}} dx $$

where $a$ and $b$ are constants. If it is not, what is the best approximation? Especially in the limit as $b \to \infty$.

In case of $a=0$, there is an analytic solution.

what is the domain of integration? For big b the integral we be dominated by the region closest to zero — tired, Nov 08 '15 at 13:36
If $b>>1$ and $a>0$ a the leading order asymptotics is given by $$e^{-b\sqrt{a}}\frac{\sqrt {\pi\sqrt{a}}}{2\sqrt{b}}$$ — tired, Nov 08 '15 at 16:47

score 2 · Accepted Answer · edited Apr 13 '17 at 12:21

2

Letting $x=a\sinh^2t$, we have $I=2\sqrt a~K_1\Big(b\sqrt a\Big):$ see Bessel function for more information.

edited Apr 13 '17 at 12:21

Community

answered Nov 08 '15 at 17:06

Lucian

1 Answers1