Is there an analytic approximation to this integral

Question

The following integral does not have an analytical form.. however, would it be possible to find an analytical approximation?

$$\int_b^c e^{-a\sqrt{x}} e^{-a \sqrt{x-b}} dx $$

with $c > b$ and $a>0$. Any help would be much appreciated.

Answer 1

The following analysis is valid for $b>0$.

By the substitution $x=b\cosh^2 y$, your integral becomes $$\int^{r}_0e^{-a\sqrt b e^y}\sinh 2y dy$$ where $r=\cosh^{-1}\sqrt{\frac cb}$.

Furthermore, substitute $u=e^y$, yielding $$b^2\left(\int^{e^r}_1ue^{-a\sqrt b u}du +\int^{e^r}_1 u^{-3}e^{-a\sqrt bu}du\right)$$

The first term can be written in closed form, and the second can be expressed in terms of incomplete gamma function.

$$\int^{e^r}_1ue^{-a\sqrt b u}du =\frac{-(a\sqrt be^r+1)e^{-a\sqrt be^r}+(a\sqrt b+1)e^{-a\sqrt b}}{a^2b}$$

$$\int^{e^r}_1 u^{-3}e^{-a\sqrt bu}du=a^2b\cdot[\Gamma(-2,a\sqrt b )-\Gamma(-2,a\sqrt b e^r)]$$

Wikipedia gives some asymptotic approximations under the section ‘asymptotic behavior’.