Exponential integral with square root in power

Question

I have been trying to solve an integral. I know that the solution exists for the form

$$1- \dfrac{2}{\mathcal{R}^2}\int_{0}^{\mathcal{R}_{\mathcal{G}}} \exp (-\Phi r^{\alpha}) r\, {\rm d}r,$$

where $\alpha>0, \mathcal{R}>0$. However, I want to solve

$$1- \dfrac{2}{\mathcal{R}^2}\int_{0}^{\mathcal{R}} \exp \left(-\Phi \left(\sqrt{r^2+h^2}\right)^{\alpha}\right) r\, {\rm d}r,$$

where $\Phi>0$, $\alpha>0, \mathcal{R}>0$. I have tried, but on substituting $t=r^2+h^2$, the lower limit of integral becomes h and I checked Table of Integrals (3.351) for an equivalent solution but no hope. Any help on this please?

Answer 1

Doing the same as Dinesh Shankar in his/her answer. $$f(R)=1- \dfrac{2}{{R}^2}\int_{0}^R \exp \left(-\Phi \left(\sqrt{r^2+h^2}\right)^{\alpha}\right)\, r\, dr$$ $$f(R)=1-\frac{2 }{\alpha\, R^2\,\Phi ^{2/\alpha }}\left( \Gamma \left(\frac{2}{\alpha },\left(h^2\right)^{\alpha /2} \Phi \right)-\Gamma \left(\frac{2}{\alpha },\left(h^2+R^2\right)^{\alpha /2} \Phi \right)\right)$$

Answer 2

Your integral is something like that

$$ I=\int x\, \mathrm{e}^{-a\left(x^2+b^2\right)^\frac{n}{2}}\,\mathrm{d}x $$

The change of variable $u=a^\frac{2}{n}\left(x^2+b^2\right)$ give us

$$I=\frac{1}{2a^\frac{2}{n}}\int\mathrm{e}^{-u^\frac{n}{2}}\,\mathrm{d}u $$

The last integral is related to the incomplete Gamma function:

$$\int\mathrm{e}^{-u^\frac{n}{2}}\,\mathrm{d}u= -\frac{2\operatorname{\Gamma}\left(\frac{2}{n},u^\frac{n}{2}\right)}{n}.$$

So,

$$ I= -\frac{\operatorname{\Gamma}\left(\frac{2}{n},u^\frac{n}{2}\right)}{na^\frac{2}{n}}=-\frac{\operatorname{\Gamma}\left(\frac{2}{n},a\left(x^2+b^2\right)^\frac{n}{2}\right)}{na^\frac{2}{n}}.$$

Answer 3

Using expansion of $e^x$ we see $$1- \dfrac{2}{\mathcal{R}^2}\int_{0}^{\mathcal{R}} \exp \left(-\Phi \left(\sqrt{r^2+h^2}\right)^{\alpha}\right) r\, {\rm d}r,$$ $$1- \dfrac{2}{\mathcal{R}^2}\int_{0}^{\mathcal{R}} \sum_{n\geq0}\dfrac{1}{n!}\left(-\Phi \left(\sqrt{r^2+h^2}\right)^{\alpha}\right)^n r\, {\rm d}r,$$ $$1- \dfrac{2}{\mathcal{R}^2} \sum_{n\geq0}\dfrac{(-\Phi )^n}{2n!}\int_{0}^{\mathcal{R}} \left(r^2+h^2\right)^{n\alpha/2} 2r\, {\rm d}r,$$ $$1- \dfrac{2}{\mathcal{R}^2} \sum_{n\geq0}\dfrac{(-\Phi )^n}{2n!(\frac{n\alpha}{2}+1)}\left[ \left(\mathcal{R}^2+h^2\right)^{\frac{n\alpha}{2}+1} -\left(h\right)^{n\alpha+2}\right]$$