Original problem: I have a problem in which i need to evaluate the integral: $$ \int_1^\infty \dfrac{\sqrt{r^2-1}e^{-\alpha r}}{r} dr\, $$ I have tried to evaluate it taking the $\alpha$ derivative, here i give you all the steps i have done: $$ \int_1^\infty \dfrac{\sqrt{r^2-1}e^{-\alpha r}}{r} dr\,=-\int d\alpha \int_1^\infty\sqrt{r^2-1}e^{-\alpha r}=-\int d\alpha \dfrac{K_1(\alpha)}{\alpha}\\ =-\frac{1}{4} G^{2,1}_{0,1}\left(\frac{\alpha}{2},\frac{1}{2} \middle| \begin{array}{c} 1 \\ -1/2,1/2,0 \\ \end{array} \right) $$ The two last steps in the last equation were obtained using Mathematica, and have been checked numerically, however i have calculated numerically the first integral for a few different values for $\alpha$ and it doesn't match the values the MeijerG function gives.
So now i think i have two possibilities:
-There is something wrong in the first step, the differentiation under integral sign doesn't work and i don't know why this could be the case.
-It's a numerical problem. Mathematica could be giving me wrong values of the integral, but i think this can not be because i have been changing the precision and the method for the numerical integral, obtaining the same results.
I want to know what is happening, i have tried similar examples with the same method and i have obtained the good results. Of course, if you have another way of evaluating this integral, don't hesitate, i would like to know it, but also i would like to know what is wrong here.
EDIT:
Comparison between numerical solution vs MeijerG solution (using differentiation under integral sign) as $\alpha$ functions:
EDIT 2:
I have found that the difference between the numeric integral and the analytic solution obtained using differentiation under integral sign is exactly $\pi/2$ for all $\alpha$. Actually I have found that for an integral of the form:
$$ \int_z^\infty \dfrac{(r^2-z)^c e^{-\alpha r}}{r} dr\, $$ Being z a positive number and $0<c<1$. The differentiation under integral sign method gives the result up to a function f(z,c).
For example:
-For z=1 and c=1/2, this function is equal to $\pi/2$
-For z=1 and c=1/3 this function is equal to $\pi/\sqrt{3}$
-For c=1/2 this function is exactly $f(z,1/2)=-1+(1+\frac{\pi}{2})z$
With this i can continue with my calculations but, does anyone know why does this happen?
Do you know a way for obtaining this function f(z,c) analytically?