I'm trying to do the integral
Integrate[ B^2*BesselK[0, ko*ρ]^2*2 π*ρ, {ρ, a, ∞}]
which I figured should be relatively simple as the integral of a Bessel Function outside of a certain circle (of radius a) from a to infinity. Mathematica throws me the answer
ConditionalExpression[(B^2 π^(3/2) MeijerG[{{}, {3/2}}, {{0, 1, 1}, {}}, a^2 ko^2])/(2 ko^2), Re[a] > 0 && Im[a] == 0 && Re[ko] > 0]
The conditional expression is expected, but I was wondering if it's possible to expand the G function in terms of other Bessel Functions, especially the modified one $K_o$. Integrating across both regions and subtracting them gives a similar answer; while the answer for the integration from $0\to \inf$ is a constant, the integral from 0 to a involves the G function again, like
Integrate[B^2*BesselK[0, ko*ρ]^2*2 π*ρ, {ρ, 0, a}]
ConditionalExpression[( B^2 π^(3/2) MeijerG[{{1}, {3/2}}, {{1, 1, 1},{0}},a^2 ko^2])/(2 ko^2), Re[a ko] > 0]
I'm hoping to get a wholly symbolic answer or at least an expansion of one, I was wondering if it's possible to have Mathematica get rid of the G function stuff in these results?
int = Integrate[B^2*BesselK[0, ko*\[Rho]]^2*2 \[Pi]*\[Rho], \[Rho]];
Limit[int, \[Rho] -> Infinity, Direction -> "FromBelow"] - (int /. \[Rho] -> a)
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