How can I analytically integrate a BesselK function

Question

I want to integrate the function below analytically, so that later on I can use the result for numerical calculations. But it seems Mathematica can not handle it the way I express it. However, if I change the argument of the Bessel function from $s^{1/2}$ to $s$, then Mathematica handles it easily.

I tried to evaluate

int1 = 
  Integrate[BesselK[1, s^(1/2)/T], {s, 4 mX^2, Infinity}, 
    GenerateConditions -> False]

and got this output:

Integrate[BesselK[1, Sqrt[s]/T], {s, 4 mX^2, Infinity}, 
  GenerateConditions -> False]

But after changing the argument of the Bessel function from $s^{1/2 }$ to $s$:

Integrate[BesselK[1, s/T], {s, 4 mX^2, Infinity}, GenerateConditions -> False]

I got this output:

BesselK[0, (4 Sqrt[1/T^2])/Sqrt[1/mX^4]]/Sqrt[1/T^2]

which I can further use for numerical evaluation.

I need some suggestions.

Answer 1

The integration limit at $\infty$ causes problems. One idea is to integrate over the range from 0 to $\infty$, and then subtract off the integral from 0 to 4 mX^2:

i0 = Integrate[BesselK[1, Sqrt[s]/T], {s, 0, Infinity}, Assumptions->T>0]
i[a_] = Integrate[BesselK[1, Sqrt[s]/T], {s, 0, a}, Assumptions->T>0 && a>0]

π T^2

2 T^2 MeijerG[{{1}, {}}, {{1/2, 3/2}, {0}}, a/(4 T^2)]

Then, the desired integral is:

int[T_, mX_] = i0 - i[4 mX^2]

π T^2 - 2 T^2 MeijerG[{{1}, {}}, {{1/2, 3/2}, {0}}, mX^2/T^2]

As a check, here's a function to integrate numerically:

nint[T_, mX_] := NIntegrate[BesselK[1, Sqrt[s]/T], {s, 4 mX^2, Infinity}]

Comparison:

int[10, 2] //N
nint[10, 2]

238.203

238.203