# How can I analytically integrate a BesselK function

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.

• The first and the second integral are not equivalent. MA can compute the first integral without the limits in terms of the Meijer function. You can substitute limits manually. Commented Jan 8, 2020 at 21:03

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

• The limit at $\infty$ appears to approach $T^2\pi$ by numerically checking large $s$ for several values of $T$. Commented Jan 9, 2020 at 8:09