# 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. Jan 8 '20 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 $T$. Jan 9 '20 at 8:09