Cannot understand the meaning of Derivative[1, 0][BesselK][-M, 2]?

Question

When I do the following integration
Integrate[(Log[x]/x)*x^M*Exp[-x-1/x],{x,0,[Infinity]},Assumptions->Element[M,PositiveIntegers]]

Mathematica return a very strange result that is

In this case what is the meaning of Derivative[1,0]. Is it some kind of dirac delta function ?

Roman · Accepted Answer

For non-integer $M$ we can get an explicit expression for this derivative, as given on the Wolfram Functions site:
f[M_] = Derivative[1, 0][BesselK][-M, 2] // FunctionExpand
(*    huge result    *)

f[0.37]
(*    -0.0179285    *)

Plot[f[M], {M, -1.5, 1.5}]

For specific values of $M$ we can get explicit expressions for this derivative directly:
Table[Derivative[1, 0][BesselK][-M, 2], {M, 0, 7}] // FunctionExpand

(*    {0,
       -1/2 BesselK[0, 2],
       -1/2 BesselK[0, 2] - BesselK[1, 2],
       -5/2 BesselK[0, 2] - 3 BesselK[1, 2],
       -10 BesselK[0, 2] - 13 BesselK[1, 2],
       -99/2 BesselK[0, 2] - 65 BesselK[1, 2],
       -575/2 BesselK[0, 2] - 381 BesselK[1, 2],
       -3863/2 BesselK[0, 2] - 2576 BesselK[1, 2]}    *)

Cannot understand the meaning of Derivative[1, 0][BesselK][-M, 2]?

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