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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

enter image description here

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

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    $\begingroup$ Would it help if you read it as $$\left.\frac{\mathrm d}{\mathrm d\nu}K_{\nu}(2)\right|_{\nu=-M}$$? $\endgroup$ – J. M.'s torpor Dec 26 '20 at 10:13
  • $\begingroup$ This is very strange, is there any way to obtain any numerical result from this expression ? $\endgroup$ – Tuong Nguyen Minh Dec 26 '20 at 10:22
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    $\begingroup$ Have you already tried supplying a specific numerical value of M, and then using N[]? $\endgroup$ – J. M.'s torpor Dec 26 '20 at 10:23
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    $\begingroup$ From the documentation: "Derivative[n1,n2,…][f] is (...) a function obtained from f by differentiating n1 times with respect to the first argument, n2 times with respect to the second argument, and so on." which is what @J.M. wrote. $\endgroup$ – Roman Dec 26 '20 at 13:31
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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}]

enter image description here

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]}    *)
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  • $\begingroup$ Thank you so much I am interested in the case of integer M :) Particularly for M = 1,2,3,4.... $\endgroup$ – Tuong Nguyen Minh Dec 27 '20 at 6:15
  • $\begingroup$ Is there any way to add the assumption that M is a positive integer into the command Derivative[1, 0][BesselK][-M, 2] ? $\endgroup$ – Tuong Nguyen Minh Dec 27 '20 at 12:18
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    $\begingroup$ @TuongNguyenMinh I don't know how to do it directly in Mathematica, but there's a formula on the Wolfram Functions site that can be simplified to your f[M_Integer?NonNegative] := -Sum[M!/(2*k!*(M-k))*BesselK[k,2], {k, 0, M-1}] // FullSimplify. $\endgroup$ – Roman Dec 27 '20 at 13:28
  • $\begingroup$ Oh this is so nice thank you ! $\endgroup$ – Tuong Nguyen Minh Dec 27 '20 at 14:06

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