Is Mathematica unaware of an integral representation of the modified Bessel function of the second kind BesselK[0, x]?

Question

Mathematica is unable to produce a result for the following definite integral:

$$\int\limits_{0}^{\infty}\cos{(x \sinh{t})} \ dt$$

Integrate[Cos[x Sinh[t]], {t, 0, Infinity}]

However, this is a standard integral of the form BesselK[0, x], the modified Bessel function of the second kind. Is Mathematica unaware of this definition?

It seems to be aware of the equivalent integral representation Integrate[Cos[x t]/Sqrt[t^2 + 1], {t, 0, Infinity}, Assumptions -> x \[Element] Reals], which returns BesselK[0, Abs[x]]. Mathematica only "knows" the special integrals that it was taught... — MarcoB, May 11 '18 at 15:34
If you add the assumption that x in Reals, then you get the message: Integrate::idiv: Integral of Cos[x Sinh[t]] does not converge on {0,[Infinity]}. — bill s, May 11 '18 at 15:51
Related: https://mathematica.stackexchange.com/q/4728/1871 (Notice the posts linked to this one. ) — xzczd, May 12 '18 at 03:24

Mariusz Iwaniuk · Answer 1 · 2018-05-11T17:09:23.493

8

Workaround:

func=Cos[x*Sinh[t]];    

InverseMellinTransform[Integrate[MellinTransform[func, x, s], {t, 0, Infinity}, 
Assumptions -> 0 < s < 1], s, x]

(* BesselK[0, x] *)

FourierCosTransform[Integrate[FourierCosTransform[func, x, s], {t, 0, Infinity}, 
Assumptions -> s > 0], s, x]

(* BesselK[0, x] *)

Summary: Obviously nothing's foolproof.

edited May 11 '18 at 17:09

answered May 11 '18 at 16:53

Mariusz Iwaniuk

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Michael E2 · Answer 2 · 2018-05-12T11:23:24.900

You can try dumb substitutions:

Integrate[
 Cos[x Sinh[t]] Dt[t] /. First@Solve[x Sinh[t] == u, t, Reals] /. {Dt[u] -> 1, Dt[x] -> 0},
 {u, 0, ∞}, Assumptions -> x > 0]
(*  BesselK[0, x]  *)

The above converts the integral to almost the same as in @MarcoB's comment.

Here's a general function to do the above more or less automatically:

ClearAll[trydumbsubs];
SetAttributes[trydumbsubs, HoldAll];
trydumbsubs[Integrate[f_, {t_, a_, b_}, Assumptions -> hyp_]] :=
  Module[{subs, sub, res = $Failed},
   subs = SortBy[
      DeleteCases[DeleteDuplicates@Cases[f, (g_)[arg_] :> arg, {0, Infinity}], t],
      -LeafCount[#] &];
   Do[
     sub = Solve[arg == u, t, Reals];
     If[ListQ[sub],
      res = Integrate[
        f*Dt[t] /. First@sub /. {Dt[u] -> 1, _Dt -> 0},
        {u, Limit[arg, t -> a, Assumptions -> hyp], 
         Limit[arg, t -> b, Assumptions -> hyp]},
        Assumptions -> hyp
        ],
      res = $Failed];
     If[FreeQ[res, Integrate | $Failed],
      Return[res]
      ],
     {arg, subs}];
   res /; FreeQ[res, Integrate | $Failed]
   ];

OP's example:

trydumbsubs@ Integrate[Cos[x Sinh[t]], {t, 0, Infinity}, Assumptions -> x > 0 && t > 0]
(*  BesselK[0, x]  *)

I tried yours code for general function(automata) and dosen't work ? — Mariusz Iwaniuk, May 12 '18 at 08:39
@MariuszIwaniuk Dumb editing mistake. Fixed now, I think. Thanks. — Michael E2, May 12 '18 at 11:29

Is Mathematica unaware of an integral representation of the modified Bessel function of the second kind BesselK[0, x]?

2 Answers2