# Integration involving Bessel function and algebraic function

While trying to compute the following integrals, Mathematica is not able to do that. Any suggestion will be helpful.

Assuming[{R ∈ Reals, α ∈ Reals},
Integrate[q^4 BesselJ[1, q ]/(1 - α^2 q^2)^2, {q, 0, ∞}]
]

Assuming[{R ∈ Reals, α ∈ Reals},
Integrate[q^4 BesselJ[1, q ]/(1 - α^2 q^2)^3, {q, 0, ∞}]
]

• What makes you expect that the integrals can be written down in closed-form? – Henrik Schumacher Sep 12 '18 at 7:30
• In the "Table of Integrals, Series, and Products" by Gradshteyn and Ryzhik, (seventh edition), there is a similar integral given in page no 679, which reads – ark Sep 12 '18 at 8:30
• @ark I do not have the seventh edition, but in the fourth edition, which I have, this integral does not exist. Can you publish a copy of the page or write out the integral itself here? – Alex Trounev Sep 12 '18 at 15:06

Don't know about the non-convergence messages for NIntegrate, but this seems to work.

\$Assumptions = \[Alpha] \[Element] Reals

int = Integrate[
q^4 BesselJ[1, q]/(1 - \[Alpha]^2 q^2)^n, {q, 0, \[Infinity]}];

ConditionalExpression[((1/(\[Alpha]^4*Gamma[n - 1]*Gamma[n]))*Pi*Csc[Pi*n]*
Abs[\[Alpha]]^(-n - 3)*(Abs[\[Alpha]]^3*Gamma[n - 1]*
((1 - 4*\[Alpha]^2*(n^2 - 3*n + 2))*BesselI[n - 2, -(I/Abs[\[Alpha]])] -
BesselI[4 - n, -(I/Abs[\[Alpha]])]) -
2*I*\[Alpha]^4*(2*Gamma[n - 1]*BesselI[3 - n, -(I/Abs[\[Alpha]])] -
Gamma[n]*BesselI[n - 1, -(I/Abs[\[Alpha]])])))/
(2^n*E^((1/2)*I*Pi*n)), Re[n] > 7/4]

Limit[int, n -> 2] // FunctionExpand

(*Piecewise[{{(I*Pi*BesselJ[0, 1/\[Alpha]])/(4*\[Alpha]^6) - (Pi*BesselY[0, 1/\[Alpha]])/
(4*\[Alpha]^6) + (I*Pi*BesselJ[1, 1/\[Alpha]])/(2*\[Alpha]^5) -
(Pi*BesselY[1, 1/\[Alpha]])/(2*\[Alpha]^5), \[Alpha] >= 0}},
-((Pi*BesselY[0, 1/\[Alpha]])/(4*\[Alpha]^6)) +
((Log[1/\[Alpha]] - Log[I/\[Alpha]])*BesselJ[0, 1/\[Alpha]])/(2*\[Alpha]^6) -
(Pi*BesselY[1, 1/\[Alpha]])/(2*\[Alpha]^5) +
((Log[1/\[Alpha]] - Log[I/\[Alpha]])*BesselJ[1, 1/\[Alpha]])/\[Alpha]^5]*)

Limit[int, n -> 3] // FunctionExpand

(*Piecewise[{{(-4*I*Pi*\[Alpha]*BesselJ[0, 1/\[Alpha]] + I*Pi*BesselJ[1, 1/\[Alpha]] +
4*Pi*\[Alpha]*BesselY[0, 1/\[Alpha]] - Pi*BesselY[1, 1/\[Alpha]])/(16*\[Alpha]^7),
\[Alpha] >= 0}}, (-12*I*Pi*\[Alpha]*BesselJ[0, 1/\[Alpha]] +
3*I*Pi*BesselJ[1, 1/\[Alpha]] + 4*Pi*\[Alpha]*BesselY[0, 1/\[Alpha]] -
Pi*BesselY[1, 1/\[Alpha]])/(16*\[Alpha]^7)]*)


Without taking limits, the results are ComplexInfinity, which may explain why NIntegrate has trouble.

• Thank You. @Bill – ark Sep 13 '18 at 9:12
          Quiet[Integrate[
q^4 BesselJ[1, q]/(1 - \[Alpha]^2 q^2)^n, {q, 0, \[Infinity]}]]

(*ConditionalExpression[(1/(\[Alpha]^4 Gamma[n]))
4^-n (-\[Alpha]^2)^(-(1/2) -
n) (2^n \[Pi] (-\[Alpha]^2)^(
n/2) (4 \[Alpha]^2 BesselI[3 - n, 1/Sqrt[-\[Alpha]^2]] -
Sqrt[-\[Alpha]^2] BesselI[4 - n, 1/Sqrt[-\[Alpha]^2]]) Csc[
n \[Pi]] +
16 (-1 + n) \[Alpha]^4 Sqrt[-\[Alpha]^2]
Gamma[3 - n] HypergeometricPFQ[{n}, {-2 + n, -1 + n}, -(1/(
4 \[Alpha]^2))]),
4 Re[n] >
7 && (Re[\[Alpha]^2] <= 0 || \[Alpha]^2 \[NotElement] Reals)]*)
for real \[Alpha]^2 intégral is divergent.

• I was looking for a compact analytical form of the mentioned integral not the Numerical integration, if possible. Thanks – ark Sep 12 '18 at 9:05
• It is dangerous to switch of the MMA error messages, especially in the case of not converging integrals! – Ulrich Neumann Sep 12 '18 at 9:52

If you try to solve the integrals numerically

int[\[Alpha]_?NumericQ, n_?NumericQ] :=NIntegrate[q^4 BesselJ[1, q]/(1 - \[Alpha]^2 q^2)^n, {q, 0, \[Infinity]} ]


the evaluation of for example

int[  1, 2]


gives error messages NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in q near {q} = {1.00277}. NIntegrate obtained 6648.614253396791 and 6097.681026759085for the integral and error estimates.

Elaborate the comment Henrik Schumacher gave...