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MathGroup Archive 2006

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BesselJ integrals

  • To: mathgroup at smc.vnet.net
  • Subject: [mg70077] BesselJ integrals
  • From: dimmechan at yahoo.com
  • Date: Mon, 2 Oct 2006 00:33:56 -0400 (EDT)

Hello to all.

Mathematica evaluates correctly the following integrals

Assuming[p>0&&r>0,Integrate[Exp[-I*p*r*Cos[u]]*{Cos[u],Sin[u]},{u,0,2Pi}]]

{(-2*I)*Pi*BesselJ[1, p*r], 0}


Assuming[p>0&&r>0,Integrate[
Exp[-I*p*r*Cos[u]]*{Cos[2*u],Sin[2*u]},{u,0,2Pi}]]
{-2*Pi*BesselJ[2, p*r], 0}


Assuming[p>0&&r>0,Integrate[
Exp[-I*p*r*Cos[u]]*{Cos[3*u],Sin[3*u]},{u,0,2Pi}]]
{(2*I)*Pi*BesselJ[3, p*r], 0}

Can someone explain me why however fails to evaluate the general
case (which is equal to {2Pi*(-I)^m*BesselJ[m,p*r],0}, cf. e.g.
McLachlan 1955)?


Assuming[p>0&&r>0&m>0,Integrate[
Exp[-I*p*r*Cos[u]]*{Cos[m*u],Sin[m*u]},{u,0,2Pi}]]
{Integrate[Cos[m*u]/E^(I*p*r*Cos[u]), {u, 0, 2*Pi}],
Integrate[Sin[m*u]/E^(I*p*r*Cos[u]), {u, 0, 2*Pi}]}

Also Mathematica fails to evaluate the following integrals (even for
given m)

Assuming[p>0&&r>0&m>0,Integrate[Exp[-I*p*r*Cos[u-v]]*{Cos[m*u],Sin[m*u]},{u,0,2Pi}]]

{Integrate[Cos[m*u]/E^(I*p*r*Cos[u - v]), {u, 0, 2*Pi}],
Integrate[Sin[m*u]/E^(I*p*r*Cos[u - v]), {u, 0, 2*Pi}]}

which (cf. e.g. McLachlan 1955) are equal to

{2Pi*(-I)^m*BesselJ[m,p*r]*Cos[v],2Pi*(-I)^m*BesselJ[m,p*r]*Sin[v]})

and as well the integral


Assuming[p>0&&r>0,Integrate[ Exp[-I*p*r*Cos[u-v]],{u,0,2Pi}]]
Integrate[E^((-I)*p*r*Cos[u - v]), {u, 0, 2*Pi}]

which is equal to 2Pi*BesselJ[0,p*r]

Can somehow "help" Mathematica to evaluate these integrals (e.g
last integral is evaluated by hand considering that the integral of
a periodic function is the same regrardless where the integration
begins)? 

Any help will be greatly appreciate. 

Thanks.


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