       some integrals containing BesselJ

• To: mathgroup at smc.vnet.net
• Subject: [mg69390] some integrals containing BesselJ
• From: dimmechan at yahoo.com
• Date: Sun, 10 Sep 2006 07:20:01 -0400 (EDT)

```Hello to all.

***Mathematica evaluates correctly the following integrals (I have
converted everything to
InputForm)

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)?

***I have tried to add these rules to Integrate. For example:

Unprotect[Integrate];
Integrate[
Exp[-I*p_*r_*Cos[u_-v_]],{u_,0,2Pi},Assumptions->{p_>0,r_>0}]:=
2*Pi*BesselJ[0,p*r]

Integrate[ Exp[-I*p*r*Cos[u-v]],{u,0,2Pi},Assumptions->{p>0,r>0}]
2*Pi*BesselJ[0, p*r]

***But I failed when I tried to add the more general rules. For example
I can't make them work
for both symbolic and numeric values of m.

Any help will be greatly appreciate.

Dimitris

```

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