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Re: How to define a specific definite integral result in Mathematica

  • To: mathgroup at smc.vnet.net
  • Subject: [mg125564] Re: How to define a specific definite integral result in Mathematica
  • From: Antonio Alvaro Ranha Neves <aneves at gmail.com>
  • Date: Mon, 19 Mar 2012 04:55:52 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <jjmv02$cf1$1@smc.vnet.net> <jjrugh$93q$1@smc.vnet.net>

Dear nanobio9,

I don't think it is a simple problem of function definition. I'll try to explain it better, using this manipulation below:

Lets call Integrand the integrand of he integral in the previous post,

Integrand = 
  Sin[a] Exp[I r Cos[b] Cos[a]] LegendreP[n, m, Cos[a]] BesselJ[m, 
    r Sin[b] Sin[a]];

Now we want to derivate it with respect to r,

D[Integrand, r] // FullSimplify

In such a way that the integral,

Integrate[FullSimplify[D[Integrand, r]], {a, 0, \[Pi]}]

is equal to the derivative with respect to r,

D[2 I^(n - m) LegendreP[n, m, Cos[b]] SphericalBesselJ[n, r], 
  r] // FullSimplify

But to do this, Mathematica has first to "learn" or "memoraize" the following integral result,

Integrate[ 
 Sin[a] Exp[I r Cos[b] Cos[a]] LegendreP[n, m, Cos[a]] BesselJ[m, 
   r Sin[b] Sin[a]], {a, 0, \[Pi]}]

is equal to

2 I^(n - m) LegendreP[n, m, Cos[b]] SphericalBesselJ[n, r] 

Thanks,
Antonio

On Thursday, March 15, 2012 6:25:37 AM UTC+1, nanobio9 wrote:
> On 3=E6=9C=8813=E6=97=A5, =E4=B8=8B=E5=8D=884=E6=99=8203=E5=88=86, Antonio =
> Alvaro Ranha Neves <ane... at gmail.com>
> wrote:
> > Dear Mathematica users,
> >
> > I'd like to use the following integral for symbolic computation,
> >
> > Integrate[
> >  Sin[a] Exp[I r Cos[b] Cos[a]] LegendreP[n, m, Cos[a]] BesselJ[m,
> >  r Sin[b] Sin[a]], {a, 0, \[Pi]}]
> >
> > whose result is
> >
> > 2 I^(n - m) LegendreP[n, m, Cos[b]] SphericalBesselJ[n, r]
> >
> > is there a way to make Mathematica "learn" this result, so that I can work with symbolic computation of the integrand?
> >
> > Thanks,
> > Antonio
> 
> I hope that I got your question correctly. If you want to manipulate
> any of {I, n, m, b, r} later, you can just say
> myIntegral[ I_, n_, m_, b_, r_]:= Integrate[what you did]
> Later you can put any expression into I or n or m and so on.
> 
> Best



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