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