Re: Integrating by parts....II

*To*: mathgroup at smc.vnet.net*Subject*: [mg8752] Re: [mg8724] Integrating by parts....II*From*: "w.meeussen" <meeussen.vdmcc at vandemoortele.be>*Date*: Sat, 20 Sep 1997 22:28:17 -0400*Sender*: owner-wri-mathgroup at wolfram.com

Adrean, first a typo: use Exp[-x^2] or E^-x^2 but not E[-x^2]. second: there is always a spot of difficulty with orthogonal functions like HermiteH, LegendreP, LaguerreL, ... if you solve the *defining* differential equation, then you can get results= in terms of Hypergeometric functions, that even sometimes do not (easily ?) satisfy the DE. DSolve[y''[x]-2x y'[x]+2 n y[x]=3D=3D0,y,x] Out[63]=3D -n 1 2 {{y -> (C[1] Hypergeometric1F1[--, -, #1 ] +=20 2 2 =20 1 n 3 2 C[2] Hypergeometric1F1[- - -, -, #1 ] #1 & )}} 2 2 2 now try to get y''[x]-2x y'[x]+2 n y[x] /.%[[1,1]] to equal 0 !?!? no way, Jose ! third: for your specific problem, set n equal to 1, 3, 5, ... and then to 0, 2, 4, ... and see how nice it is! It's even nicer if you split the integration into {x,-Infinity,0} and {x,0,Infinity}. fourth: this is again a nice example where a future Add-On could do the induction (on n) for us. ****************************************************************************= *** Explicit polynomials are given for non=ADnegative integers n.=20 The Hermite polynomials satisfy the differential equation \!\(TraditionalForm\`y\^\[Prime]\[Prime] - 2 x y\^\[Prime] + 2 n y =3D= 0\). =20 They are orthogonal polynomials with weight function=20 \!\(TraditionalForm\`e\^\(-x\^2\)\) in the interval (0,\[Infinity]).=20 HermiteH[n, x] is an entire function of x with no branch=20 cut discontinuities.=20 ****************************************************************************= *** At 02:48 19-09-97 -0400, Adrean Webb wrote: >My appologies - I submitted a query yesterday asking om how to integrate >by parts to solve a particulary function. I overlooked the simple fact >that n was still undefined and it would iterate undefinitely.=20 > >Can Mathematica still solve the function below? > > f[n] =3D Integrate[Cos[x]*E[-x^2]*HermiteH[n,x],{x,-Inf,Inf}]=20 > >Any help would be appreciated. >Thanks, Adrean > > > > Dr. Wouter L. J. MEEUSSEN wm.vdmcc at pophost.eunet.be w.meeussen.vdmcc at vandemoortele.be