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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
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