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Re: Series problem
*To*: mathgroup at smc.vnet.net
*Subject*: [mg4123] Re: Series problem
*From*: Paul Abbott <paul at earwax.pd.uwa.edu.au>
*Date*: Wed, 5 Jun 1996 01:39:14 -0400
*Organization*: University of Western Australia
*Sender*: owner-wri-mathgroup at wolfram.com
Tommy Nordgren wrote:
> I want to expand the Cosine only in the expression:
>Cos[b x] Exp[-x^2]/(k^2+x^2) into a taylor series around 0.
How about
Cos[b x] Exp[-x^2]/(k^2+x^2) /. Cos[a_] :> Normal[Cos[a] + O[x]^3]
2 2
b x
1 - -----
2
-------------
2
x 2 2
E (k + x )
Note that delayed replacement (:>) is required for otherwise the series
on the right-hand side of the rule is evaluated befor the replacement
operation is carried out.
> Making the series expansion in terms of b don't work, because
> Mathematica
> can't integrate the resulting series expansion in terms of x.
Yes it can:
Cos[b x] Exp[-x^2]/(k^2+x^2) + O[b]^3
2 2
1 x b 3
------------- - --------------- + O[b]
2 2
x 2 2 x 2 2
E (k + x ) 2 E (k + x )
It is best to Map the integration operation over each term in this
expression:
(Integrate[#, {x,-Infinity,Infinity}]& /@
Normal[%]) // PowerExpand
2
k
E Pi (1 - Erf[k])
------------------- +
k
2
2 k 1 2
b E k Sqrt[Pi] (2 Sqrt[Pi] + Gamma[-(-), 0, k ])
2
---------------------------------------------------
4
> Are there any way to handle this except by introducing a new
> representation for function series.
Another way is to use parametric differentiation. Noting that
D[Exp[-a x^2]/(k^2+x^2),a]
2
x
-(---------------)
2
a x 2 2
E (k + x )
then the integral:
gen[a_] = Integrate[Exp[-a x^2]/(k^2+x^2),
{x,-Infinity,Infinity}] // PowerExpand
2
a k
E Pi (1 - Erf[Sqrt[a] k])
-----------------------------
k
generates all the terms in the series expansion of Cos[b x]:
gen[1]
2
k
E Pi (1 - Erf[k])
-------------------
k
gen'[1]
2
k
-Sqrt[Pi] + E k Pi (1 - Erf[k])
Hence another (simpler) representation for the integral is
Sum[b^(2n)/(2n)! Derivative[n][gen][1], {n,0,1}]
2 2
2 k k
b (-Sqrt[Pi] + E k Pi (1 - Erf[k])) E Pi (1 - Erf[k])
-------------------------------------- + -------------------
2 k
Cheers,
Paul
_________________________________________________________________
Paul Abbott
Department of Physics Phone: +61-9-380-2734
The University of Western Australia Fax: +61-9-380-1014
Nedlands WA 6907 paul at physics.uwa.edu.au
AUSTRALIA http://www.pd.uwa.edu.au/Paul
_________________________________________________________________
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