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.

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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