       Re: Series problem

• To: mathgroup at smc.vnet.net
• Subject: [mg4116] Re: Series problem
• From: withoff (David Withoff)
• Date: Wed, 5 Jun 1996 01:38:00 -0400
• Organization: Wolfram Research, Inc.
• Sender: owner-wri-mathgroup at wolfram.com

```In article <4ojdna\$5i8 at dragonfly.wolfram.com> f85-tno at telesto.nada.kth.se
(Tommy Nordgren) writes:
>
> 	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.
> Computing the series in terms of x don't work, because the exponetial
and
> the divisor will be expanded as well, when the series of the Cosine is
> multiplied by the other factors.
> Making the series expansion in terms of b don't work, because
Mathematica
> can't integrate the resulting series expansion in terms of x.
> Are there any way to handle this except by introducing a new
representation
> for function series.
> (The problem I'm currently interested in is finding a series for the
function
> f[b_,k_] = Integrate[Cos[b x]
Exp[-x^2]/(k^2+x^2),{x,-Infinity,Infinity}],
> which is valid for small b)
> --
>
-------------------------------------------------------------------------
> Tommy Nordgren                    "Home is not where you are born,
> Royal Institute of Technology      but where your heart finds peace."
> Stockholm                         Tommy Nordgren - The dying old crone
>
--------------------------------------------------------------------------
>

===============================================

Will something like this work?

In:= integrand = Expand[Normal[
Series[Cos[b x] Exp[-x^2]/(k^2+x^2), {b, 0, 4}] ] ]

2  2              4  4
1              b  x              b  x
Out= ------------- - --------------- + ----------------
2                 2                  2
x    2    2       x    2    2        x    2    2
E   (k  + x )   2 E   (k  + x )   24 E   (k  + x )

In:= Integrate[integrand, {x, -Infinity, Infinity}]

2
2             2  k        2
-Sqrt[Pi]   Sqrt[Pi]   b  Sqrt[Pi]   b  E   Sqrt[k ] Pi
Out= --------- + -------- - ----------- + ------------------ +
4          2            2               2
2 k          k          4 k

2                    2
k        2        4  k   2       2
E   Sqrt[k ] Pi   b  E   k  Sqrt[k ] Pi
>    --------------- + --------------------- -
2                   24
k

2
2  k        2                   3       2
3 b  E   Sqrt[k ] Sqrt[Pi] Gamma[-(-), 0, k ]
2
>    --------------------------------------------- -
8

2
k        2                   3       2
3 E   Sqrt[k ] Sqrt[Pi] Gamma[-(-), 0, k ]
2
>    ------------------------------------------ -
2
4 k

2
4  k   2       2                   3       2
b  E   k  Sqrt[k ] Sqrt[Pi] Gamma[-(-), 0, k ]
2
>    ----------------------------------------------
32

Dave Withoff
Research and Development
Wolfram Research

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