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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
> f85-tno at nada.kth.se
>
--------------------------------------------------------------------------
>
===============================================
Will something like this work?
In[7]:= 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[7]= ------------- - --------------- + ----------------
2 2 2
x 2 2 x 2 2 x 2 2
E (k + x ) 2 E (k + x ) 24 E (k + x )
In[8]:= Integrate[integrand, {x, -Infinity, Infinity}]
2
2 2 k 2
-Sqrt[Pi] Sqrt[Pi] b Sqrt[Pi] b E Sqrt[k ] Pi
Out[8]= --------- + -------- - ----------- + ------------------ +
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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