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MathGroup Archive 1996

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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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