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MathGroup Archive 2007

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Re: FourierTransform and removable singularities

  • To: mathgroup at smc.vnet.net
  • Subject: [mg75404] Re: FourierTransform and removable singularities
  • From: Roman <rschmied at gmail.com>
  • Date: Sat, 28 Apr 2007 06:00:40 -0400 (EDT)
  • References: <f0kbmk$qvt$1@smc.vnet.net><f0pk9v$1t0$1@smc.vnet.net>

David, Peter:

I like this workaround best, using Peter's Piecewise[] function.
Still, some manual work required, but pretty quick through defining
something like

fixfunction[f_,x0_] := With[{f0=Limit[f[x],x->x0]},
Piecewise[{{f0,x0}},f[#]]&]

where you can just "fix" a function's removable singularity at x0,
e.g.,

fixfunction[Sin[#]/#&, 0]

Thanks for the help!
Roman.


On Apr 27, 11:21 am, Peter Pein <pet... at dordos.net> wrote:
> David W.Cantrell schrieb:
> ...
>
> > Perhaps the best way to handle you problem would be to have the sine
> > cardinal function
>
> > |                {  1          if  x = 0,
> > |    sinc(x)  =  {
> > |                {  sin(x)/x   otherwise
>
> > implemented in Mathematica. But defining that function yourself, it does
> > not work as desired with FourierTransform.
>
> > David W. Cantrell
>
> Hi David,
>
> sorry, I did not believe this. And indeed:
>
> In[1]:= Off[General::spell];
>         Sinc[t_] := Piecewise[{{1, t == 0}}, Sin[t]/t]
>
> In[3]:= FourierTransform[DiracDelta[t]*Sinc[t], t, w]
> Out[3]= 1/Sqrt[2*Pi]
>
> In[4]:= TrigToExp[FourierTransform[
>             DiracDelta[t - b] * Sinc[omega*(t - b)], t, w]]
> Out[4]= E^(I*b*w)/Sqrt[2*Pi]
>
> In[5]:= $Version
> Out[5]= "5.2 for Linux x86 (64 bit) (June 20, 2005)"
>
> Why isn't this the wanted result? Or did you have other (more
> complicated) FourierTransform[]s in mind?
>
> Regards,
> Peter




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