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Re: Hilbert transform bug in 7.0.3?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg100878] Re: Hilbert transform bug in 7.0.3?
*From*: dh <dh at metrohm.com>
*Date*: Wed, 17 Jun 2009 04:36:26 -0400 (EDT)
*References*: <h19i5r$p4g$1@smc.vnet.net>
Hi Nacho,
your problem comes from the fact that you are using PrincipalValue ->
True together with DiracDelta. Consider the simplest case:
Integrate[DiracDelta[x], {x, -Infinity, Infinity}]
Integrate[DiracDelta[x], {x, -Infinity, Infinity},PrincipalValue -> True]
The first integral gives 1 in agreement with the definition of
DiracDelta. The second integral is actually a limit of two integrals.
One from -Infinity to epsilon. The second from epsilon to Infinity. For
epsilon > 0 both integrals are zero. Therefore, the limit is also zero.
Daniel
Nacho wrote:
> Hello.
>
> I've trying the Hilbert Transform defined in Mathworld as:
>
> HilbertTransform[f_, x_, y_, assum___?OptionQ] :=
> Integrate[f/(x - y), {x, -Infinity, Infinity},
> PrincipalValue -> True, assum]/Pi
>
>
> I've trying to transform some functions but DiracDelta[x] seems to
> fail:
>
> In[5]:= HilbertTransform[DiracDelta[x],x,y]
> Out[5]= 0
>
> But it should be -1/(Pi y) according with Mathworld or 1/(Pi y)
> according with Wikipedia, but not just 0. Older versions seems to
> work, as you can see in Mathworld's notebook.
>
> Is this a bug? Any other way to calculate Hilbert Transforms in
> 7.0.3?
>
> Thanks.
>
>
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