Re: Hilbert transform bug in 7.0.3?
- To: mathgroup at smc.vnet.net
- Subject: [mg100916] Re: Hilbert transform bug in 7.0.3?
- From: did <didier.oslo at hotmail.com>
- Date: Thu, 18 Jun 2009 04:52:45 -0400 (EDT)
- References: <h19i5r$p4g$1@smc.vnet.net>
On Jun 17, 3:50 am, Nacho <ncc1701... at gmail.com> 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.
Try something like:
HilbertTransform[f_, x_, y_] :=
Integrate[(f[y + x] - f[y - x])/x, {x, -Infinity,
Infinity},Assumptions -> y \[Element] Reals]/Pi/2;
HilbertTransform[DiracDelta, x, y]
and adapt it for the general case you need.