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.