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. > >