I thought there is no Fourier transform exist for log(1-g*i*x)/(i*x-d), where g and d are positive real numbers, "i" is the imaginary unit, x is a real number on (-inf, +inf), am I right? If we allow the generalized Fourier transforms such as dirac delta function and Swartz functions, does it have a generalized FT? Thanks!
help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)?
Started by ●July 1, 2007
Reply by ●July 1, 20072007-07-01
"Vista" <abc@gmai.com> writes:> I thought there is no Fourier transform exist for log(1-g*i*x)/(i*x-d), > > where g and d are positive real numbers, "i" is the imaginary unit, x is a > real number on (-inf, +inf), > > am I right?This function is not in L^1, but it is in L^2, so the L^2 version of this Fourier transform does exist.> If we allow the generalized Fourier transforms such as dirac delta function > > and Swartz functions, > > does it have a generalized FT?Yes, certainly it is a tempered distribution, and as such it has a Fourier transform. -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2
Reply by ●July 1, 20072007-07-01
"Robert Israel" <israel@math.MyUniversitysInitials.ca> wrote in message news:rbisrael.20070702020320$37e5@news.ks.uiuc.edu...> "Vista" <abc@gmai.com> writes: > >> I thought there is no Fourier transform exist for log(1-g*i*x)/(i*x-d), >> >> where g and d are positive real numbers, "i" is the imaginary unit, x is >> a >> real number on (-inf, +inf), >> >> am I right? > > This function is not in L^1, but it is in L^2, so the L^2 version of this > Fourier transform does exist. > >> If we allow the generalized Fourier transforms such as dirac delta >> function >> >> and Swartz functions, >> >> does it have a generalized FT? > > Yes, certainly it is a tempered distribution, and as such it has a Fourier > transform. > -- > Robert Israel israel@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada V6T 1Z2Thanks Robert! What's the expression of its tempered distribution form? Where to read more about these stuffs? Much appreciated!
Reply by ●July 2, 20072007-07-02
Vista :
> I thought there is no Fourier transform exist for log(1-g*i*x)/(i*x-d),
>
> where g and d are positive real numbers, "i" is the imaginary unit, x is a
> real number on (-inf, +inf),
>
> am I right?
>
> If we allow the generalized Fourier transforms such as dirac delta function
> and Swartz functions,
>
> does it have a generalized FT?
>
> Thanks!
Hello Vista.
Very interesting the threeads based to your queries.
In Mma 5.2 we have
In[42]:=
PowerExpand[Log[(1 - I*g*x)/(-d + I*x)]]
(FourierTransform[#1, x, s] & ) /@ %
Out[42]=
-Log[-d + I*x] + Log[1 - I*g*x]
Out[43]=
(I*E^(s/g)*(Log[-(I/g)] - Log[I/g] + I*Pi*Sign[s]))/(Sqrt[2*Pi]*s) +
(E^(d*s)*(Pi - I*(Log[(-I)*d] - Log[I*d])*Sign[s]))/
(Sqrt[2*Pi]*Abs[s])
Dimitris
Reply by ●July 2, 20072007-07-02
"dimitris" <dimmechan@yahoo.com> wrote in message news:1183354955.382758.45910@w5g2000hsg.googlegroups.com...> > Vista : >> I thought there is no Fourier transform exist for log(1-g*i*x)/(i*x-d), >> >> where g and d are positive real numbers, "i" is the imaginary unit, x is >> a >> real number on (-inf, +inf), >> >> am I right? >> >> If we allow the generalized Fourier transforms such as dirac delta >> function >> and Swartz functions, >> >> does it have a generalized FT? >> >> Thanks! > > Hello Vista. > Very interesting the threeads based to your queries. > > In Mma 5.2 we have > > In[42]:= > PowerExpand[Log[(1 - I*g*x)/(-d + I*x)]] > (FourierTransform[#1, x, s] & ) /@ % > > Out[42]= > -Log[-d + I*x] + Log[1 - I*g*x] > > Out[43]= > (I*E^(s/g)*(Log[-(I/g)] - Log[I/g] + I*Pi*Sign[s]))/(Sqrt[2*Pi]*s) + > (E^(d*s)*(Pi - I*(Log[(-I)*d] - Log[I*d])*Sign[s]))/ > (Sqrt[2*Pi]*Abs[s]) > > Dimitris >Interesting! I didn't know Mathematica can do this sort of FT... I will give it a try! Thanks a lot!
Reply by ●July 2, 20072007-07-02
On Jul 1, 10:42 pm, dimitris <dimmec...@yahoo.com> wrote:> Vista : > > > I thought there is no Fourier transform exist for log(1-g*i*x)/(i*x-d), > > > where g and d are positive real numbers, "i" is the imaginary unit, x is a > > real number on (-inf, +inf), > > > am I right? > > > If we allow the generalized Fourier transforms such as dirac delta function > > and Swartz functions, > > > does it have a generalized FT? > > > Thanks! > > Hello Vista. > Very interesting the threeads based to your queries. > > In Mma 5.2 we have > > In[42]:= > PowerExpand[Log[(1 - I*g*x)/(-d + I*x)]]My reading of the original question was that it involved (in Mma format) Log[1-I*g*x]/(-d+I*x), not Log[(1+I*g*x)/(-d+I*x)]. Does Mma find a Fourier transform of this one? -- Robert Israel israel@math.MyUniversitysInitials.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2
Reply by ●July 2, 20072007-07-02
<israel@math.ubc.ca> wrote in message news:1183396435.238553.204280@m37g2000prh.googlegroups.com...> On Jul 1, 10:42 pm, dimitris <dimmec...@yahoo.com> wrote: >> Vista : >> >> > I thought there is no Fourier transform exist for log(1-g*i*x)/(i*x-d), >> >> > where g and d are positive real numbers, "i" is the imaginary unit, x >> > is a >> > real number on (-inf, +inf), >> >> > am I right? >> >> > If we allow the generalized Fourier transforms such as dirac delta >> > function >> > and Swartz functions, >> >> > does it have a generalized FT? >> >> > Thanks! >> >> Hello Vista. >> Very interesting the threeads based to your queries. >> >> In Mma 5.2 we have >> >> In[42]:= >> PowerExpand[Log[(1 - I*g*x)/(-d + I*x)]] > > My reading of the original question was that it involved (in Mma > format) > Log[1-I*g*x]/(-d+I*x), not Log[(1+I*g*x)/(-d+I*x)]. Does Mma find a > Fourier > transform of this one? > > -- > Robert Israel israel@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada V6T 1Z2 > > >I also found it was wrong. And I was unable to find the FT using MMA.
Reply by ●July 3, 20072007-07-03
isr...@math.ubc.ca :> On Jul 1, 10:42 pm, dimitris <dimmec...@yahoo.com> wrote: > > Vista : > > > > > I thought there is no Fourier transform exist for log(1-g*i*x)/(i*x-d), > > > > > where g and d are positive real numbers, "i" is the imaginary unit, x is a > > > real number on (-inf, +inf), > > > > > am I right? > > > > > If we allow the generalized Fourier transforms such as dirac delta function > > > and Swartz functions, > > > > > does it have a generalized FT? > > > > > Thanks! > > > > Hello Vista. > > Very interesting the threeads based to your queries. > > > > In Mma 5.2 we have > > > > In[42]:= > > PowerExpand[Log[(1 - I*g*x)/(-d + I*x)]] > > My reading of the original question was that it involved (in Mma > format) > Log[1-I*g*x]/(-d+I*x), not Log[(1+I*g*x)/(-d+I*x)]. Does Mma find a > Fourier > transform of this one? > > -- > Robert Israel israel@math.MyUniversitysInitials.ca > Department of Mathematics http://www.math.ubc.ca/~israel > University of British Columbia Vancouver, BC, Canada V6T 1Z2Sorry but during the process of converting the expression from Maple to Mma convention I did a mistake. Unfortunately there is not a command in Mathematica similar to convert("Mma expression",FromMma)... Dimitris
Reply by ●July 3, 20072007-07-03
Vista :
> <israel@math.ubc.ca> wrote in message
> news:1183396435.238553.204280@m37g2000prh.googlegroups.com...
> > On Jul 1, 10:42 pm, dimitris <dimmec...@yahoo.com> wrote:
> >> Vista :
> >>
> >> > I thought there is no Fourier transform exist for log(1-g*i*x)/(i*x-d),
> >>
> >> > where g and d are positive real numbers, "i" is the imaginary unit, x
> >> > is a
> >> > real number on (-inf, +inf),
> >>
> >> > am I right?
> >>
> >> > If we allow the generalized Fourier transforms such as dirac delta
> >> > function
> >> > and Swartz functions,
> >>
> >> > does it have a generalized FT?
> >>
> >> > Thanks!
> >>
> >> Hello Vista.
> >> Very interesting the threeads based to your queries.
> >>
> >> In Mma 5.2 we have
> >>
> >> In[42]:=
> >> PowerExpand[Log[(1 - I*g*x)/(-d + I*x)]]
> >
> > My reading of the original question was that it involved (in Mma
> > format)
> > Log[1-I*g*x]/(-d+I*x), not Log[(1+I*g*x)/(-d+I*x)]. Does Mma find a
> > Fourier
> > transform of this one?
> >
> > --
> > Robert Israel israel@math.MyUniversitysInitials.ca
> > Department of Mathematics http://www.math.ubc.ca/~israel
> > University of British Columbia Vancouver, BC, Canada V6T 1Z2
> >
> >
> >
>
> I also found it was wrong. And I was unable to find the FT using MMA.
Mma 5.2 fails.
As I was informed in Mma 6 you get
In[47]:= FourierTransform[Log[1 - I*g*x]/(-d + I*x), x, s,
Assumptions -> g > 0 && d > 0]
Out[47]= 0
Hope it helps,
Dimitris
Reply by ●July 3, 20072007-07-03
dimitris :
> Vista :
> > <israel@math.ubc.ca> wrote in message
> > news:1183396435.238553.204280@m37g2000prh.googlegroups.com...
> > > On Jul 1, 10:42 pm, dimitris <dimmec...@yahoo.com> wrote:
> > >> Vista :
> > >>
> > >> > I thought there is no Fourier transform exist for log(1-g*i*x)/(i*x-d),
> > >>
> > >> > where g and d are positive real numbers, "i" is the imaginary unit, x
> > >> > is a
> > >> > real number on (-inf, +inf),
> > >>
> > >> > am I right?
> > >>
> > >> > If we allow the generalized Fourier transforms such as dirac delta
> > >> > function
> > >> > and Swartz functions,
> > >>
> > >> > does it have a generalized FT?
> > >>
> > >> > Thanks!
> > >>
> > >> Hello Vista.
> > >> Very interesting the threeads based to your queries.
> > >>
> > >> In Mma 5.2 we have
> > >>
> > >> In[42]:=
> > >> PowerExpand[Log[(1 - I*g*x)/(-d + I*x)]]
> > >
> > > My reading of the original question was that it involved (in Mma
> > > format)
> > > Log[1-I*g*x]/(-d+I*x), not Log[(1+I*g*x)/(-d+I*x)]. Does Mma find a
> > > Fourier
> > > transform of this one?
> > >
> > > --
> > > Robert Israel israel@math.MyUniversitysInitials.ca
> > > Department of Mathematics http://www.math.ubc.ca/~israel
> > > University of British Columbia Vancouver, BC, Canada V6T 1Z2
> > >
> > >
> > >
> >
> > I also found it was wrong. And I was unable to find the FT using MMA.
>
> Mma 5.2 fails.
> As I was informed in Mma 6 you get
>
> In[47]:= FourierTransform[Log[1 - I*g*x]/(-d + I*x), x, s,
> Assumptions -> g > 0 && d > 0]
>
> Out[47]= 0
>
> Hope it helps,
> Dimitris
Hello Vista nad Robert.
I can't justify Mma 6 result.
But I found the following more useful in the
sense of generalized functions. I look forward to seeing
your comments.
In[234]:=
D[Log[1 - I*g*x]/(-d + I*x), g]
FourierTransform[%, x, s, Assumptions -> g > 0 && d > 0]
Integrate[%, g, Assumptions -> d > 0]
Out[234]=
-((I*x)/((-d + I*x)*(1 - I*g*x)))
Out[235]=
((E^(s/g) - d*E^(d*s)*g)*Sqrt[2*Pi]*UnitStep[-s])/(g*(-1 + d*g))
Out[236]=
E^(d*s)*Sqrt[2*Pi]*(ExpIntegralEi[(-d + 1/g)*s] - Log[-1 +
d*g])*UnitStep[-s]
Regards
Dimitris
