Science > Physics > help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)?
| Topic: |
Science > Physics |
| User: |
"Vista" |
| Date: |
01 Jul 2007 07:36:35 PM |
| Object: |
help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)? |
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!
.
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| User: "dimitris" |
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| Title: Re: help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)? |
02 Jul 2007 12:42:35 AM |
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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
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| User: "Vista" |
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| Title: Re: help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)? |
02 Jul 2007 06:52:55 AM |
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"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!
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| User: "" |
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| Title: Re: help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)? |
02 Jul 2007 12:13:55 PM |
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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
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada V6T 1Z2
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| User: "Vista" |
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| Title: Re: help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)? |
02 Jul 2007 07:38:29 PM |
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<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
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.
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| User: "dimitris" |
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| Title: Re: help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)? |
03 Jul 2007 01:05:56 AM |
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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
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
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| User: "dimitris" |
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| Title: Re: help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)? |
03 Jul 2007 01:35:03 AM |
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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
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
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| User: "dimitris" |
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| Title: Re: help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)? |
03 Jul 2007 01:47:12 AM |
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dimitris :
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
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
This thread appeared to a couple of forums.
I replied to sci.math.
There I didn't see Maxim Rytin's reply.
But his solution is more complete than mine.
Dimitris
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| User: "dimitris" |
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| Title: Re: help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)? |
03 Jul 2007 12:15:24 AM |
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:
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
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada V6T 1Z2
Sorry 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
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| User: "Robert Israel" |
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| Title: Re: help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)? |
01 Jul 2007 09:34:10 PM |
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"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
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada V6T 1Z2
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| User: "Vista" |
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| Title: Re: help! what is the Fourier transform of log(1-g*i*x)/(i*x-d)? |
01 Jul 2007 09:52:06 PM |
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"Robert Israel" <> 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
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada V6T 1Z2
Thanks Robert!
What's the expression of its tempered distribution form?
Where to read more about these stuffs?
Much appreciated!
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