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Unilateral Laplace transform of 1/t

Started by vv September 30, 2008
An electrical engineering textbook has an excercise problem requesting
the Laplace transform of 1/t, and wanted it to be done using Laplace
transform properties.  Since this raised some concern, I looked up
Gradshteyn and Ryzhik (6th ed.).  On p. 1100 it states that t^nu has
transform gamma(nu + 1)/s^(nu + 1) but requires nu > -1.  I guess if
it is not listed in G&R, it doesn't exist :-).  Interestingly, 1/t has
Fourier transform.  I just wanted confirmation that the problem as
stated in the book is not correct.

For the curious: the book is Lathi's "Signals and Linear Systems".

--vv

"vv" <vanamali@netzero.net> schrieb im Newsbeitrag 
news:a65dfc9b-12ab-4b35-992a-20c70660936d@79g2000hsk.googlegroups.com...
> An electrical engineering textbook has an excercise problem requesting > the Laplace transform of 1/t, and wanted it to be done using Laplace > transform properties. Since this raised some concern, I looked up > Gradshteyn and Ryzhik (6th ed.). On p. 1100 it states that t^nu has > transform gamma(nu + 1)/s^(nu + 1) but requires nu > -1. I guess if > it is not listed in G&R, it doesn't exist :-). Interestingly, 1/t has > Fourier transform. I just wanted confirmation that the problem as > stated in the book is not correct. > > For the curious: the book is Lathi's "Signals and Linear Systems".
In case of the unilateral LT, 1/t is given for t>0. How do you assume 1/t for t<0 when calculating the FT? Salviati
On Tue, 30 Sep 2008 04:02:47 -0700 (PDT), vv <vanamali@netzero.net>
wrote:

>An electrical engineering textbook has an excercise problem requesting >the Laplace transform of 1/t, and wanted it to be done using Laplace >transform properties. Since this raised some concern, I looked up >Gradshteyn and Ryzhik (6th ed.). On p. 1100 it states that t^nu has >transform gamma(nu + 1)/s^(nu + 1) but requires nu > -1. I guess if >it is not listed in G&R, it doesn't exist :-). Interestingly, 1/t has >Fourier transform. I just wanted confirmation that the problem as >stated in the book is not correct. > >For the curious: the book is Lathi's "Signals and Linear Systems".
"Linear Systems and Signals" ?
> >--vv
Perhaps(?) the point of the exercise is to suggest that evaluating the integral Int[0..oo] 1/t exp[-s t] dt is "difficult" (not obvious), whereas using L{1} = 1/s and the property L{f(t) / t} = Int[s..oo] F(u) du makes it simple to determine the nature of the transform.
> In case of the unilateral LT, 1/t is given for t>0. How do you assume 1/t > for t<0 when calculating the FT? > > Salviati
Mea culpa. The LT I am after is 1/t for t > 0, and I shouldn't have connected it to the FT of the two-sided 1/t. Cary:
>Perhaps(?) the point of the exercise is to suggest that evaluating >the integral > > Int[0..oo] 1/t exp[-s t] dt > >is "difficult" (not obvious), whereas using L{1} = 1/s > >and the property L{f(t) / t} = Int[s..oo] F(u) du > >makes it simple to determine the nature of the transform.
All this assumes that the transform exists to begin with, which itself seems questionable. --vv
In article <9o94e4d8rhh7eddjqm8q882nb2s8nn2iap@4ax.com>,
 Cary <cary@domain.invalid> wrote:

> On Tue, 30 Sep 2008 04:02:47 -0700 (PDT), vv <vanamali@netzero.net> > wrote: > > >An electrical engineering textbook has an excercise problem requesting > >the Laplace transform of 1/t, and wanted it to be done using Laplace > >transform properties. Since this raised some concern, I looked up > >Gradshteyn and Ryzhik (6th ed.). On p. 1100 it states that t^nu has > >transform gamma(nu + 1)/s^(nu + 1) but requires nu > -1. I guess if > >it is not listed in G&R, it doesn't exist :-). Interestingly, 1/t has > >Fourier transform. I just wanted confirmation that the problem as > >stated in the book is not correct. > > > >For the curious: the book is Lathi's "Signals and Linear Systems". > > "Linear Systems and Signals" ? > > > > >--vv > > Perhaps(?) the point of the exercise is to suggest that evaluating > the integral > > Int[0..oo] 1/t exp[-s t] dt > > is "difficult" (not obvious), whereas using L{1} = 1/s > > and the property L{f(t) / t} = Int[s..oo] F(u) du > > makes it simple to determine the nature of the transform.
Maybe the exercise was supposed to "suggest" this, but it's not so. What do you get for Int[s..oo] du/u ? -- David C. Ullrich
On Tue, 30 Sep 2008 14:29:40 -0500, "David C. Ullrich"
<dullrich@sprynet.com> wrote:

>In article <9o94e4d8rhh7eddjqm8q882nb2s8nn2iap@4ax.com>, > Cary <cary@domain.invalid> wrote: > >> On Tue, 30 Sep 2008 04:02:47 -0700 (PDT), vv <vanamali@netzero.net> >> wrote: >> >> >An electrical engineering textbook has an excercise problem requesting >> >the Laplace transform of 1/t, and wanted it to be done using Laplace >> >transform properties. Since this raised some concern, I looked up >> >Gradshteyn and Ryzhik (6th ed.). On p. 1100 it states that t^nu has >> >transform gamma(nu + 1)/s^(nu + 1) but requires nu > -1. I guess if >> >it is not listed in G&R, it doesn't exist :-). Interestingly, 1/t has >> >Fourier transform. I just wanted confirmation that the problem as >> >stated in the book is not correct. >> > >> >For the curious: the book is Lathi's "Signals and Linear Systems". >> >> "Linear Systems and Signals" ? >> >> > >> >--vv >> >> Perhaps(?) the point of the exercise is to suggest that evaluating >> the integral >> >> Int[0..oo] 1/t exp[-s t] dt >> >> is "difficult" (not obvious), whereas using L{1} = 1/s >> >> and the property L{f(t) / t} = Int[s..oo] F(u) du >> >> makes it simple to determine the nature of the transform. > >Maybe the exercise was supposed to "suggest" this, but >it's not so. What do you get for Int[s..oo] du/u ?
Same as considering Int[0..oo] 1/t exp[-s t] dt. Transform doesn't exist. To the OP: Post the exact statement of the problem from the textbook you referenced so we can consider what the author had in mind.
> > > &#2013266080;Int[0..oo] 1/t exp[-s t] dt > > >is "difficult" (not obvious), whereas using &#2013266080;L{1} = 1/s > > >and the property &#2013266080;L{f(t) / t} = Int[s..oo] F(u) du >
A good way to attack this using only elementary integrals when you have a 1/t as part of the integrand over a semiinfinite interval is to replace "1/t" with integral[0,oo] exp(-s*x) dx, thus you end up with a double integral and proceed from there. Clay
On Oct 1, 1:18 am, Cary <c...@domain.invalid> wrote:
> To the OP: Post the exact statement of the problem from the textbook > you referenced so we can consider what the author had in mind.
I don't have the book handy, but it is along the following lines: Knowing the transform of x(t) = 1 for t >= 0 to be 1/s, use the integration property to obtain the transform of x(t)/t. This leads to Int[s..oo] du/u, which I suspected didn't exist. I guess the textbook author was not too careful when framing this problem. If someone has the Solutions Manual to this book and post the solution given there, maybe it will throw more light on what the author had in mind. --vv
"vv" <vanamali@netzero.net> schrieb im Newsbeitrag 
news:30f4dc97-d8e6-42f9-88d7-e7c0890015ed@q35g2000hsg.googlegroups.com...
> On Oct 1, 1:18 am, Cary <c...@domain.invalid> wrote: >> To the OP: Post the exact statement of the problem from the textbook >> you referenced so we can consider what the author had in mind. > > I don't have the book handy,
Could you please specify the book. You wrote "Signals and Linear Systems". Our library has "Linear Systems and Signals", 1992 and "Signal Processing and Linear Systems", 1998, both Berkeley Cambridge Press. Salviati but it is along the following lines:
> Knowing the transform of x(t) = 1 for t >= 0 to be 1/s, use the > integration property to obtain the transform of x(t)/t. This leads to > Int[s..oo] du/u, which I suspected didn't exist. I guess the textbook > author was not too careful when framing this problem. If someone has > the Solutions Manual to this book and post the solution given there, > maybe it will throw more light on what the author had in mind. > > --vv
On Oct 1, 1:15&#2013266080;am, vv <vanam...@netzero.net> wrote:
> On Oct 1, 1:18 am, Cary <c...@domain.invalid> wrote: > > > To the OP: &#2013266080;Post the exact statement of the problem from the textbook > > you referenced so we can consider what the author had in mind. > > I don't have the book handy, but it is along the following lines: > Knowing the transform of x(t) = 1 for t >= 0 to be 1/s, use the > integration property to obtain the transform of x(t)/t. &#2013266080;This leads to > Int[s..oo] du/u, which I suspected didn't exist. &#2013266080;I guess the textbook > author was not too careful when framing this problem. &#2013266080;If someone has > the Solutions Manual to this book and post the solution given there, > maybe it will throw more light on what the author had in mind. > > --vv
Are you looking for something that works for all s? If you consider integral(1/t) for t = 0 to 1 is infinite (=-ln(0)), it looks like with s not in the left half plane you are guaranteed a problem. Dirk