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#26 2011-08-12 06:03:54

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Laplace Transform of tan(t)

Yes, that's what I observed too...

Anyway, the output is not useful.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#27 2011-08-12 06:16:03

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

Re: Laplace Transform of tan(t)

Also there is a Inverse Laplace Transform that could be tried on that answer. 5 to 1 says it does not return tan(t).


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#28 2011-08-12 06:25:37

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Laplace Transform of tan(t)

Yes, bur I'll not try that!
I checked numerical results for s=1, they didn't match.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#29 2011-08-12 06:27:33

zetafunc.
Guest

Re: Laplace Transform of tan(t)

alexfloo, Physics Forums wrote:

Let me start by saying I've never seen a Cauchy principal value integral. All my comments are from a few minutes of research.

The Cauchy principal value integral is essentially a method of assigning mathematically useful values to divergent improper integrals. Wikipedia, for instance, defines the Laplace transform in terms of a Lebesgue integral. The key point really is that the Lebesgue and the Cauchy principle value are *not* the same, so in the traditional sense, what Wolfram|Alpha computed is not actually the Laplace transform of the tangent, which doesn't exist, because the Lebesgue integral diverges. However, (it is likely that - again, I've never seen this integral before) the function Wolfram computed has many of the useful properties of the Laplace Transform, and is therefore a reasonable substitute.

Posted here if you might find it useful. If this is true then the Inverse Laplace transform of that function won't return tan(t)...

#30 2011-08-12 06:34:46

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

Re: Laplace Transform of tan(t)

That is almost like Borel summation. Some Sums that diverge ( Ramanujan sums ) converge in a Borel sense.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#31 2011-08-12 06:40:29

zetafunc.
Guest

Re: Laplace Transform of tan(t)

What I want to know though is how you can change the integral so that it does not diverge...

#32 2011-08-12 06:41:23

gAr
Member
Registered: 2011-01-09
Posts: 3,479

Re: Laplace Transform of tan(t)

Hmmm, okay.

It's my EOD, see you both later.
Good discussion.


"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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#33 2011-08-12 06:41:37

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

Re: Laplace Transform of tan(t)

Simplest way might be by truncating it. But why? It would not be the Laplace Transform.

See you later gAr!


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#34 2011-08-12 06:45:08

zetafunc.
Guest

Re: Laplace Transform of tan(t)

You mean a Laplace transform such that the transform is valid for some interval [a,b] provided that the interval does not contain any singularities? That might work but I agree it would not be the Laplace transform. Is there a way you can do it without including any singularities?

#35 2011-08-12 06:48:20

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

Re: Laplace Transform of tan(t)

Hi zetafunc.;

Not for tan(t) which has an infinite amount of them. You would have to go from 0 to pi / 2. Then it might converge, but what does it represent?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#36 2011-08-12 06:48:37

zetafunc.
Guest

Re: Laplace Transform of tan(t)

I have a whiteboard at home and on it still has written the result of performing integration by parts twice on tan(t) defined in terms of e and i.

What I wanted to ask you earlier was if i can be treated the same way as a real constant can? In other words, can you say that ∫if(t)dt = i∫f(t)dt?

#37 2011-08-12 06:49:52

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

Re: Laplace Transform of tan(t)

Yes, move the i right through it is a constant.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#38 2011-08-12 06:51:34

zetafunc.
Guest

Re: Laplace Transform of tan(t)

Do you know anything about Cauchy principal values? As you said it is similar to Borel sums where you can turn a diverging summation into a converging one that can hold true for all values.

#39 2011-08-12 06:54:43

zetafunc.
Guest

Re: Laplace Transform of tan(t)

Thanks, I can get on with performing another IBP now now that I know factoring out the i is okay...

Sorry, I am just a bit new to Laplace transforms, I feel bad because I don't really have a conceptual understanding of what I'm doing so to speak; all I know how to do is use it to solve ordinary differential equations and to turn a function of t into a function of s. I've read that it can change it from the time-domain into the frequency-domain but I don't know what that means or how that makes sense.

#40 2011-08-12 06:55:03

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

Re: Laplace Transform of tan(t)

Hi;

I am sorry but I do not. Borel sumation is something physicists use more, I think?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#41 2011-08-12 06:56:58

zetafunc.
Guest

Re: Laplace Transform of tan(t)

Borel summations are used by physicists? Hmm... what would they use them for? (sorry for going off-topic)

#42 2011-08-12 07:01:24

zetafunc.
Guest

Re: Laplace Transform of tan(t)

Never mind, found it:

Wikipedia wrote:

Borel summation finds application in perturbation expansions in quantum field theory. In particular in 2-dimensional Euclidean field theory the Schwinger functions can often be recovered from their perturbation series using Borel summation (Glimm & Jaffe 1987, p. 461). Some of the singularities of the Borel transform are related to instantons and renormalons in quantum field theory (Weinberg 2005, 20.7).

#43 2011-08-12 07:02:02

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

Re: Laplace Transform of tan(t)

As near as I understand it was convenient for certain series that diverged to be convergent for quantum mechanics. I do not know any more about that. Except that using Borel summation a sequence of positive numbers when added, can sum to a negative number!


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#44 2011-08-12 07:04:43

zetafunc.
Guest

Re: Laplace Transform of tan(t)

Thanks for the reply -- are you saying that Borel summation can mean a sequence of positive numbers can sum to a negative number?

#45 2011-08-12 07:05:48

zetafunc.
Guest

Re: Laplace Transform of tan(t)

Maybe I should post the problem on lots of other forums, no one seems to be responding on PF. Maybe someone knows how to apply Cauchy principal values here...

#46 2011-08-12 07:14:42

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

Re: Laplace Transform of tan(t)

You could do that but remember you can not force this one to have the answer you want. It will always come up the same, divergent.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#47 2011-08-12 07:15:34

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

Re: Laplace Transform of tan(t)

-- are you saying that Borel summation can mean a sequence of positive numbers can sum to a negative number?

Ramanujan did a couple of sums like that.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#48 2011-08-13 03:37:52

zetafunc.
Guest

Re: Laplace Transform of tan(t)

Still no luck with converting it to a Cauchy principal value integral... I'm not sure how to set it up because tan(t) is periodic with singularities every interval of π. Do you think I could work backwards from W|A's result and see where that gets if I calculate the inverse Laplace transform of that by hand? I know I shouldn't completely trust Wolfram Alpha's result but the direction that this problem has taken interests me as I didn't think you could find a way to define the Laplace transform for a non-piecewise continuous periodic function...

#49 2011-08-13 03:47:03

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,771

Re: Laplace Transform of tan(t)

Hi;

You could try that but I think the Alpha answer is gibberish. What if I am right?


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#50 2011-08-13 03:54:39

zetafunc.
Guest

Re: Laplace Transform of tan(t)

Well, I'll never know until I try... I don't know how to use a computer to do it either, and taking the inverse Laplace transform of all those digamma functions looks sticky.

I just tried integration by parts 4 times using the definition of tan in terms of e and i and it seems to follow a similar pattern to just taking the improper integral of e[sup]-st[/sup]tan(t)dt. That would make sense. Unless my IBP is wrong...

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