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**clumsy shark****Member**- Registered: 2012-01-07
- Posts: 13

Which of fourier representations is suitable for f(x) = tan (x) : fourier trigonometric series, fourier half-range expansion , oe fourier integral and why ?

Well I searched and found that :

1- tan x cannot be expanded as a Fourier series .Since tan x not satisfies Dirichlets

conditions.(tan x has infinite number of infinite discontinuous).

2- the pic (sorry posting links is not allowed for me )

the first answer is clear for me but I can't understand the pic I go confused :S

THNX in advance

*Last edited by clumsy shark (2013-05-20 23:26:57)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,478

Hi clumsy shark;

I am not following that well. You are right about the FourierSeries for tan(x). But what part of the derivation of that series has you stumped?

**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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**clumsy shark****Member**- Registered: 2012-01-07
- Posts: 13

Thanks bobbym for replying

Well I can't get how he made this expansion ?? Is it allowed ? I see those two answers contradicting I understand well that tan(x) is unbounded at x = pi/2 and fourier series is only applicable to periodic functions where f (x) is defined for all real x

how could he manage it and what does his phrase 'This series has very poor convergence properties (look at x= pi/4). So the equality

is only in some sense of generalized functions.' means ??

I wanna know what misunderstanding I suffer here

sorry for my bad language

Thnx ^_^

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,478

I agree with all your complaints about

So the equality

is only in some sense of generalized functions.'

I have no idea what his vague wording means.

Let me check that series for correctness.

That is indeed a series that represents tan(x). The poor convergence he is talking about at the endpoints is caused by the well known Gibbs phenomena for Fourier series. The blue line is tan(x) and the red is a truncated form of his sum. That type of behaviour is standard for a Fourier fit.

The paper is from a lecturer at MIT.

He may be correct in his derivation too because your understanding of a Dirichlet condition might need some adjustment. There is an interesting discussion of that at

http://forum.allaboutcircuits.com/showt … hp?t=10266

it might help a little. From the viewpoint of a practical numerical analyst I would go with Kreyszigs definition which would just handle tan(x) as another piecewise function which are fit by Fourier series all the time.

Follow this link to 10.1.2

http://fractional-calculus.com/termwise … tive_1.pdf

To sum up that is only a Fourier series in some vague sense that no one is willing to share. Again, from a practical standpoint I accept the Fourier series of it but do not use such a garbage series for computation!

**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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**clumsy shark****Member**- Registered: 2012-01-07
- Posts: 13

Thank You bobbym ^_^ That was really such an annoying matter, I got it .. THANKS

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,478

Hi;

It is very confusing to me also. I do wish he was clearer or I was smarter.

**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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**clumsy shark****Member**- Registered: 2012-01-07
- Posts: 13

Oh yea then it must be the second

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,478

What me get smarter? Fat chance of that happening. They say you can spit in one hand and wish in the other and watch which one will fills up faster.

Are you attending Jerome's classes?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**clumsy shark****Member**- Registered: 2012-01-07
- Posts: 13

No , I'm from Egypt. Do you know it ?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,478

Egypt? Yes, I do but just from reading.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**clumsy shark****Member**- Registered: 2012-01-07
- Posts: 13

^^ It's been a pleasure to know someone dedicated like You ^_^

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 86,478

Hi;

Thanks for saying that but I rarely do a post where I do not learn something new and I meet people from other places.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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