Discussion about math, puzzles, games and fun.   Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫  π  -¹ ² ³ °

You are not logged in.

## #1 2013-05-20 23:26:28

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

### Fourier representations for tan x

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 Dirichlets
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

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

Offline

## #2 2013-05-21 07:21:03

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Fourier representations for tan x

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

## #3 2013-05-21 09:39:45

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

### Re: Fourier representations for tan x

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

Thnx ^_^

Offline

## #4 2013-05-21 11:49:09

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Fourier representations for tan x

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

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.

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

## #5 2013-05-21 20:30:53

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

### Re: Fourier representations for tan x

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

Offline

## #6 2013-05-21 20:32:17

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Fourier representations for tan x

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

## #7 2013-05-21 21:07:49

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

### Re: Fourier representations for tan x

Oh yea then it must be the second

Offline

## #8 2013-05-21 21:12:13

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Fourier representations for tan x

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?

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

## #9 2013-05-21 21:29:11

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

### Re: Fourier representations for tan x

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

Offline

## #10 2013-05-21 21:42:03

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Fourier representations for tan x

Egypt? Yes, I do but just from reading.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

## #11 2013-05-21 22:03:14

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

### Re: Fourier representations for tan x

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

Offline

## #12 2013-05-21 22:06:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: Fourier representations for tan x

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.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline