Talk:Fourier series

Vibrating Disc doesn't look right

it states that this is a variable when integer is constant —Preceding unsigned comment added by 117.196.34.205 (talk) 16:16, 14 December 2009 (UTC)

I would be surprised if the animation of the vibrating disc is correct. If it is a circular disk, as depicted, the solution should have circular symmetry. I understand that this kind of symmetry preservation doesn't always hold, but in this fairly simple case I think it should.

I learned that vibrating disks have solutions that look like radial Bessel functions anyhow. —Preceding unsigned comment added by 129.105.207.213 (talk) 04:12, 2 May 2008 (UTC)

Too technical

All this is too much math for a common user trying to understand or get a clue of what Fourier did. I know, we are not all math people, but more instroduction is necesary nad then you can start with formulas. 192.35.17.15 (talk) 16:00, 19 December 2007 (UTC)

The article as a whole is disjointed and probably too detailed, but the intro looks OK to me. If you don't understand it, the internal links, like Fourier analysis, provide background material which you might need. If you can't understand the background material either, you probably aren't in the target group for an article like this one. If you do understand the article, but think you can do better, you are free to show us the way.

--Bob K (talk) 22:47, 20 December 2007 (UTC)

I can imagine a more layperson-oriented intro, but it would certainly take some work. Michael Hardy (talk) 01:09, 21 December 2007 (UTC)

I took a stab at it. Loisel (talk) 20:32, 8 January 2008 (UTC)

Putting a lay person's introduction at the start of the article is fine. However you should not simplify the article at the expense of the technical user. Loisel: I think the edits you have made do just this. In particular the choice of a particular period and a particular integration regime instead of the more general $[t_0,t_0+T]$ could (a) leave more technical users who don't know much about Fourier series with the impression that the integration region isn't movable by an arbitrary offset, (b) annoy those technical users who wanted the formula in the more general form and will now have to convert it themselves (which if they are intending to actually use the formulas will likely be quite a large number of such users, and (c) confuse intermediate users who wish to make the Fourier series of a function with a period of other than $2\pi$ but don't understand the details well enough to change the given formula.

If you really feel there is a class of user out there who will be unable to understand the more general form of the equations and yet will want more than a lay person's non-mathematical introduction to the topic, then maybe make a 'simple case' section where you give them in the form you have used in your edit.

Another criticism (sorry!) is that with respect to the lay person's introduction, I would have though that a simple modern description of the technique ranks higher than a historical account of its development --- so I'm not convinced about moving the 'Historical Development' section to the top of the article.

But really my largest gripe is just the removal of the more general form; as a technical user of Wikipedia I personally love the fact that you can get gritty detailed descriptions of mathematical techniques that you wouldn't find in other encyclopedias.

137.222.187.157 (talk) 12:25, 9 January 2008 (UTC)

Dear Anonymous,

You are very interested in the T-periodic case. I have made some notes to that effect in the T-periodic section, giving explicit formulae. However, please note that the T-periodic cse is not "the general" case. The general case is the Hilbert space orthonormal basis approach. However, without going to an abstract Hilbert space, there are interesting Fourier series which are not on intervals, see spherical harmonic.

Sincerely,

Loisel (talk) 22:29, 9 January 2008 (UTC)

I agree, the 2pi period constraint threw me off quite a bit. It's the reason for the equations, the explanation of what's being done, that's missing. Simplifying shouldn't even be necessary, except to show what happens when you plug in certain circumstances. In other words, it's not the math that's wrong, it's the article. 97.123.87.61 (talk) 18:16, 17 October 2009 (UTC)

In regards to a comment above, awhile back: you should not simplify the article at the expense of the technical user ... I understand your general points about article quality, but an encyclopedia can't be all things to everyone. (There's an old chestnut about "Try to make everyone happy, you make no one happy.") Besides, the more technical readers can always skim/skip the intro. Wikilinks are fine for further detail or branching topics, but an article should stand on its own. I think the intro reads well at the moment but could use some tweaking. For instance, what general field of math does this fall under? I haven't studied diffy-q's since college, but I always thought this was the domain of calculus -- yet that term does not appear in the article. -PrBeacon (talk) 07:46, 6 October 2010 (UTC)

Would it be appropriate to add CAT:Multivariable Calculus, Differential equations and/or Partial differential equations? -PrBeacon (talk) 07:56, 6 October 2010 (UTC)

No, No and No. Fourier series are in one variable, ODEs and PDEs are analyzed using the Lagrange tranform or the Fourier transform, not the series. There is a connection from the series to Sturm-Liouville, but that does not deserve a category.--LutzL (talk) 13:33, 6 October 2010 (UTC)

Ok, what branch of mathematics would you suggest? -PrBeacon (talk) 01:03, 7 October 2010 (UTC)

A great many of the scientific articles in Wikipedia seem to be addressed primarily to fellow specialists, or at any rate students already well-versed in the subject matter. That's great for professionals, but why cannot the articles of an enyclopedia for the general public unpack the technical terms as it goes along? I don't see why such unpacking would have to come at the expense of technical precision or comprehensiveness. —Preceding unsigned comment added by 72.225.204.182 (talk) 06:55, 9 December 2010 (UTC)

I think that's what wiki-links are for. If you find terms that you don't understand, and they're not linked to an article that explains them, let us know here. Dicklyon (talk) 07:50, 9 December 2010 (UTC)

Very first example

There will be no π in the denominator in the final result since it's canceled out.
look at http://planetmath.org/?op=getobj&from=objects&id=4718 Thanks. —Preceding unsigned comment added by 70.119.103.136 (talk) 04:14, 24 May 2010 (UTC)

NOTE now fixed; this error was introduced by a very recent, incorrect edit. 128.95.41.29 (talk) 02:51, 26 May 2010 (UTC)

Quote isn't actually Fourier's?

We have a quote under the heading Revolutionary Article attributed to Fourier. However, if you check up the reference with the link provided, you'll see that the words aren't actually his. Despite the article being listed as "par M. Fourier", it's written in the third person with constant references to "M. Fourier" did this/that etc. On the first page, there is a footnote:

Cet Article, que nous avons déjà signalé dans l'Avant-Propos du Tome I, n'est pas de Fourier. Signé de l'initialo P, il a été écrit par Poisson, qui était un des rédacteurs du Bulletin des Sciences pour la partie mathématique. A raison de l'intérêt historique qu'il présente comme étant le premier écrit où l'on ait fait connaître la théorie de Fourier, nous avons eru devoir le reproduire intégralement.

In my own loose, and possibly mildly inaccurate translation:

This article, as we have already indicated in the "Avant-Propos" of Book 1, is not by Fourier. Signed with the initial P, it has been written by Poisson, who was one of the editors of the Bulletin des Sciences for the mathematical part. For reason of historical interest, it is presented as though it were the first writing where the theory of Fourier was made known, [we have had to reproduce it in full?] (is "eru" supposed to be "eu"?)

Not quite sure what to make of this - sounds like a rather bizarre practice, but I believe that the original memoire may have been lost. Anybody got any ideas on how this should be correctly attributed? —Preceding unsigned comment added by Tcnuk (talk • contribs) 15:56, 9 June 2010 (UTC)

I've put a footnote to this effect into the article. If anybody can shine any light on this mystery, I'd be delighted to hear from them... Tcnuk (talk) 12:03, 10 June 2010 (UTC)

The last sentence: Because of the historic interest that it presents as being the first .... —Tamfang (talk) 16:22, 9 December 2010 (UTC)

Not "eru, not "eu", it's "nous avons cru devoir le reproduire intégralement", "we believed it to be our duty to reproduce it in full".--LutzL (talk) 19:20, 9 December 2010 (UTC)

And the translation of the last sentence is slightly wrong: "For reasons of historical interest, as it appears to be the first text to publicate the theory of Fourier, we deemed it indispensable to publish it in full."--LutzL (talk) 19:31, 9 December 2010 (UTC)

And perhaps it should be "in its original form" or similar for "integralement".--LutzL (talk) 20:26, 9 December 2010 (UTC)

On decembre 21st 1807, Fourier presented his current work in something like a seminar at the "Academie des Sciences". The base of this presentation was a "memoire", perhaps hand-written, that was probably given to some mathematicians, but not published. Obviously, Poisson found it interesting enough to publish his notes of this memoire in 1808 on 4 pages in the "Nouveau Bulletin des Sciences, par la Societe Philomathique", <a href="http://www.archive.org/stream/nouveaubulletind11807soci#page/112/mode/2up"> thanks to google available online</a>. This was reprinted as the earliest published occurence of the computation of (Fourier)trigonometric series coefficients by integrals and the notorious claim that the series converges to the given function. The computation of coefficients of trigonometric polynomials by something like the discrete Fourier transform was known some decades earlier.--LutzL (talk) 21:18, 14 January 2011 (UTC)

Poisson's 1808 publication of a 4-page brief summary and comment on Fourier's memoir/presentation to the Institut doesn't qualify as a first publication of the memoir, although it did help protect Fourier from anyone who might have wished to publish before Fourier was able to, and thereby compete for priority. Although Fourier included a version of his earlier 1807 and 1811 work in his 1922 book, the first real publication of the 1807 memoir manuscript is given below, and yes, the 1807 manuscript did go missing for several decades. Fourier, Jean-Baptiste Joseph. "1811 mathematics prize paper." In Joseph Fourier, 1768-1830; a survey of his life and work, based on a critical edition of his monograph on the propagation of heat, presented to the Institut de France i 1807., by Ivor Grattan-Guinness, 30-440. Cambridge, MA: The MIT Press, 1972. Fourier did turn in a 234-page hand-written copy of his 1807 memoir to the Institute. However, Grattan-Guinness thinks that Fourier may have presented without the aid of the usual extract, since none survives [For new information and discussion on this point, please see note below by AJR]. To the previous contributor, the notoriousness of Fourier's claim faded as and when it became known that all practical data satisfies the requirements for convergence. Even Fourier himself, in using the words "arbitrary function" also said that the function had to be finite in extent, thereby presaging Dirichlet's conditions. In effect Fourier was saying that the function could be arbitrary, as long as it was integrable. Fourier's daring gave mathematics the necessary kick in the pants, but it took over 100 years for the bruise to fully go away.  ;-)

AJR: John Herivel's book, Joseph Fourier: The man and the physicist, published in 1975, in other words 3 years after I. Grattan-Guinness' landmark publication of Fourier's 1807 memoir, has this:

J Herivel, Page 318: Chapt XXI Fourier to an unknown correspondent, around 1808-9

"I have the honour to send you two notes concerning the memoir on heat. The first(Note 1) is the one that was read at the Institut in place of the reading of the memoir... [presumably due to the excessive length of the actual memoir, comprising 234 sides!, in other words Fourier is referring to the extract. Therefore Fourier *did* read an extract, rather than presenting without using one].

J Herivel, Page 320: Note 1. This was the abstract (extrait) of Fourier's 1807 memoir which has been retained in the MS. 1851 of the library of the Nationale École des Ponts et Chaussées, Paris. It must therefore be dated 1807, and not 1809 as suggested by Grattan-Guiness (3), p. 497.

When I looked in Grattan-Guinness, I immediately saw that this would not be a simple correction, since discussion of the extrait in q uestion appears in multiple locations, and is quite embedded in the discussion. For example, see

Grattan-Guinness, page 24 (lines 14-18, 22-24); page 26 (lines 6 and 24).

Grattan-Guinness, Page xii Abbreviations of Titles of Key Works by Fourier

4. "Extrait," for the paper sent by Fourier to the Institut de France in October 1808. [I had long wondered about this, since as far as I knew there was only the 1807 extract -AJR]

5. "Notes," for the set of footnotes to the Extrait.

I have not seen a correction published by Ivor Grattan-Guinness, nor any acknowledgement of the problem. I also have yet to find any further discussion by John Herivel. One of them must be more correct than the other in this matter.

Formula incorrect

I believe the formula giving the conversion of coefficients into exponential form is incorrect (and the equivalent problem exists in the other direction as well). it suggests that c₀ = a₀/2. This should be c₀ = a₀. since a₀ cos(0x) + b₀ sin(0x) = a₀, and c₀ e^i0x = c₀.

99.52.193.237 (talk) 16:09, 13 January 2011 (UTC)

You appear to be assuming something like a_n cos(nx) + b_n sin(nx) = c_n e^inx.  Try proving it.

--Bob K (talk) 18:39, 14 January 2011 (UTC)

What I'm suggesting (in more detail) is that a₀ cos(0x) + b₀ sin(0x) = c₀ cos(0x) + c₀ i sin(0x), by Euler's formula. The sine terms drop out so: a₀ cos(0x) = c₀ cos(0x) and c₀ = a₀. —Preceding unsigned comment added by 99.52.193.237 (talk) 17:04, 16 January 2011 (UTC)

And that is totally correct, except that the constant term of the real form of the series is given as $\frac{a_0}2$. This is done to have a unified integral formula for the real coefficients. And to be consistent with the conversion in the opposite direction given above the discussed one. Write that down explicitely for the case n=0.--LutzL (talk) 17:12, 16 January 2011 (UTC)

Might it be useful to include this formula: $a_{0} = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)dx$ in the article as the general form for a₀? —Preceding unsigned comment added by Brvman (talk • contribs) 23:49, 4 May 2011 (UTC)

Formula and graph axis labelling

Sorry to ask such a stupid question, but shouldn't the graph axes in the "Plot of a periodic identity function" associated with Example 1 be the same as the formula states? I.e. instead of x(t) and t (which appear nowhere in the formula or the coefficient calcs), shouldn't it be labelled f(x) and x? Or am I missing something obvious? (I'm no mathematician, as you can probably tell) Cephas Borg (talk) 14:52, 7 December 2011 (UTC)

You're not missing anything. I'd guess the figure was originally created for a different purpose or different text... not an unusual occurrence at Wikipedia. It's not a perfect fit, but it's close enough that nobody has bothered to fix it. I don't think anyone would complain if you want to fix it yourself.

--Bob K (talk) 18:16, 7 December 2011 (UTC)

Thank Fourier it wasn't just me! I'm currently learning to modify the original source. Thanks for the encouragement, Bob. (This is my first real Wikipedia contribution, woo hoo! :). Cephas Borg (talk) 02:49, 8 December 2011 (UTC)

One thing I should have checked and mentioned is the section File usage on other wikis at http://commons.wikimedia.org/wiki/File:Sawtooth_pi.svg. Luckily, all it lists is this article. (Whew!) Thanks for your help, and good luck with your first "adventure".

--Bob K (talk) 01:20, 9 December 2011 (UTC)