Talk:Jacobi elliptic functions

French references

Hello. Sorry to remove good-faith additions, but I believe the French references will be of limited use to readers of an English wiki. Best wishes, Robinh 21:28, 8 September 2007 (UTC)

I don't see that that makes sense. Lot's of people have studied French. I think even more English speaking people have studied French than Spanish. Michael Hardy (talk) 21:09, 24 June 2010 (UTC)

Confusing Phrase

Reading the introduction, I came across the phrase "The Jacobi elliptic functions have historical importance with also many features that show up important structure". This sounds like it could be important, but I can't understand it. Does anyone know what the phrase means? David Schwein (talk) 05:00, 30 August 2009 (UTC)

"In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e.g. the equation of the pendulum—also see pendulum (mathematics)). They also have useful analogies to the functions of trigonometry, as indicated by the matching notation sn for sin. The Jacobi elliptic functions occur more in practical problems than the Weierstrass elliptic functions. They were introduced by Carl Gustav Jakob Jacobi, around 1830."

I've noticed that these sentences have still not been changed. If I'm not mistaken, it seems to be a grammar issue that was never resolved. Long and precise sentences can become cumbersome and difficult to follow, even when writing them. What I am wondering is if it should be phrased like the following.

"In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions and auxiliary theta functions that occur in important structures, have direct relevance to some applications (see pendulum (mathematics)), and have a good deal of historical importance. They have useful analogies to trigonometric functions, as indicated by similar notation (e.g. sn and sin). These functions, introduced by Carl Gustave Jakob Jacobi circa 1830, occur more frequently than the Weierstrass elliptic functions in practical problems."

If someone who is an expert, or at least someone who is quite knowledgeable on this subject, could make it known if this version preserves the meaning of the current introduction, I would appreciate it. Also, if it does so, feel free to place the above in the article and, if it does not, to edit it in a manner in which it will preserve the proper meaning.168.28.180.32 (talk) 06:17, 6 December 2011 (UTC)

I've discovered what happened. Someone later moved this from another section to where it is now and it got butchered in the process. I'm splicing the current version with the original wording now.168.28.180.32 (talk) 04:58, 10 December 2011 (UTC)

Show some images?

It would be nice to have some graphical representation of at least some of these functions. —Preceding unsigned comment added by 141.5.32.68 (talk) 10:08, 29 July 2010 (UTC)

Confusion in first three sections

The introduction is very inaccurate and confusing. "corresponds to an arrow drawn from one corner of a rectangle to another" doesn't make any sense to me - what is the correspondence? An integral? If yes, over which function? And what is an arrow? I don't know that as a mathematical term, and the usual interpretation of an "arrow" doesn't make much sense to me in this context either. The introduction doesn't tell us either what the arguments of the elliptic functions are (the corners of the rectangle?)

The section "notation" says that an elliptic function takes two arguments, and the next section just talks about "sn u", which looks like one argument to the casual reader. --141.5.32.68 (talk) 16:36, 6 December 2010 (UTC)

On the contrary, I find this explanation clear, useful, and well written (that is easy to read). AlainD (talk) 20:18, 13 February 2014 (UTC)

Move to "Jacobi elliptic functions"

The phrase "Jacobi elliptic functions" is far more used than "Jacobi's elliptic functions", see these Google Scholar searches: [1] and [2]; as well as these Google Books searches: [3] and [4]. So I moved the page to Jacobi elliptic functions. -- Crowsnest (talk) 18:02, 1 February 2011 (UTC)

Mistakes

Looks to me that the whole section "Definition in terms of trigonometry" is wrong. Could someone please double-check and delete or modify? The arc-length is computed as $$\int r(\theta) d\theta$$ but it should be $$\int \sqrt{r(\theta)^2+r'(\theta)^2}d\theta.$$ This gives a more complicated integral.--> see edit next to diagram below. 37.250.2.216 (talk) 05:23, 11 June 2016 (UTC)

Disputing Definition as trigonometry

1. phi is not arccos(x) - since x = r cos phi. Strictly speaking your phi=arccos(x').

2. I have calculated for the ellipse a=1, b>1 a few values for u, sn u and cn u using Wolfram's Elliptic[phi,m], JacobiSN[u,m] and JacobiCN[u,m] functions (see http://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html and http://mathworld.wolfram.com/JacobiEllipticFunctions.html).

E.g. for m=0,7500 (b=2,0000), phi=pi/6 --> sin(phi)=0.5000; cos(phi)=0.8660; r(phi,m)=1.1094; x(phi,m)=0.9608; y(phi,m)=0.5547; y/b=0.2774; u(phi,m)= Elliptic[phi,m] = 0.5422; sn(u,m)= JacobiSN[u,m]= 0.5000; cn(u,m)= JacobiCN[u,m] = 0.8660.

added July 25: JacobiDN[u,m]=0.9014; cn/dn=0.9608; sn/dn=0.5547.

Clearly sn u = sin phi ≠ y/b and cn u = cos phi ≠ x !

This is invalid because you have started with a value for phi. Try starting with a value for x. Then put phi = arccos(x). — Preceding unsigned comment added by Stephen William Wynn (talk • contribs) 09:21, 30 July 2016 (UTC)

added July 25: It is also clear that for phi=pi/6 the formulae x = cn/dn and y = sn/dn are correct, and should replace the old x = cn and y = b*sn. The same conclusion holds for the values phi=0, pi/12, pi/4, pi/3, 5pi/12 and pi/2 - if you are interested in the excel with all checks I can send or publish it.

3. Checking the history of this section it appears it has been taken from http://www.und.edu/instruct/schwalm/MAA_Presentation_10-02/handout.pdf with the title “Elliptic functions as trigonometry,” presented by W. Schwalm, North Central Section, Math. Assn. Am., Moorhead MN, October 25-26, 2002. (See his CV in http://www.und.edu/instruct/schwalm/VitaSchwalm.pdf).

This handout.pdf states without proof that the ellipse with half axes a>1 and b=1 has the following "Trigonometry of the ellipse" (Schwalm's equations 4 and 5):

sn(u,k) = y (4)

cn(u,k) = x/a (5)

Here argument u is defined as u ≡ ∫rdθ running from the point P = (1,0) in his fig. 1, corresponding to θ=0, to a larger angle, say θ=φ, represented by the point Q in his fig. 1.

Both equations (4) and (5) are false. Why? With the angle φ = φ(u) defining the point Q the sn u and cn u functions are originally defined (by Jacobi himself) as sin φ(u) and cos φ(u) respectively. On the other hand the y-coordinate of Q is r(φ(u),m)sin φ(u), which is only equal to sin φ(u) when r(φ(u),m)=1, i.e. when φ=π/2. Also, x/a = (r(φ(u),m)/a) cos φ(u) is equal to cos φ(u) only if r(φ(u),m) = a, i.e. when φ=0.

Edit august 01: (4) and (5) are true for x' and y" in daigram below.

4. I would like to adjust the WP-section using the idea of the unit circle in remark 1 above. July 27: done

5. Picture with unit ellipse, unit circle and points P, P', P", .... could clarify things greatly. Edit: see rough diagram below

LWJdO (talk) 12:08, 21 July 2016 (UTC)

6. First concept of Definition as trigonometry moved to online article. Edit July 27: Integrated Schwalm's x=cn, y=b sn.

Diagram for Definition as trigonometry

To make the section better understandable, a diagram can say more than 1000 words.

sketch with P --> u(P), phi, P' and P"

LWJdO (talk) 16:54, 29 July 2016 (UTC)

Edit: The interpretation of u(P) = u(phi,m) as 'arc length' of the ellipse is not correct, it is only the 'angular part' ∫r(θ,m)dθ of the total arc length ∫sqrt(r(θ,m)²+(dr(θ,m)/dθ)²)dθ --> I renamed 'elliptic arc length' u(P) to 'angular (elliptic) arc length' in the article.

Twin equation

For x = cn(iu',m)/dn(iu',m), y=sn(iu',m)/dn(iu',m)

x^2 + (1-m) y^2 = 1

From Jacobi's imaginary transformation. Since

cn(iu',m) = 1/cn(u',1-m)

sn(iu',m) = i sn(u',1-m)/cn(u',1-m)

dn(iu',m) = dn(u',1-m)/cn(u',1-m)

x = 1/dn(u',1-m)

y = i sn(u',1-m)/dn(u',1-m)

x^2 + (1-m) y^2

= (1 - (1-m) sn(u',1-m)^2)/dn(u',1-m)^2 = 1

Q. E. D. — Preceding unsigned comment added by Stephen William Wynn (talk • contribs) 15:19, 2 August 2016 (UTC)

New diagrams and tables

The three diagrams following "definition as trigonometry" need more work. We need b > 1, and u is not the arc length. But I like seeing the twelve Jacobi elliptic functions on the same diagram. The first new table "Jacobi Elliptic Functions pq[u,m] ... " is excellent. E.g. x,y,r are better definitions of cd,sd,nd than the cumbersome cn/dn, sn/dn, 1/dn. Stephen William Wynn (talk) 09:55, 28 December 2017 (UTC)

Well, good. I don’t think there’s any mention of u being the arc length now, but the diagrams need to have the red arc length and red u changed to black so there is no implication that it is, and I will do that soon. Not that I disagree, but I was wondering why b>1 is needed. For 0<b<1, the ellipse is compressed in the y direction while for b>1 it’s elongated and I don’t see a qualitative difference. Also, I agree, Glaisher’s pq notation is best, the twelve are on equal footing and the restriction to the original three is a historical artifact. I think Neville’s book is the best, It’s dense but methodical and readable while Cayley is opaque to me at my level. PAR (talk) 01:17, 25 January 2018 (UTC)

Update - the red has been removed from the ellipse and hyperbola diagrams, and e.g. "ss=1" has replaced unit lengths. PAR (talk) 11:19, 25 January 2018 (UTC)

We need b>1, so y is the major axis as in the above diagram, because m and 1-m must be between 0 and 1. Stephen William Wynn (talk) 11:41, 26 January 2018 (UTC)

Ok, I will look into replacing the present m=-1 diagram with, say, m=2/3. I am studying these functions for a particular application, so I haven't read the literature extensively enough to understand the statement "m and 1-m must be between 0 and 1". Generally m (and 1-m) can be any complex number and the restriction you mention, as I understand it, is a convenience, rather than a requirement (?). PAR (talk) 12:50, 26 January 2018 (UTC)

Update - I replaced the m = -1 diagram with an m=1/2 diagram, giving y as the major axis. I don't think we need both, do you? PAR (talk) 13:32, 26 January 2018 (UTC)

In your first diagram you have changed b to 1.41421 and m to 0.5. In your third diagram about the hyperbola you need to change b to i 1.41421, m to 1.5 to make this compatible. (We need m between 0 and 1 for real b.) I don't understand "I don't think we need both,". Stephen William Wynn (talk) 12:39, 27 January 2018 (UTC)

Ok, but just so I understand, for the hyperbola, we need b=1.414 i just because it would be helpful for comparison purposes, it would be nice, but its not a requirement. When I said we don't need both, I meant we don't need two diagrams, one with a Y major axis ellipse, another with an X major axis ellipse. PAR (talk) 08:48, 30 January 2018 (UTC)

Having x the major axis is the convention and is aesthetic, but when introducing elliptic functions it is not valid. It produces problems as illustrated by Professor Schwalm: For this reason I don't think m=3 in your third diagram is valid. It would make x the major axis. We need the modulus of b to be > 1. Stephen William Wynn (talk) 12:11, 30 January 2018 (UTC)

I read the article you mentioned, and it is very good, but I did not see where he expressed a problem with an X-major axis ellipse. If I missed it, could you point it out to me? Also, the Jacobi functions pq(u,m) are well defined for any complex u and m, so pq(u,m) for m=3 is well-defined and the line segments in the diagram accurately depict the values of the various functions.

Since b^2=1/(1-m), b can be any complex number as well, and need not be constrained to have its modulus > 1. When m=3, b is $i/{\sqrt {2}}$, no problem. This results in a hyperbola rather than an ellipse. For b purely real, an ellipse results, while for b purely imaginary, a hyperbola results (ignoring degenerate cases). These are the only two cases in which x and y are real somewhere. For b generally complex, the resulting curves involve complex x and y.

I think the functions were originally defined for real values, resulting in a lot of restrictions on u and m, which disappeared when they were extended to the complex plane in both arguments.

You say "I read the article you mentioned, and it is very good" He defines cn, sn, dn by x/a, y, r/a. Not very good compared to Wikipedia x/r, y/r, 1/r. Stephen William Wynn (talk) 17:37, 14 March 2018 (UTC)

I agree with that, but I got some insights from that article. PAR (talk) 00:49, 24 February 2018 (UTC)

In my opinion his article is nonsense from start to finish, and it goes wrong right at the start by making x the major axis. You say "but I did not see where he expressed a problem with an X-major axis ellipse." He obviously does not realise there is a problem. Stephen William Wynn (talk) 18:28, 14 March 2018 (UTC)

I don't understand, what exactly is the problem with an x-major axis? The u and m arguments can generally be any complex number. PAR (talk) 04:15, 16 March 2018 (UTC)

The problem is that making x the major axis produces the wrong r ($cos^{2}$ instead of $sin^{2}$) in the elliptic integral of the first kind expression for the definition of cn and sn from integral r dtheta. This is what started me on this topic, I observed that turning Professor Schwalm's diagram through 90 degrees produces the correct r. Stephen William Wynn (talk) 11:35, 17 March 2018 (UTC)

There's two issues here - the use of $x^{2}/a^{2}+y^{2}=1$ or $x^{2}+y^{2}/b^{2}=1$ as the equation of the ellipse, and making the x-axis the major or minor axis, and the two are not the same. If a and b are real,

• If $x^{2}/a^{2}+y^{2}=1$ is the equation for the ellipse, then a<1 makes x the minor axis, a>1 makes x the major axis.

• If $x^{2}+y^{2}/b^{2}=1$ is the equation for the ellipse, then b<1 makes x the major axis, b>1 makes x the minor axis.

Since a and b can be any complex number, there is no problem with making the x-axis the major axis. The equation

$u=\int _{0}^{\phi }{\frac {d\theta }{\sqrt {1-k^{2}\sin ^{2}(\theta )}}}$

is good no matter what the value of k (which is simply related to a or b).

I agree with you in the sense that Schwam's article choice of $x^{2}/a^{2}+y^{2}=1$ as the equation of the ellipse is a bad choice if you want to understand the analogy between trig functions and the Jacobi elliptic functions. I played around with that a little, and then ignored it. I don't think its mathematically wrong, its just not helpful.

I agree that if you use his development, $rd\theta$ comes out to be $1/{\sqrt {1-k^{2}\cos ^{2}(\theta )}}$, but that's not wrong, if you change the upper limit of integration when you integrate it. He never goes into what that upper limit is in detail, and its kind of a mess if you do. Using the a equation for the ellipse is not the same as making x the major axis, so I think your statement "The problem is that making x the major axis produces the wrong r" is not correct. It's that using the a equation produces a messy version of r. Whether or not the x-axis is major or not has nothing to do with it.

What I found interesting was the relationships he listed for the elliptic functions which I have not seen before.PAR (talk) 18:46, 18 March 2018 (UTC)

On Wikipedia under "Quarter period" it says "m is a real number, 0 ≤ m ≤ 1". So whether m = 1 - 1/a^2 or m = 1 - 1/b^2 we must have 1 ≤ a and b. We cannot have a<1 or b<1. For the definition of cn and sn we must obtain the identical expression as in "definition as inverses of elliptic integrals". Which we do not starting from $x^{2}/a^{2}+y^{2}=1$. Therefore $x^{2}/a^{2}+y^{2}=1$ is wrong for the definition of Jacobi elliptic functions. Stephen William Wynn (talk) 16:24, 29 March 2018 (UTC)