**Jacobi elliptic functions** are needed for writing down explicit solutions of the free asymmetric top, and I decided to define them in a way that somehow resembles the way in which ordinary sinus and cosinus functions are defined in trigonometry. In my Polish blog (Sinus zdeformowany, Cosinus zdeformowany, Zdeformowna jedynka)I used for this purpose the method proposed by William A. Schwalm, that I have mentioned in my previous post on Jacobi.

Yesterday professor Schwalm kindly informed me about his debate with one of the Wikipedia librarians, one responsible for the “Definition as trigonometry” section of the Wikipedia entry on Jacobi elliptic functions. After exchanging several emails with prof. Schwalm I decided, if only just for change, to present here **the method used in Wikipedia**, a variation of the method used by W. A. Schwalm in his handout notes “**Elliptic Functions sn, cn, dn, as Trigonometry**“ and in his book “**Lectures on Selected Topics in Mathematical Physics: Elliptic Functions and Elliptic Integrals**“.

I must, however, say right away that I am not happy with what I have found in Wikipedia. I think that **what is in Wikipedia is, at least, very misleading**. And I will explain toward the end of this post why I think it is so.

Let’s go to work. We start with drawing an ellipse

(1)

and we choose one with and

Wikipedia for some reason avoids showing any such drawing, so what I am presenting here is a drawing much like in Schwalm’s “**Elliptic Functions sn, cn, dn, as Trigonometry**“, but adapted to Wikipedia article conventions.

Define

(2)

Consider a point on the ellipse, with

(3)

(4)

Using the equation of the ellipse we can eliminate and get for

(5)

We have and Therefore and so or Thus But Therefore and we can calculate

Notice that we are in agreement with the definition that can be found, for instance, in “**Handbook of Mathematical Functions**“, Ed. Milton Abramowitz and Irene A. Stegun:

We are also in agreement with the definition of Jacobi Elliptic Functions in Wolfram MathWorld.

The advantage of using the picture above is that the integrand in Eq. (7) has a simple interpretation as of Eq. (5).

Thus so far so good. **But the Wikipedia tries to do even better, and that is where it gets worse**. Here is the bad part, extract from Wikipedia:

First they introduce **undefined** “*angular component of the arc length*” and “*angular arc length*“. There is no such thing as “*angular component of the arc length*“, it does not make any sense. In mathematics we should avoid using undefined terms. Introducing such term is a crime – unless the meaning is obvious. But here the meaning is not obvious at all. **The fact that “angular arc length” is in quotation marks is not an excuse**. It is, in my opinion, **misleading**. Then there is a mysterious sentence about “*angular arc length*” and “*total arc length of hyperbolas*“. Then the formula (6) is interpreted as “angular *arc length for the ellipse*“. That is very misleading. The arc length for ellipse, like for any other curve, is defined as the integral

(9)

For our ellipse it can be transformed into

(10)

which is certainly** not the same** as **as suggested by Wikipedia**.

We have

For we substitute now to get

Adding together:

In order to check Eq. (10) and get the circumference of the ellipse (for ), I have numerically calculated the integral (10) from to and multiplied by to get . On the other hand Ramanujan suggested a simple approximate formula for the perimeter of an ellipse

(11)

Using Ramanujan formula I get essentially the same answer . But if I would use “*angular arc length for the ellipse*” **of Wikipedia**, I would get as the answer – **evidently wrong**.

Of course there is yet another way of calculating the perimeter of the ellipse. We can parametrize ellipse as

The parameter is then **not the same** as the angle above. But now for , for the perimeter we easily get

(12)

It is good to know, but it is irrelevant for our purpose.