# Jacobi Elliptic sn – the case of a stuttered sinus

In my last post I introduced Jacobi elliptic sinus, the function with real, and One has to be careful with notation here. Let us have a look at the definition as it is given on Wolfram’s pages:

It is almost the same, but not the same as one from “Handbook of Mathematical Functions“, Ed. Milton Abramowitz and Irene A. Stegun:

Here Abramowitz and Stegun write simply , but later on they use the notation where of Wolfram. Wolfram’s Mathematica is using JacobiSN(u,m). Maple is using
Matlab warns the user:

We will use
The parameter is sometimes called the “modulus”. The shape of the function depends on the value of When we have just ordinary sinus:

At the other extreme, for sn is nothing else but the hyperbolic tangent:

In between, for , it is what I would dare to call a “stuttering sinus“. This stuttering is not seen at all for It looks just as if the period became a little longer:

and it is hard to notice for

But for we get

It almost looks like the hyperbolic tangent, but when we zoom out we can see the stuttering:

Clear case of stuttering

The graphs looks somewhat like that of a rectangular signal. There are “flips” and then there are longer and longer periods when the function is almost constant – between the flips. This is the main characteristic of Dzhanibekov effect: there are almost pure rotation periods, and sudden flips when the axis suddenly reverses the direction. But for all values of the function is periodic (for we have an exception – we have infinite period).

For the period is then slowly grows, but for very close to it becomes very sensitive to the value of For this reason every repetition of the Dzhanibekov effect as seen in the movies taken in space would probably give a different period.

## 17 thoughts on “Jacobi Elliptic sn – the case of a stuttered sinus”

1. To use latex in comments you should write, at the beginning of your comment []. Then we can see latex at work

Syntax explained here.

2. Bjab says:

“In between, for 0<m<1, it is what I would dare to call a “stuttering sinus“"
Poetry piece.

1. Yes. In fact: not a very ambitious. But: blogs are good places for publishing poetry pieces. On my blog I will sometimes make jokes, sometimes write something surrealistic. Blogs are personal. I am not pretending to know a lot. I often make mistakes. Blogs are for sharing ideas. Wikipedia has different goals and therefore different duties.

1. Bjab says:

“Wikipedia has different goals”
Wikipedia has the goals of its owners.
(a good example of this rule was salon24.pl)

3. Bjab says:

“for m=1, sn is nothing else but the hyperbolic sinus”

“It almost looks like the hyperbolic sinus”

I was used to the different appearance of the graph of hyperbolic sinus.

1. The graphs for m=1 and m=0.9999999 are different. Though they loook almost indistinguishable for u from 0 to 10. The differences clearly show up for, say, u=20.

1. Bjab says:

I wanted to write:
I was used to the different appearance of the graph of hyperbolic sinus”

The graphs you presented don’t look like graphs of hypebolic sinus functions.

1. You are right. It was my error. It looks like hyperbolic tangent, not hyperbolic sinus. Thanks!!!

1. Bjab says:

“for m=1, sn is nothing else but the hyperbolic tangent”
Would you prove it?

4. @BJAB

““for m=1, sn is nothing else but the hyperbolic tangent”
Would you prove it?”

OK I did it. And it was a good exercise. Thank you for this question.
For we have

Then, from the definition So I asked Mathematica to find the integral.
I got

It looks good. It can be verified by differentiation, and for we get as it should be.(Wikipedia gives the answer in a somewhat different form) It looks complicated, but I knew what the answer should be anyway. The answer simplified to

Then I asked to calculate using the trigonometric identity

5. Bjab says:

Ark,
“2 attempts remaining.”

It is rather frustrating.

Can you do something with it?

1. It is a security feature that I yet have to learn how to use it. Will try to increase from 2 to 3 :)

1. Bjab says:

Well, it would be great if after correct loging the counter was reset.

6. Bjab says:

Ark,
thank you for the proof.
I’ve checked it. I’ve found one misspelling. I think that there should be 2 in numerator

1. Bjab says:

I can see that you can intervene in former comments. That gives you a lot of power.

1. I installed a plugin that should allow users to edit their comments during 5 minutes after posting. It is better than nothing.