According to I Ching, hexagram 10, when we are innocent, the tiger will not bite us even when we step on tiger’s tail.

There is a danger. Tigers are wild animals.

But, if we examine the signs carefully, there will be great good fortune.

A couple of days ago, when writing the post Jacobi Elliptic SN – The case of inverted modulus, and then writing Jacobi amplitude- realism or cubism, I stepped on tiger’s tail. Fortunately the tiger just looked at me with a forgiving smile and did not bite. I was lucky. Sure,** I will have to correct the misleading pieces in those previous posts**. But, thanks to **good qcomments from BJAB**, I have progressed with my understanding. The reading in Hexagram 10 includes:

The Judgement

Trading: Stepping on the tail of a tiger.

But it does not bite one. Success.

The Symbol

The heaven above and the lake below.

Symbolize Treading.

The superior man differentiates between high and low,

And thus fixes the minds of people.

Well, differentiating between high and low means **making it clear how to define Jacobi elliptic functions for in terms of those with ** However, before fixing the mind of people I surely had to fix my own mind. This post is about how I did it …. till now. I must say that **I was searching through all the available sources, and could not find a clear formulation**. Perhaps it is one of those things that all experts know but do not bother to explain. Which would be however hard to understand, because in textbooks they often explain so many completely elementary issues…

Let us recall from Jacobi Elliptic SN – The case of inverted modulus:

The original name of Jacobi elliptic sn function was

sinus amplitudinis. One contemplates the integral(1)

and calls the amplitude of where Then is defined as that is as – that is how

sinus amplitudiniswas invented.First let us contemplate the integral 1. If the integral is well defined (produces a finite number) for all finite values of The integrand is always positive, therefore is strictly increasing function of The inverse function exists, is well defined.

So far, so good, but then I continued with

But what if ? From the definition we see that is antisymmetric function of therefore let us restrict ourselves to nonnegative. If is smaller that the critical value defined by , the integral is finite and is an increasing function of For approaching the integral becomes infinite. Therefore the inverse function is well defined for all also for

**his was wrong**. This was stepping on tiger’s tail. The sentence “For approaching the integral becomes infinite” i

**s false**. I wrote it because I thought it was true, but I did not care to verify. For the integral

**remains finite for**

Nevertheless **the rest of the derivation in that post is OK**, as long as is restricted to be the value of the integral. Therefore, for , the formula

(2)

holds, except that we have derived it for such that

In order to be sure that this is what Wolfram’s Mathematica software “knows”, I checked:

I do not know which algorithm is being used by Mathematica for calculating sn function for (I suspect it is using complex Weierstrass functions), but evidently our definition above is correct (up to negligible numerical errors).

Once we are done with the sn function, we can now define the am function for :

(4)

We verify if this definition can be used for reproducing the plot of am in Jacobi amplitude- realism or cubism

And again we see that the definition above agrees with the one used by Mathematica – the difference between the two is zero. Our plot was correct. In fact, to be sure I asked to do the same plot by Maple. Here is the result:

Maple is using the notation JacobiAM(u,k), where

Still good. Finally, what about Jacobi elliptic *cosinus amplitudinis*? Can we define it as also for ?

Again checking with Mathematica:

Good. Of course one can ask: since when Mathematica is a judge of what is a good definition of a mathematical function? It should not be so. But here we have to take into account the fact that **mathematical monographs are surprisingly silent on the subject. It is like a conspiracy! Much like with UFO!**

Now, I will have to fix the misleading parts in previous posts while hoping that I am on the right track now.

I did some more reading. Below is the text from Greenhiil, The Applications of Elliptic Functions, 1892

Very informative, but, as you can see – it is fuzzy, to say the least

Why no one would write clearly, with all the assumptions and defintions? I just can’t believe it!

Ark,

have you seen my last comments under your former post?

Yes but I do not see how can use it?

““for ems>1 look like sinusoids. Are they?

“1 look like sinusoids. Are they?”

Good question. What is the period there? We need to think.

“What is the period there?”The period is the same as the period of sn(u) (for 1/m) divided by k.

But is it sinusoid, that is the question.

First it would be good to present a picture where ellipse with its r(theta) was replaced by hyperbola with its r(theta) and showing maksimum (phi c).

Then our integral, u(phi), would be the “angular arc lenth” (I know you don’t like this name) of an arc, that is, of an hyperbola.

@BJAB

It is not a sinusoid. Here is the graph for m=1.00001

But, when m grows, it looks more and more like a sinusoid.

“It is not a sinusoid.”Thank you.

Ark,

now I have two requests.

First for graph of am(u,m) for m from minus 4 to 4 (and u from -10 to 10).

Second for graph of am(u,k) for k from minus 2 to 2 (and u from -10 to 10).

@BJAB

The two graphs you have requested

Thanks for the two graphs.

I think that the second graph resembles the plaster model you criticized the other day.

“I think that the second graph resembles the plaster model”

According to the philosophy of cubism – perhaps :)

“ Philosophy of Cubism

The Cubists were more intrigued by form than color. They wanted to develop a new of seeing entirely. They were interested in changing viewpoint as it was effected by space or time. The point of Cubism was to show all viewpoints simultaneously. “

According to shape which is the same (despite of scale and domain (u>0) which can be attuned).

But you were right. Here is the plot with k from -4 to 4 and u from 0 to 10

I have updated (at the bottom) the post Jacobi amplitude- realism or cubism on the plaster model.