This one is another purely technical post. The point is that I am not very happy with all the existing expositions about Jacobi elliptic functions. I want to organize it my way which, I am sure, is better than all other ways. Of course what is better and what not depends on the interests of the particular person and on the purpose of the study. I am not completely sure what my purpose is, but somehow I am navigating and, at some point, I will be able to shout: Tierra! Tierra! But not yet. So I continue.
First: a summary of the essential properties of the elliptic functions
In fact the properties below are sufficient to completely define these elliptic functions. This is how these functions are defined in an old but good (and freely avilable) monograph by Dixon, The elementary properties of the elliptic functions, with examples, 1894.
Quite often the argument is omitted, it simplifies the notation. Sometimes we use sometimes For instance in the formulas above should be written as but some people will write it as where After some little training there should be no confusion.
Previously, we have also found the addition formula for We could also prove the addition formulas for and but I will not do it. I am just copying, for instance Eqs (48)-(50) in Jacobi Elliptic Functions, on Wolfram’s site
We have also defined
and, for we have defined the function as the inverse function of : If then Moreover
The function is called “the incomplete elliptic integral of the first kind“. From the shape of the function
it is clear that
Thus we can write and therefore This way we can build the following table
Then from addition formulas we obtain:
The case of m>1
When the expression under the square root in the integrand of the integral (8) defining goes through the negative values. Thus , for is, in general, a complex number. During the discussion under the last post Period of a pendulum, the following conjecture has appeared:
I do not have a rigorous proof of that conjecture. What I have is a plot, using Mathematica, of the numerical difference between the left and the right hand side of Eq. (9). Here it is:
As you can see it looks like a solid zero! One can also verify that both, the real and the imaginary part of the right hand side, as a function of satisfy the differential equation that is supposed to satisfy – Eq. (19) in Complete Elliptic Integral of the First Kind:
But that is not yet a complete proof. Nevertheless I will assume that the above conjecture is true. To proof that the real part of is right is, in fact, easy. Here is how I do it: Let be the angle for which Then the real part of is given by
Introduce defined by Then and thus Moreover Since we obtain
I suspect that the proof for the imaginary part should not be much more complicated, but I did not succeeded doing it yet. Therefore at this point I will make a break. Till tomorrow.
The conjectured formula is well known. In the tables by I. S. Gradshteyn, I. M. Ryzhik we find
There is a condition but there is no such restriction in the source that is quoted – MO 131. Some has added this condition later, for security reasons. We extend it to the case of Then, solving for we get our formula.
In fact I can almost get this formula by changes of integration variables, except that one has to be careful what to do with the square root of a negative number. A convention is needed. The sign of the imaginary part that I am getting depends on this convention.
Update 2, Sunday Jan. 29
I decided to share my derivation for the imaginary part. It is a continuation of the real part derivation at the end of the post. The imaginary part is given by the integral
Here the expression under the square root is negative all the time. So we write
and therefore (since )
Now we change the integration variable by setting Then and so
where is the positive value of such that where
Now we do the same trick that we did in the real case. We introduce so that
This is possible since the integration is within the range of where Then so
Here and there I was skipping explanation and justifications, but these small cracks in the proof can be easily filled in.
The conjecture appears to have been finally proved. So, it is not a conjecture any more, it is a proven property! Uff!