THE ELLIPTIC EXPONENTIAL NUMBER E(1, i)

THE ELLIPTIC EXPONENTIAL NUMBER: E(1, i)


  I hereby report a certain number of significant interest that I have recently found in the course of my investigations in the theory of elliptic functions and theta functions.

The following objects are well-known.

The exponential function: e(iu) = cos u + i sin u

The exponential number: e := e(1) = 2.7182818284 (approximately)


Now let E(iu, τ) denote the elliptic exponential function: E(iu, τ) := cn(u, τ) + i sn(u, τ).

I have found the following object by calculations which took me several weeks to complete (I omit them from this note.):

The Gaussian elliptic exponential number: E(i) := E(1, i) = 3.0252897893 (approximately).

It can be shown that  lim  E(iu, τ) = e(iu)  as q approaches 0, where q = e(iπτ), Im(τ) > 0.
                               
As τ ranges over the upper half plane, E(1, τ) takes various values. For τ=i, the lattice is that of the Gaussian integers.

                                     

This explains the name 'Gaussian elliptic exponential number' of E(i) := E(1, i).

There is a class of elliptic exponential numbers, each of them corresponding to a specific value of τ in the upper half plane. 

Somjit Datta, Ph.D

May 13, 2018.
Calcutta, India