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 va...