Elliptic Integral of the Second Kind

Ryan Hirst

A new formula for the incomplete elliptic integral of the second kind

Placeholder for a paper on the elliptic integrals of the first and second kind, based upon formulas of Gauss, Legendre, and Cayley. I will show efficient formulas for all cases. In particular, I will show that the incomplete Integral of the Second Kind can be iterated in a manner analogous to the First, leading to a succession of increasingly accurate formulas. The paper is in two main parts: I. The Derivation and implementation of the new integral formulas. II. Establish a sound mathematical context A lot of nonsense has been written about the Integral of the Second Kind. Most recently, Semjon Adlaj published a paper in Notices of the AMS claiming - without justification- a new iterative solution for the complete integral, alongside a number of self-aggrandizing, false claims. For the practical mathematician, this will not do. We need to know if Cayley's formula is safe for arbitrarily small k' (it is), or if we should instead use Adlaj's (we should not). And we must be sure the formulas are correct. After the first section, it will be plain that this problem was already solved by Gauss, Legendre and Cayley; all that remained was to carry out the instructions. We will have in hand the tools to assess and correct Adlaj's claims, while offering superior numerical methods. Some numerical methods in the second part rely upon a knowledge of in-place numerical differentiation, which does not appear to be widely understood. I will present separately an efficient method for numerical differentiation: exact finite differences which pass smoothly into the limit, without the cumbersome apparatus of complex-step differentiation. Peace