We undertake a thorough investigation of the moments of Ramanujan’s alternative elliptic integrals and of related hypergeometric functions. Along the way we are able to give some surprising closed forms for Catalan-related constants and various new hypergeometric identities.

Landen formulas, which connect Jacobi elliptic functions with different modulus parameters, were first obtained over two hundred years ago by making a suitable quadratic transformation of variables in elliptic integrals. We obtain and discuss significant generalizations of the celebrated Landen formulas. Our approach is based on some recently obtained periodic solutions of physically interestin...

This paper develops a recursive method for computing moments of 2D objects described by elliptic Fourier descriptors (EFD). To this end, Green’s theorem is utilized to transform 2D surface integrals into 1D line integrals and EFD description is employed to derive recursions for moments computations. A complexity analysis is provided to demonstrate space and time efficiency of our proposed techn...

I am presenting a survey of regularity results for both minima of variational integrals, and solutions to non-linear elliptic, and sometimes parabolic, systems of partial differential equations. I will try to take the reader to the Dark Side...

The symbol bootstrap has proven to be a powerful tool for calculating polylogarithmic Feynman integrals and scattering amplitudes. In this Letter, we initiate the elliptic integrals. Concretely, of twelve-point two-loop double-box integral in four dimensions, which depends on nine dual-conformal cross ratios. We obtain alphabet, contains 100 logarithms as well simple integrals, via Schubert-typ...

In this article, we prove the existence of minimizers for integrals associated with a second-order elliptic problem. For this three-dimensional optimization problem, the admissible set is a rearrangement class of a given function.

The mean length of finite clusters is derived exactly for the case of directed compact percolation near a damp wall. We find that the result involves elliptic integrals and exhibits similar critical behaviour to the dry wall case.