From Wallis to Ramanujan, in companionship with RW Gosper

Hypergeometric function

In mathematics, the Gaussian or ordinary hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by and .

Generalized hypergeometric function

In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series.

Euler method

In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1870).

From Wallis to Ramanujan, in companionship with RW Gosper

One- and two-step semi-Lagrangian integrators for arbitrary Lagrangian-Eulerian-finite element two-phase flow simulations

On the nonlinear Dirichlet-Neumann method and preconditioner for Newton's method

On the nonlinear Dirichlet-Neumann method and preconditioner for Newton's method

One- and two-step semi-Lagrangian integrators for arbitrary Lagrangian-Eulerian-finite element two-phase flow simulations