Weighted forms of Euler's theorem

ar X iv : m at h / 05 10 05 4 v 2 [ m at h . H O ] 1 7 A ug 2 00 6 EULER AND THE PENTAGONAL NUMBER THEOREM

In this paper we give the history of Leonhard Euler's work on the pentagonal number theorem, and his applications of the pentagonal number theorem to the divisor function, partition function and divergent series. We have attempted to give an exhaustive review of all of Euler's correspondence and publications about the pentagonal number theorem and his applications of it. Comprehensus: In hoc di...

Euler's Method for O.D.E.'s

The first method we shall study for solving differential equations is called Euler's method, it serves to illustrate the concepts involved in the advanced methods. It has limited use because of the larger error that is accumulated with each successive step. However, it is important to study Euler's method because the remainder term and error analysis is easier to understand. Theorem (Euler's Me...

18.312 Lecture 10: Introduction to Partitions and Two Proofs of Euler's Theorem

Weighted quadrature rules with binomial nodes

In this paper, a new class of a weighted quadrature rule is represented as --------------------------------------------  where  is a weight function,  are interpolation nodes,  are the corresponding weight coefficients and denotes the error term. The general form of interpolation nodes are considered as   that  and we obtain the explicit expressions of the coefficients  using the q-...

A Geometric Approach to Regular Perturbation Theory with an Application to Hydrodynamics

The Lyapunov-Schmidt reduction technique is used to prove a persistence theorem for fixed points of a parameterized family of maps. This theorem is specialized to give a method for detecting the existence of persistent periodic solutions of perturbed systems of differential equations. In turn, this specialization is applied to prove the existence of many hyperbolic periodic solutions of a stead...