Functions and Analysis

1702 Submissions

[3] viXra:1702.0305 [pdf] submitted on 2017-02-24 10:49:41

The Gamma Function

Authors: James Bonnar
Comments: 163 Pages.

The Gamma Function is the finest book dedicated to the topic of the Gamma function. Written in an easily understandable manner, the book is well-suited for advanced undergraduates in science and mathematics. The book is concise and thorough, covering the most important aspects of the Gamma function. The Gamma function has important applications in probability theory, combinatorics and most, if not all, areas of physics. A large number of proofs and derivations of theorems and identities are covered in the book including: Analytic continuation of the factorials, properties via complex analysis, convexity and the Gamma function, the Beta function, Wallis's integrals, Wallis's product, product & reflection formulas, half-integer values, digamma and polygamma functions, series expansions, Euler-Mascheroni integrals, duplication & multiplication formulas, the Gamma and zeta function relationships, Hankel's contour integral representation, Stirling's formula, the Weierstrass factor theorem and the Mittag-Leffler theorem.
Category: Functions and Analysis

[2] viXra:1702.0119 [pdf] submitted on 2017-02-09 07:58:42

Lagrangian Solid Modeling

Authors: Matthew Marko
Comments: 114 pages including supplementary code

The author demonstrates a stable Lagrangian solid modeling method, tracking the interactions of solid mass particles, rather than using a meshed grid. This numerical method avoids the problem of tensile instability often seen with Smooth Particle Applied Mechanics by having the solid particles apply stresses expected with Hooke's law, as opposed to using a smoothing function for neighboring solid particles. This method has been tested successfully with a bar in tension, compression, and shear, as well as a disk compressed into a flat plate, and the numerical model consistently matched the analytical Hooke's law as well as Hertz contact theory for all examples. The solid modeling numerical method was then built into a 2-D model of a pressure vessel, which was tested with liquid water particles under pressure and simulated with Smoothed Particle Hydrodynamics. This simulation was stable, and demonstrated the feasibility of Lagrangian specification modeling for Fluid Solid Interactions.
Category: Functions and Analysis

[1] viXra:1702.0039 [pdf] submitted on 2017-02-03 03:27:16

Proof to Euler's Formula

Authors: Carl-Gustav Hedenby
Comments: 1 Page. -

The author proves Euler's formula for the imaginary exponential without reverting to series expansions.
Category: Functions and Analysis