Functions and Analysis

Erratum to "Tables of Integral Transforms" by A. Erdelyi, W. Magnus, F. Oberhettinger & F. G. Tricomi (1953), p. 61 (4)

Authors: Richard J. Mathar
Comments: 3 Pages.

The integral (4) on page 61 in the "Tables of Integral Transforms", the Fourier Cosine Transform of a product of a Gaussian and a symmetric sum of two Parabolic-Cylinder Functions, is erroneous. A more general integral is derived here.
Category: Functions and Analysis

On the Equation F(x)=x^2+exp(-2x)-1=0

Authors: Edgar Valdebenito
Comments: 5 Pages.

In this note we give solution of the equation f(x)=x^2+exp(-2x)-1=0
Category: Functions and Analysis

On the Integral Inequality of Some Trigonometric Functions in $mathbb{r}^n$

Authors: Theophilus Agama
Comments: 6 Pages.

In this note, we prove the inequality begin{align}bigg| int limits_{|a_n|}^{|b_n|} int limits_{|a_{n-1}|}^{|b_{n-1}|}cdots int limits_{|a_1|}^{|b_1|}cos bigg(frac{sqrt[4s]{sum limits_{j=1}^{n}x^{4s}_j}}{||vec{a}||^{4s+1}+||vec{b}||^{4s+1}}bigg)dx_1dx_2cdots dx_nbigg| leq frac{bigg|prod_{i=1}^{n}|b_i|-|a_i|bigg|}{|Re(langle a,b angle)|}onumberend{align}and begin{align}bigg|int limits_{|a_n|}^{|b_n|} int limits_{|a_{n-1}|}^{|b_{n-1}|}cdots int limits_{|a_1|}^{|b_1|}sin bigg(frac{sqrt[4s]{sum limits_{j=1}^{n}x^{4s}_j}}{||vec{a}||^{4s+1}+||vec{b}||^{4s+1}}bigg)dx_1dx_2cdots dx_nbigg| leq frac{bigg|prod_{i=1}^{n}|b_i|-|a_i|bigg|}{|Im(langle a,b angle)|}onumberend{align}under some special conditions.
Category: Functions and Analysis

Numerical Derivatives

Authors: Horacio Useche
Comments: 50 Pages. In Spanish. Cálculo de derivadas mediante métodos numéricos (Calculation of derivatives using numerical methods)

The idea of ​​this work is to present the software that allows us to quickly and numerically calculate values ​​of f 0, f 00 , f 000 and f IV at the points where they are required, especially thinking about the estimation of the error in problems that involve differential equations. ordinary and partial differential equations. The calculation of these values ​​by means of numerical methods is of great. It helps in solving these problems, as it saves a lot of time. The routines presented have been written in Google Inc.'s Go language, following our policy of making the most of "21st century C", which is a very fast, comfortable tool with sufficient accuracy for the proposed applications. We hope that this study will be useful for professional mathematicians as well as scientists from other areas and engineers who need to calculate the error in their equations or the rates of change associated with a whole potential of physical applications.

La idea de este trabajo es presentar el software que nos permite calcular rápida y numéricamente valores de f 0, f 00 , f 000 y f IV en los puntos donde se les requiera, sobre todo pensando en la estimación del error en problemas que involucran ecuaciones diferenciales ordinarias y ecuaciones en derivadas parciales. El cálculo de estos valores mediante métodos numéricos es de gran ayuda en la resolución de estos problemas, pues ahorra mucho tiempo. Las rutinas presentadas han sido escritas en lenguaje Go de Google Inc., siguiendo nuestra polı́tica de usufructuar al máximo el “C del siglo XXI”, que es una herramienta muy rápida, cómoda y con la exactitud suficiente para las aplicaciones propuestas. Esperamos que este estudio sea de utilidad tanto para matemáticos profe-sionales como cientı́ficos de otrás áreas e ingenieros que requieran calcular el error en sus ecuaciones o las ratas de cambio asociadas con todo un potencial de aplicaciones fı́sicas.
Category: Functions and Analysis