Maple and Mathematica: A Problem Solving Approach for Mathematics / Edition 2

by Inna K. Shingareva, Carlos Lizïrraga-Celaya

Overview

In the history of mathematics there are many situations in which cal- lations were performed incorrectly for important practical applications. Let us look at some examples, the history of computing the number— began in Egypt and Babylon about 2000 years BC, since then many mathematicians have calculated— (e. g. , Archimedes, Ptolemy, Vi‘ ete, etc. ). The first formula for computing decimal digits of— was disc- ered by J. Machin (in 1706), who was the first to correctly compute 100 digits of—. Then many people used his method, e. g. , W. Shanks calculated— with 707 digits (within 15 years), although due to mistakes only the first 527 were correct. For the next examples, we can mention the history of computing the fine-structure constant— (that was first discovered by A. Sommerfeld), and the mathematical tables, exact - lutions, and formulas, published in many mathematical textbooks, were not verified rigorously [25]. These errors could have a large effect on results obtained by engineers. But sometimes, the solution of such problems required such techn- ogy that was not available at that time. In modern mathematics there exist computers that can perform various mathematical operations for which humans are incapable. Therefore the computers can be used to verify the results obtained by humans, to discovery new results, to - provetheresultsthatahumancanobtainwithoutanytechnology. With respectto our example of computing?, we can mention that recently (in 2002) Y. Kanada, Y. Ushiro, H. Kuroda, and M.

Product Details

ISBN-13:	9783211994313
Publisher:	Springer Vienna
Publication date:	09/29/2009
Edition description:	2nd ed. 2009
Pages:	484
Product dimensions:	6.10(w) x 9.20(h) x 0.60(d)

Table of Contents

I Foundations of Maple and Mathematica.- Maple.- Mathematica.- II Mathematics: Maple and Mathematica.- Graphics.- Algebra.- Linear Algebra.- Geometry.- Calculus and Analysis.- Complex Functions.- Special Functions and Orthogonal Polynomials.- Integral and Discrete Transforms.- Mathematical Equations.- Numerical Analysis and Scientific Computing.