Quantum calculus

Victor G. Kac, Pokman Cheung

112 pages, parution le 21/01/2002

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Résumé

Simply put, quantum calculus is ordinary calculus without taking limits. This undergraduate text develops two types of quantum calculi, the q-calculus and the h-calculus. As this book develops quantum calculus along the lines of traditional calculus, the reader discovers, with a remarkable inevitability, many important notions and results of classical mathematics.

This book is written at the level of a first course in calculus and linear algebra and is aimed at undergraduate and beginning graduate students in mathematics, computer science, and physics. It is based on lectures and seminars given by Professor Kac over the last few years at MIT.

Contents

Introduction
1 q-Derivative and h-Derivative 1
2 Generalized Taylor's Formula for Polynomials 5
3 q-Analogue of (x - a)[superscript n], n an Integer, and q-Derivatives of Binomials 7
4 q-Taylor's Formula for Polynomials 12
5 Gauss's Binomial Formula and a Noncommutative Binomial Formula 14
6 Properties of q-Binomial Coefficients 17
7 q-Binomial Coefficients and Linear Algebra over Finite Fields 21
8 q-Taylor's Formula for Formal Power Series and Heine's Binomial Formula 27
9 Two Euler's Identities and Two q-Exponential Functions 29
10 q-Trigonometric Functions 33
11 Jacobi's Triple Product Identity 35
12 Classical Partition Function and Euler's Product Formula 37
13 q-Hypergeometric Functions and Heine's Formula 43
14 More on Heine's Formula and the General Binomial 47
15 Ramanujan Product Formula 50
16 Explicit Formulas for Sums of Two and of Four Squares 56
17 Explicit Formulas for Sums of Two and of Four Triangular Numbers 60
18 q-Antiderivative 64
19 Jackson Integral 67
20 Fundamental Theorem of q-Calculus and Integration by Parts 73
21 q-Gamma and q-Beta Functions 76
22 h-Derivative and h-Integral 80
23 Bernoulli Polynomials and Bernoulli Numbers 85
24 Sums of Powers 90
25 Euler-Maclaurin Formula 92
26 Symmetric Quantum Calculus 99
App.: A List of q-Antiderivatives 106
Literature 109
Index

L'auteur - Victor G. Kac

Victor Kac is Professor of Mathematics at MIT. He is an author of 4 books and over a hundred research papers. He was awarded the Wigner Medal for his work on Kac-Moody algebras that has numerous applications to mathematics and theoretical physics. He is a honorary member of the Moscow Mathematical Society.

L'auteur - Pokman Cheung

Pokman Cheung graduated from MIT in 2001 after three years of undergraduate studies. He is presently a graduate student at Stanford University.

Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Victor G. Kac, Pokman Cheung
Parution	21/01/2002
Nb. de pages	112
Format	15,5 x 23,5
Couverture	Broché
Poids	200g
Intérieur	Noir et Blanc
EAN13	9780387953410
ISBN13	978-0-387-95341-0