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Numbers And Functions

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Numbers And Functions

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Author : R. P. Burn
language : en
Publisher: Cambridge University Press
Release Date : 2000-08-28

Numbers And Functions written by R. P. Burn and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-08-28 with Mathematics categories.

This work should aid students in the transition from studying calculus in schools to studying mathematical analysis at university. It helps them tackle a sequence of problems to concepts, definitions and proofs of classical real analysis.

Numbers And Functions

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Author : R. P. Burn
language : en
Publisher: Cambridge University Press
Release Date : 2015-02-19

The transition from studying calculus in schools to studying mathematical analysis at university is notoriously difficult. In this third edition of Numbers and Functions, Professor Burn invites the student reader to tackle each of the key concepts in turn, progressing from experience through a structured sequence of more than 800 problems to concepts, definitions and proofs of classical real analysis. The sequence of problems, of which most are supplied with brief answers, draws students into constructing definitions and theorems for themselves. This natural development is informed and complemented by historical insight. Carefully corrected and updated throughout, this new edition also includes extra questions on integration and an introduction to convergence. The novel approach to rigorous analysis offered here is designed to enable students to grow in confidence and skill and thus overcome the traditional difficulties.

Bernoulli Numbers And Zeta Functions

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Author : Tsuneo Arakawa
language : en
Publisher: Springer
Release Date : 2014-07-11

Bernoulli Numbers And Zeta Functions written by Tsuneo Arakawa and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-07-11 with Mathematics categories.

Two major subjects are treated in this book. The main one is the theory of Bernoulli numbers and the other is the theory of zeta functions. Historically, Bernoulli numbers were introduced to give formulas for the sums of powers of consecutive integers. The real reason that they are indispensable for number theory, however, lies in the fact that special values of the Riemann zeta function can be written by using Bernoulli numbers. This leads to more advanced topics, a number of which are treated in this book: Historical remarks on Bernoulli numbers and the formula for the sum of powers of consecutive integers; a formula for Bernoulli numbers by Stirling numbers; the Clausen–von Staudt theorem on the denominators of Bernoulli numbers; Kummer's congruence between Bernoulli numbers and a related theory of p-adic measures; the Euler–Maclaurin summation formula; the functional equation of the Riemann zeta function and the Dirichlet L functions, and their special values at suitable integers; various formulas of exponential sums expressed by generalized Bernoulli numbers; the relation between ideal classes of orders of quadratic fields and equivalence classes of binary quadratic forms; class number formula for positive definite binary quadratic forms; congruences between some class numbers and Bernoulli numbers; simple zeta functions of prehomogeneous vector spaces; Hurwitz numbers; Barnes multiple zeta functions and their special values; the functional equation of the doub le zeta functions; and poly-Bernoulli numbers. An appendix by Don Zagier on curious and exotic identities for Bernoulli numbers is also supplied. This book will be enjoyable both for amateurs and for professional researchers. Because the logical relations between the chapters are loosely connected, readers can start with any chapter depending on their interests. The expositions of the topics are not always typical, and some parts are completely new.

Numbers And Functions

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Author : R. P. Burn
language : en
Publisher: Cambridge University Press
Release Date : 1993-12-09

The transition from studying calculus in schools to studying mathematical analysis at university is notoriously difficult. In this book, Dr Burn follows a route that proved successful with A Pathway to Number Theory and Groups: A Path to Geometry. He invites the student reader to tackle each of the key concepts in turn, progressing from experience (using computers for graph drawing where appropriate) through a structured sequence of several hundred problems to concepts, definitions and proofs of classical real analysis. The sequence of problems, which all have solutions supplied, draws students into constructing definitions and theorems for themselves. This natural development is informed by historical insight and complemented by historical discussion. The sequence also takes into account recent research which has shown how intuitive ideas about numbers, limits, functions and infinity may be at odds with the standard definitions. The novel approach to rigorous analysis offered here is designed to enable students to grow in confidence and skill and thus overcome the traditional difficulties. Teachers in sixth forms will find that questions at the beginning of every chapter provide ways of preparing those at school for university mathematics. Lecturers in universities will be challenged to rethink their conventions about the best way to introduce the central ideas of analysis to undergraduates.

Numbers And Functions Theory Formulation And Python Codes

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Author : Gui-rong Liu
language : en
Publisher: World Scientific
Release Date : 2024-06-18

Numbers And Functions Theory Formulation And Python Codes written by Gui-rong Liu and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-06-18 with Mathematics categories.

This unique volume covers two fundamental elements of computational methods — numbers and functions. It provides an in-depth discussion of the behaviors of numbers, including both real and complex numbers. The discussion leads to the important closure properties of numbers, ensuring solution consistence and existence, and also possible failure in numerical computations in science and engineering.This text then introduces types of functions that take numbers as independent variables and produce a single number. Approaches for constructing inverse functions are also provided. Frequently used basis functions are introduced, with detailed studies on their properties and behaviors. Techniques are presented for constructing basis functions and their use in function approximation in computational methods.

Famous Functions In Number Theory

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Author : Bowen Kerins
language : en
Publisher: American Mathematical Soc.
Release Date : 2015-10-15

Famous Functions In Number Theory written by Bowen Kerins and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-10-15 with Education categories.

Designed for precollege teachers by a collaborative of teachers, educators, and mathematicians, Famous Functions in Number Theory is based on a course offered in the Summer School Teacher Program at the Park City Mathematics Institute. But this book isn't a "course" in the traditional sense. It consists of a carefully sequenced collection of problem sets designed to develop several interconnected mathematical themes, and one of the goals of the problem sets is for readers to uncover these themes for themselves. Famous Functions in Number Theory introduces readers to the use of formal algebra in number theory. Through numerical experiments, participants learn how to use polynomial algebra as a bookkeeping mechanism that allows them to count divisors, build multiplicative functions, and compile multiplicative functions in a certain way that produces new ones. One capstone of the investigations is a beautiful result attributed to Fermat that determines the number of ways a positive integer can be written as a sum of two perfect squares. Famous Functions in Number Theory is a volume of the book series "IAS/PCMI-The Teacher Program Series" published by the American Mathematical Society. Each volume in that series covers the content of one Summer School Teacher Program year and is independent of the rest. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Examples And Problems In Advanced Calculus Real Valued Functions

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Author : Bijan Davvaz
language : en
Publisher: Springer Nature
Release Date : 2020-12-11

Examples And Problems In Advanced Calculus Real Valued Functions written by Bijan Davvaz and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-12-11 with Mathematics categories.

This book includes over 500 most challenging exercises and problems in calculus. Topical problems and exercises are discussed on set theory, numbers, functions, limits and continuity, derivative, integral calculus, Rolle’s theorem, mean value theorem, optimization problems, sequences and series. All the seven chapters recall important definitions, theorems and concepts, making this book immensely valuable to undergraduate students of engineering, mathematics, statistics, computer science and basic sciences.

Number Theory In Function Fields

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Author : Michael Rosen
language : en
Publisher: Springer Science & Business Media
Release Date : 2013-04-18

Number Theory In Function Fields written by Michael Rosen and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-04-18 with Mathematics categories.

Elementary number theory is concerned with the arithmetic properties of the ring of integers, Z, and its field of fractions, the rational numbers, Q. Early on in the development of the subject it was noticed that Z has many properties in common with A = IF[T], the ring of polynomials over a finite field. Both rings are principal ideal domains, both have the property that the residue class ring of any non-zero ideal is finite, both rings have infinitely many prime elements, and both rings have finitely many units. Thus, one is led to suspect that many results which hold for Z have analogues of the ring A. This is indeed the case. The first four chapters of this book are devoted to illustrating this by presenting, for example, analogues of the little theorems of Fermat and Euler, Wilson's theorem, quadratic (and higher) reciprocity, the prime number theorem, and Dirichlet's theorem on primes in an arithmetic progression. All these results have been known for a long time, but it is hard to locate any exposition of them outside of the original papers. Algebraic number theory arises from elementary number theory by con sidering finite algebraic extensions K of Q, which are called algebraic num ber fields, and investigating properties of the ring of algebraic integers OK C K, defined as the integral closure of Z in K.

Algebraic Numbers And Algebraic Functions

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Author : P.M. Cohn
language : en
Publisher: CRC Press
Release Date : 1991-09-01

Algebraic Numbers And Algebraic Functions written by P.M. Cohn and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1991-09-01 with Mathematics categories.

This book is an introduction to the theory of algebraic numbers and algebraic functions of one variable. The basic development is the same for both using E Artin's legant approach, via valuations. Number Theory is pursued as far as the unit theorem and the finiteness of the class number. In function theory the aim is the Abel-Jacobi theorem describing the devisor class group, with occasional geometrical asides to help understanding. Assuming only an undergraduate course in algebra, plus a little acquaintance with topology and complex function theory, the book serves as an introduction to more technical works in algebraic number theory, function theory or algebraic geometry by an exposition of the central themes in the subject.

Arithmetical Functions

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Author : Komaravolu Chandrasekharan
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Arithmetical Functions written by Komaravolu Chandrasekharan and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.

The plan of this book had its inception in a course of lectures on arithmetical functions given by me in the summer of 1964 at the Forschungsinstitut fUr Mathematik of the Swiss Federal Institute of Technology, Zurich, at the invitation of Professor Beno Eckmann. My Introduction to Analytic Number Theory has appeared in the meanwhile, and this book may be looked upon as a sequel. It presupposes only a modicum of acquaintance with analysis and number theory. The arithmetical functions considered here are those associated with the distribution of prime numbers, as well as the partition function and the divisor function. Some of the problems posed by their asymptotic behaviour form the theme. They afford a glimpse of the variety of analytical methods used in the theory, and of the variety of problems that await solution. I owe a debt of gratitude to Professor Carl Ludwig Siegel, who has read the book in manuscript and given me the benefit of his criticism. I have improved the text in several places in response to his comments. I must thank Professor Raghavan Narasimhan for many stimulating discussions, and Mr. Henri Joris for the valuable assistance he has given me in checking the manuscript and correcting the proofs. K. Chandrasekharan July 1970 Contents Chapter I The prime number theorem and Selberg's method § 1. Selberg's fonnula . . . . . . 1 § 2. A variant of Selberg's formula 6 12 § 3. Wirsing's inequality . . . . . 17 § 4. The prime number theorem. .

Numbers And Functions

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