CRC Concise Encyclopedia of Mathematics [part 1 of 4] - E. Weisstein (ed.) (1999) WW.pdf - Mathematics - widez2

MATHEMATICS

CRC ConciseEncyclopedia

CRCConciseEncyclopedia

A-ICS

Eric W, Weisstein

0 cp-

CRC Press

Boca Raton London New York Washington, D.C,

Library of Congress Cataloging-in-Publication

Data

Weisstein, Eric W.

The CRC concise encyclopedia of mathematics / Eric W. Weisstein.

p. cm.

Includes bibliographical references and index.

ISBN o-8493-9640-9 (alk. paper)

1. Mathematics- -Encyclopedias. I. Title.

QA5.W45 1998

5 10’.3-IX21

98-22385

CIP

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources

are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and

the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying,

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The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific

permission must be obtained in writing from CRC Press LLC for such copying.

Direct all inquiries to CRC Press LLC, 2000 Corporate Blvd., N.W., Boca Raton, Florida 33431.

Trademark Notice: Prod

without intent to infringe.

uct or corporate

may be trademarks or registered trademarks, and are only used for identification and explanation,

0 1999 by CRC Press LLC

No claim to original U.S. Government works

International Standard Book Number O-8493 -9640-9

Library of Congress Card Number 98-22385

Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

Printed on acid-free paper

Introduction

The CRC Concise Encyclopedia

of ibfuthemutics

is a compendium

of mathematical

definitions,

formulas,

figures, tabulations,

and references. It is written in an informal

style intended to make it accessible to a broad

spectrum

of readers with a wide range of mathematical

backgrounds

and interests.

Although

mathematics

a fascinating

subject, it all too frequently

is clothed in specialized jargon and dry formal exposition

that make

many interesting

and useful mathematical

results inaccessible

to laypeople.

This problem

is often further

compounded

by the difficulty

in locating

concrete and easily unders+ood

examples.

To give perspective

a subject, I find it helpful to learn why it is useful, how it is connected

to other areas of mathematics

and

science, and how it is actually

implemented.

While a picture may be worth a thousand

words, explicit

examples are worth at least a few hundred!

This work attempts

to provide enough details to give the reader

a flavor for a subject without

getting lost in minutiae.

While absolute rigor may suffer somewhat,

I hope

the improvement

in usefulness and readability

will more than make up for the deficiencies

of this approach.

The format of this work is somewhere

between a handbook,

a dictionary,

and an encyclopedia.

It differs

from existing dictionaries

of mathematics

in a number of important

ways. First,, the entire text and all

the equations

and figures are available

in searchable

electronic

form on CD-ROM.

Second, the entries are

extensively

cross-linked

and cross-referenced,

not only to related entries but also to many external

sites

on the Internet,.

This makes locating

information

very convenient.

It also provides a highly efficient way

to “navigate”

from one related concept to another,

a feature that is especially

powerful in the electronic

version. Standard

mathematical

references, combined

with a few popular ones, are also given at the end of

most entries to facilitate

additional

reading and exploration.

In the interests of offering abundant

examples,

this work also contains a large number of explicit formulas and derivations,

providing

a ready place to locate

a particular

formula,

as well as including

the framework

for understanding

where it comes from.

The selection

of topics in this work is more extensive

than in most mathematical

dictionaries

(e.g.,

Borowski and Borwein’s

Harper-Collins

Dictionary

of Mathematics

and Jeans and Jeans’ Muthematics

Dictio-

nary). At the same time, the descriptions

are more accessible than in “technical”

mathematical

encyclopedias

(e.g., Hazewinkel’s

Encyclopaedia

of Mathematics

and Iyanaga’s

Encyclopedic

Dictionary

of Mathematics).

While the latter remain models of accuracy and rigor, they are not terribly

useful to the undergraduate,

research scientist, or recreational

mathematician.

In this work, the most useful, interesting,

and entertaining

(at least t o my mind) aspects of topics are discussed in addition

to their technical

definitions.

For example,

in my entry for pi (n), the definition

in terms of the diameter

and circumference

of a circle is supplemented

by a great many formulas and series for pi, including

some of the amazing discoveries of Ramanujan.

These

formulas are comprehensible

to readers with only minimal

mathematical

background,

and are interesting

both those with and without formal mathematics

training.

However, they have not previously

been collected

in a single convenient

location.

For this reason, I hope that, in addition

to serving as a reference source, this

work has some of the same flavor and appeal of Martin

Gardner’s

delightful

Scientific

American

columns.

Everything

in this work has been compiled

by me alone. I am an astronomer

by training,

but have picked

up a fair bit of mathematics

along the way. It never ceases to amaze me how mathematical

connections

weave their way through the physical sciences. It frequently

transpires

that some piece of recently acquired

knowledge

turns out, to be just what I need to solve some apparently

unrelated

problem.

I have therefore

developed

the habit of picking up and storing away odd bits of information

for future use. This work has

provided

a mechanism

for organizing

what has turned out to be a fairly large collection

of mathematics.

have also found it very difficult to find clear yet accessible explanations

of technical

mathematics

unless I

already have some familiarity

with the subject. I hope this encyclopedia

will provide jumping-off

points for

people who are interested

in the subjects listed here but who, like me, are not necessarily

experts.

The encyclopedia

has been compiled

over the last 11 years or so, beginning

in my college years and

continuing

during graduate school. The initial document

was written in Microsoj? Word@ on a Mac Plus@

computer,

and had reached about 200 pages by the time I started graduate

school in 1990. When Andrew

Treverrow made his OLQX program

available for the Mac, I began the task of converting

all my documents

to 7&X, resulting in a vast improvement

in readability.

While undertaking

the Word to T&X conversion, I also

began cross-referencing

entries, anticipating

that eventually

I would be able to convert, the entire document

CRC Concise Encyclopedia of Mathematics [part 1 of 4] - E. Weisstein (ed.) (1999) WW.pdf

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