Foundation mathematics for the physical sciences

In so far as content is concerned, the inclusion of some additional introductorymaterial such as powers, logarithms, the sinusoidal and exponential functions, inequalitiesand the handlin

Foundation Mathematics for the Physical Sciences

This tutorial-style textbook develops the basic mathematical tools needed by first- and year undergraduates to solve problems in the physical sciences Students gain hands-on experiencethrough hundreds of worked examples, end-of-section exercises, self-test questions and homeworkproblems

second-Each chapter includes a summary of the main results, definitions and formulae Over 270 workedexamples show how to put the tools into practice Around 170 self-test questions in the footnotesand 300 end-of-section exercises give students an instant check of their understanding More than

450 end-of-chapter problems allow students to put what they have just learned into practice.Hints and outline answers to the odd-numbered problems are given at the end of each chapter

Complete solutions to these problems can be found in the accompanying Student Solution Manual.

Fully worked solutions to all the problems, password-protected for instructors, are available atwww.cambridge.org/foundation

K F R i l e y read mathematics at the University of Cambridge and proceeded to a Ph.D there

in theoretical and experimental nuclear physics He became a Research Associate in elementaryparticle physics at Brookhaven, and then, having taken up a lectureship at the Cavendish Laboratory,Cambridge, continued this research at the Rutherford Laboratory and Stanford; in particular he wasinvolved in the experimental discovery of a number of the early baryonic resonances As well

as having been Senior Tutor at Clare College, where he has taught physics and mathematics forover 40 years, he has served on many committees concerned with the teaching and examining ofthese subjects at all levels of tertiary and undergraduate education He is also one of the authors of

200 Puzzling Physics Problems (Cambridge University Press, 2001).

M P H o b s o n read natural sciences at the University of Cambridge, specialising in theoreticalphysics, and remained at the Cavendish Laboratory to complete a Ph.D in the physics of starformation As a Research Fellow at Trinity Hall, Cambridge, and subsequently an Advanced Fellow

of the Particle Physics and Astronomy Research Council, he developed an interest in cosmology, and

in particular in the study of fluctuations in the cosmic microwave background He was involved inthe first detection of these fluctuations using a ground-based interferometer Currently a UniversityReader at the Cavendish Laboratory, his research interests include both theoretical and observational

aspects of cosmology, and he is the principal author of General Relativity: An Introduction for

Physicists (Cambridge University Press, 2006) He is also a Director of Studies in Natural Sciences

at Trinity Hall and enjoys an active role in the teaching of undergraduate physics and mathematics

Foundation Mathematics for the Physical Sciences

K F RILEY

University of Cambridge

M P HOBSON

University of Cambridge

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,

S˜ao Paulo, Delhi, Dubai, Tokyo, Mexico City

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

Information on this title: www.cambridge.org/9780521192736

C

K Riley and M Hobson 2011

This publication is in copyright Subject to statutory exception

and to the provisions of relevant collective licensing agreements,

no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2011

Printed in the United Kingdom at the University Press, Cambridge

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data

Riley, K F (Kenneth Franklin), 1936–

Foundation mathematics for the physical sciences : a tutorial guide / K F Riley, M P Hobson.

Additional resources for this publication: www.cambridge.org/foundation

Cambridge University Press has no responsibility for the persistence or

accuracy of URLs for external or third-party internet websites referred to

in this publication, and does not guarantee that any content on such

websites is, or will remain, accurate or appropriate.

vii Contents

9.2 Addition, subtraction and multiplication of vectors 332

10.7 The determinant of a matrix 386

ix Contents

13.2 The Dirac δ-function and Heaviside step function 541

D Summation convention 681

a variety of basic methods of proof, though the third edition of MMPE also extended

the range, but not the general level, of the areas to which the methods developed in thebook could be applied Recent feedback suggests that still further adjustments would bebeneficial In so far as content is concerned, the inclusion of some additional introductorymaterial such as powers, logarithms, the sinusoidal and exponential functions, inequalitiesand the handling of physical dimensions, would make the starting level of the book bettermatch that of some of its readers

To incorporate these changes, and others aimed at increasing the user-friendliness of the

text, into the current third edition of MMPE would inevitably produce a text that would be

too ponderous for many students, to say nothing of the problems the physical productionand transportation of such a large volume would entail

For these reasons, we present under the current title, Foundation Mathematics for the

Physical Sciences, an alternative edition of MMPE, one that focuses on the earlier part

of a putative extended third edition It omits those topics that truly are ‘methods’ andconcentrates on the ‘mathematical tools’ that are used in more advanced texts to build upthose methods The emphasis is very much on developing the basic mathematical conceptsthat a physical scientist needs, before he or she can narrow their focus onto methods thatare particularly appropriate to their chosen field

One aspect that has remained constant throughout the three editions of MMPE is the

general style of presentation of a topic – a qualitative introduction, physically basedwherever possible, followed by a more formal presentation or proof, and finished withone or two full-worked examples This format has been well received by reviewers, andthere is no reason to depart from its basic structure

In terms of style, many physical science students appear to be more comfortable withpresentations that contain significant amounts of explanation or comment in words, ratherthan with a series of mathematical equations the last line of which implies ‘job done’ Wehave made changes that move the text in this direction As is explained below, we alsofeel that if some of the advantages of small-group face-to-face teaching could be reflected

in the written text, many students would find it beneficial

In keeping with the intention of presenting a more ‘gentle’ introduction to level mathematics for the physical sciences, we have made use of a modest number ofappendices These contain the more formal mathematical developments associated with

university-xi

the material introduced in the early chapters, and, in particular, with that discussed in theintroductory chapter on arithmetic and geometry They can be studied at the points in themain text where references are made to them, or deferred until a greater mathematicalfluency has been acquired.

As indicated above, one of the advantages of an oral approach to teaching, apparent tosome extent in the lecture situation, and certainly in what are usually known as tutorials,1

is the opportunity to follow the exposition of any particular point with an immediateshort, but probing, question that helps to establish whether or not the student has graspedthat point This facility is not normally available when instruction is through a writtenmedium, without having available at least the equipment necessary to access the contents

of a storage disc

In this book we have tried to go some way towards remedying this by making a standard use of footnotes Some footnotes are used in traditional ways, to add a comment or

non-a pertinent but not essentinon-al piece of non-additionnon-al informnon-ation, to clnon-arify non-a point by restnon-ating

it in slightly different terms, or to make reference to another part of the text or an external

source However, about half of the more than 300 footnotes in this book contain a question

for the reader to answer or an instruction for them to follow; neither will call for a lengthyresponse, but in both cases an understanding of the associated material in the text will berequired This parallels the sort of follow-up a student might have to supply orally in asmall-group tutorial, after a particular aspect of their written work has been discussed.Naturally, students should attempt to respond to footnote questions using the skills andknowledge they have acquired, re-reading the relevant text if necessary, but if they areunsure of their answer, or wish to feel the satisfaction of having their correct responseconfirmed, they can consult the specimen answers given in AppendixF Equally, footnotes

in the form of observations will have served their purpose when students are consistentlyable to say to themselves ‘I didn’t need that comment – I had already spotted and checkedthat particular point’

There are two further features of the present volume that did not appear in MMPE.

The first of these is that a small set of exercises has been included at the end of eachsection The questions posed are straightforward and designed to test whether the studenthas understood the concepts and procedures described in that section The questions arenot intended as ‘drill exercises’, with repeated use of the same procedure on marginallydifferent sets of data; each concept is examined only once or twice within the set Thereare, nevertheless, a total of more than 300 such exercises The more demanding questions,and in particular those requiring the synthesis of several ideas from a chapter, are thosethat appear under the heading of ‘Problems’ at the end of that chapter; there are more than

450 of these

The second new feature is the inclusion at the end of each chapter, just before theproblems begin, of a summary of the main results of that chapter For some areas, thistakes the form of a tabulation of the various case types that may arise in the context ofthe chapter; this should help the student to see the parallels between situations which

in the main text are presented as a consecutive series of often quite lengthy pieces ofmathematical development It should be said that in such a summary it is not possible to

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1 But in Cambridge are called ‘supervisions’!

xiii Preface

state every detailed condition attached to each result, and the reader should consider thesummaries as reminders and formulae providers, rather than as teaching text; that is thejob of the main text and its footnotes Fortunately, in this volume, occasions on whichsubtle conditions have to be imposed upon a result are rare

Finally, we note, for the record, that the format and numbering of the problems

associ-ated with the various chapters have not been changed significantly from those in MMPE,

though naturally only problems related to included topics are retained This means thatabbreviated solutions to all odd-numbered problems can be found in this text Fully worked

solutions to the same problems are available in the companion volume Student Solution

Manual for Foundation Mathematics for the Physical Sciences; most of them, except for

those in the first chapter, can also be found in the Student Solution Manual for MMPE.

Fully worked solutions to all problems, both odd- and even-numbered, are available toaccredited instructors on the password-protected website www.cambridge.org/foundation.Instructors wishing to have access to the website should contact solutions@cambridge.orgfor registration details

1 Arithmetic and geometry

The first two chapters of this book review the basic arithmetic, algebra and geometry ofwhich a working knowledge is presumed in the rest of the text; many students will have

at least some familiarity with much, if not all, of it However, the considerable choicenow available in what is to be studied for secondary-education examination purposesmeans that none of it can be taken for granted The reader may make a preliminaryassessment of which areas need further study or revision by first attempting the

problems at the ends of the chapters Unlike the problems associated with all otherchapters, those for the first two are divided into named sections and each problem dealsalmost exclusively with a single topic

This opening chapter explains the basic definitions and uses associated with some

of the most common mathematical procedures and tools; these are the componentsfrom which the mathematical methods developed in more advanced texts are built So

as to keep the explanations as free from detailed mathematical working as possible –and, in some cases, because results from later chapters have to be anticipated – somejustifications and proofs have been placed in appendices The reader who chooses toomit them on a first reading should return to them after the appropriate material hasbeen studied

The main areas covered in this first chapter are powers and logarithms, inequalities,sinusoidal functions, and trigonometric identities There is also an important section onthe role played by dimensions in the description of physical systems Topics that arewholly or mainly concerned with algebraic methods have been placed in the secondchapter It contains sections on polynomial equations, the related topic of partial

fractions, and some coordinate geometry; the general topic of curve sketching is

deferred until methods for locating maxima and minima have been developed in

in nature The same is true of a discussion of the necessary and sufficient conditions fortwo mathematical statements to be equivalent

The algebraic rules for combining different powers of the same quantity, i.e combining

expressions all of the form a n, but with different exponents in general, are summarised bythe four equations

x  y = y  x for all pairs of objects in the class; loosely speaking, it does not matter

in which order the two objects appear As examples, for real numbers, addition (x + y =

y + x) and multiplication (x × y = y × x) are commutative, but subtraction and division are not; the latter two fail to be commutative because x − y = y − x and x/y = y/x.

The same is true with regard to combining powers: when stands for multiplication

and x and y are a m and a n , then, since a m × a n = a n × a m, the operation of multiplication

is commutative; but, when stands for division and x and y are as before, the operation

is non-commutative because a m ÷ a n = a n ÷ a m It might be added that not all forms of

multiplication are commutative; for example, if x and y are matrices A and B, then, in

general, AB and BA are not equal (see Chapter10)

Associativity

Using the notation of the previous two paragraphs, the operation is said to be associative

if (x  y)  z = x  (y  z) for all triples of objects in the class; here the parentheses

indicate that the operations enclosed by them are the first to be carried out within each

grouping Again, as simple examples, for real numbers, addition [(x + y) + z = x + (y +

z )] and multiplication [(x × y) × z = x × (y × z)] are associative, but subtraction and division are not Subtraction fails to be associative because (x − y) − z = x − (y − z), i.e x − y − z = x − y + z; division fails in a similar way.

3 1.1 Powers

Corresponding results apply to the operations of combining powers In summary, the

multiplication of powers is both commutative and associative; the division of them is

neither.1

Given the rules set out above for combining powers, and the fact that any non-zero

value divided by itself must yield unity, we must have, on setting n = m in (1.3), that

1= a m ÷ a m = a m −m = a0.

Thus, for any a= 0,

The case in which a = 0 is discussed later, when logarithms are considered

Result (1.7) has already taken us away from our original construction of a power, as

the notion of multiplying no factors of a together and obtaining unity is not altogether

intuitive; rather we must consider the process of forming a n as one of multiplying unity n

times by a factor of a.

Another consequence of result (1.3), taken together with deduction (1.7), can now be

found by setting m= 0 in (1.3) Doing this shows that, for a= 0,

1

In words, the reciprocal of a n is a −n The analogy with the construction in the previous

paragraph is that a −n is formed by dividing unity n times by a factor of a.

Rule (1.4) allows us to assign a meaning to an when n is a general rational number,

i.e n can be written as n = p/q where p and q are integers; n itself is not necessarily an

integer In particular, if we take n to have the form n = 1/m, where m is an integer, then

the second equality in (1.4) reads

This shows that the quantity a 1/m when raised to the mth power produces the quantity a.

This, in turn, implies that a 1/m must be interpreted as the mth root of a, otherwise denoted

bym√

a With this identification, the first equality in (1.4) expresses the compatible result

that the mth root of a m is a.

For more general values of p and q, we have that

(a 1/q)p = a p/q = (a p

which states that the pth power of the qth root of a is equal to the qth root of the pth

power of a.

It should be noted that, so long as only real quantities are allowed, a must be confined

to positive values when taking roots in this way; the need for this will be clear from

considering the case of a negative and m an even integer It is possible to find a valid

answer for a 1/m with a negative if m is an odd integer (or, more generally, if the q in

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1 Consider for each of the following operations whether it is commutative and/or associative: (i) a  b = a2+ b2 ,

(ii) a  b = +√a2+ b2, (iii) a  b = a b ; a and b are real positive numbers.

n = p/q is an odd integer), as is shown by the calculation



−1 27

4

= 1 × 81 = 81.

However, both a and p/q could be more general expressions whose signs and values are

not fixed, and great care is needed when using anything other than explicit numericalvalues

Having established a meaning for a m when m is either an integer or a rational fraction,

we would also wish to attach a mathematical meaning to it when m is not confined to either

of these classes, but is any real number Obviously, any general m that is expressed to a

finite number of decimal places could be considered formally, but very inconveniently, as

a rational fraction; however, there are infinitely many numbers that cannot be expressed

in this way,√

2 and π being just two examples.

This is hardly likely to be a problem for any physically based situation, in which therewill always be finite limits on the accuracy with which parameters and measured valuescan be determined But, in order to fill the formal gap, a definition of a general power that

uses the logarithmic function is adopted for all real values of m The general properties of

logarithms are discussed in Section1.2, but we state here one that defines a general power

of a positive quantity a for any real exponent m:

where ln a is the logarithm to the base e of a, itself defined by

and e is the value of the exponential function when its argument is unity As it happens,

e itself is irrational (i.e it cannot be expressed as a rational fraction of the form p/q) and

the first seven of the never-ending sequence of figures in its decimal representation are

but with this choice the logarithm is known as a natural logarithm.

As a numerical example, consider the value of 70.3 This would normally be found

directly as 1.792 78 by making a few keystrokes on a basic scientific calculator But

what happens inside the calculator essentially follows the procedure given above, and

it is instructive to compute the separate steps involved.2 Set algorithms are used for

calculating natural logarithms, ln x, and evaluating exponential functions, e x, for general

values of x First the value of ln 7 is found as 1.945 91 This is then multiplied by 0.3 to yield 0.583 773 and then, as the final step, the value of e 0.583 773 is calculated

as 1.792 78

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2 It is suggested that you do so on your own calculator.

5 1.1 Powers

Because so many natural relationships between physical quantities express one quantity

in terms of the square of another,3the most commonly occurring non-integral power that

a physical scientist has to deal with is the square root For practical calculations, with data

always of limited accuracy, this causes no difficulty, and even the simplest pocket calculator

incorporates a square-root routine But, for theoretical investigations, procedures that are

exact are much to be preferred; so we consider here some methods for dealing with

expressions involving square roots

Written as a power, a square root is of the form a 1/2, but for the present discussion

we will use the notation√

a If a is the square of a rational number, then√

ais itself a

rational number and needs no special attention However, when a is not such a square,√

a

is irrational and new considerations arise It may be that a happens to contain the square

of a rational number as a factor; in such a case, the number may be taken out from under

the square root sign, but that makes no substantial difference to the situation For example:

32

128

343 = 32

87

2

7 = 127

2

7;

we started with√

128/343 which is irrational and, although some simplification has been

effected, the resulting expression, √

2/7, is still irrational It is almost as if rational and

irrational numbers were different species Square roots that are irrational are particular

examples of surds This is a term that covers irrational roots of any order (of the form a 1/n

for any positive integer n), though we are concerned here only with n= 2 and will use

the term ‘surd’ to mean a square root that is irrational

To emphasise the apparent rational–irrational distinction, consider the simple equation

a + b√p = c + d√p,

where a, b, c and d are rational numbers, whilst√

pis irrational and non-zero We can

show that the rational and irrational terms on the two sides can be separately equated, i.e.

a = c and b = d To do this, suppose, on the contrary, that b = d Then the equation can

be rearranged as

√

d − b .

But the RHS4of this equality is the finite ratio of two rational numbers and so is itself

rational; this contradicts the fact that√

pis irrational and so shows it was wrong to suppose

that b = d, i.e b must be equal to d It then follows immediately, from subtracting b√p

from both sides, that, in addition, a is equal to c To summarise:

a + b√p = c + d√p ⇒ a = c and b = d. (1.13)

An important tool for handling fractional expressions that involve surds in their

denom-inators is the process of rationalisation This is a procedure that enables an expression of

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3 As examples, using standard symbols, T = 1mv2, W = RI2, U= 1CV2, u= 10E2 + 1B2/µ0.

4 The need to refer to the ‘left-hand side’ or the ‘right-hand side’ of an equation occurs so frequently throughout this

book, that we almost invariably use the abbreviation LHS or RHS.

the general form

a + b√p

c + d√p , with a, b, c and d rational, to be converted into the (generally) more convenient form

e + f √p; normally there is no gain to be made unless, though √p is irrational, p itself

is rational The basis of the procedure is the algebraic identity (x + y)(x − y) = x2− y2.This identity is used to remove the√

pfrom the denominator, after both numerator and

denominator have been multiplied by c − d√p (note the minus sign) Mathematically,

the procedure is as follows:

a + b√p

c + d√p =

(a + b√p) (c − d√p) (c + d√p) (c − d√p) =

ac − bdp + (bc − ad)√p

This is of the stated form, with the finite5 rational quantities e and f given by e=

(ac − bdp)/(c2− d2p ) and f = (bc − ad)/(c2− d2p)

As an example to illustrate the procedure, consider the following

Example Solve the equation

a + b√28= 4+ 3

√7

for a and b, (i) by obtaining simultaneous equations for a and b and (ii) by using rationalisation.

(i) Cross-multiplying the given equation and using several of the properties of powers listed at thestart of this section, we obtain

(ii) Following the rationalisation procedure, the calculation is

a + b√28=4+ 3

√7

√7

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5 Explain why they cannot be infinite.

7 1.2 Exponential and logarithmic functions

3 Rationalisation can be extended to expressions of the form (a + b√q)/(c + d√p) to

produce the form e + f √p + g√q + h√pq Apply the procedure to

4 Determine whether each of the operations  defined below is commutative and/or

associative:

(a) a  b = the highest common factor (h.c.f.) of positive integers a and b.

(b) For real numbers a, b, c etc., a  b = a + ib, where i2 = −1 Would your

con-clusion be different if a, b, c etc could be complex?

(c) For all non-negative integers including zero

When discussing powers of a real number in the previous section, we made somewhat

premature references to logarithms and the exponential function In this section we

intro-duce these ideas more formally and show how a natural mathematical choice for the ‘base’

of logarithms arises This use of the word ‘base’ is related to the idea of a number base

for counting systems, which in everyday life is taken as 10, and in the internal structure

of computing systems is binary (base 2), though other bases such as octal (base 8) and

hexadecimal (base 16) are frequently used at the interface between such systems and

everyday life.6

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6 The Ultimate Answer to Life, the Universe and Everything is 42 when expressed in decimal Confirm that in other

bases it is 101010 (binary), 52 (octal) and 2A (hexadecimal).

a u

u

01

Figure 1.1 The variation of a u for fixed a > 1 and −∞ < u < +∞.

In the context of logarithms, the word base will be identified with the quantity we have hitherto denoted by a in expressions of the form a m It will become apparent that any posi-

tive value of a will do, but we will find that for mathematical purposes the most convenient choice, and therefore the ‘natural’ one, is for a to have the value denoted by the irrational number e, which is numerically equal to 2.718 281 in ordinary decimal notation.

The usefulness of logarithms for practical calculations depends on the propertiesexpressed in Equations (1.2) and (1.3), namely

These two equations provide a way of reducing multiplication and division calculations

to the processes of addition and subtraction (of the corresponding indices) respectively.Before proceeding to this aspect, however, we first define logarithms and then establishsome of their general properties

1.2.1 Logarithms

We start by noting that, for a fixed positive value of a and a variable u, the quantity

a u is a monotonic function of u, which, for a > 1, increases from zero for u large and negative, passes through unity at u = 0, and becomes arbitrarily large as u becomes large

and positive This is illustrated in Figure1.1 For a < 1, the behaviour of auis the reverse

of this, but we will restrict our attention to cases in which a > 1 and a uis a monotonically

increasing function of u.

Since a u is monotonic and takes all values between 0 and +∞, for any particular

positive value of a variable x, we can find a unique value, α say, such that

9 1.2 Exponential and logarithmic functions

From setting x= 1, and using result (1.7), it follows that

= a nloga x ⇒ loga x n = n log a x. (1.19)

It will be clear that, even for a fixed x, the value of a logarithm will depend upon

the choice of base As a concrete example: log10100= 2, whilst log2100= 6.644 and

loge100= 4.605.

The connection between the logarithms of the same quantity x with respect to two

different bases, a and b, is

This can be proved by repeated use of (1.16) as follows:

blogb x = x = aloga x = (blogb a)loga x = blogb a×loga x

Equating the two indices at the extreme ends of the equality chain yields the stated result

Now setting x = b in (1.20), and recalling that logb b= 1, shows that

logb a= 1

In theoretical work it is not usually necessary to consider bases other than e, but for some

practical applications, in engineering in particular, it is useful to note that

log10x = log10e× loge x ≈ 0.4343 log e x,

loge x = loge10× log10x ≈ 2.3026 log10x.

At this point, a comment on the notation generally used for logarithms employing the

various bases is appropriate Except when dealing with the theory of complex variables,

where they have other specialised meanings, the functions ln x and log x are normally

used to denote loge xand log10x , respectively; Log x is another alternative for log10x

Logarithms employing any base other than e or 10 are normally written in the same way

as we have used hitherto

1.2.2 The exponential function and choice of logarithmic base

Equations (1.14) and (1.15) give clear hints as to how the use of logarithms can be made

to turn multiplication and division into addition and subtraction, but they also indicate that

it does not matter which base a is used, so long as it is positive and not equal to unity We

have already opted to use a value for a that is greater than 1, but this still leaves an infinity

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7 The power n need not be an integer, nor need it be positive, e.g log10(0.3) −2.7 = −2.7 log100.3 = (−2.7) ×

(−0.5229) = 1.412 In brief, 0.3 −2.7= 101.412 = 25.81.

of choices The universal choice of mathematicians is the so-called ‘natural’ choice of e which, as noted previously, has the value 2.718 281

To see why this is a natural choice requires the use of some elementary calculus, asubject not covered until later in this book (Chapter3) However, if the reader is alreadyfamiliar with the notions of derivatives and integrals and the relationships between them,

knows (or accepts) that the derivative of x n is nx n−1, and understands the chain rule forderivatives, then he or she will be able to follow the derivation given in AppendixA Ifnot, the discussion given below can still be followed, though the three major properties

that make e a preferred choice for the logarithmic base will have to be taken on trust until

the relevant parts of Chapter3have been studied

We start by defining the exponential function exp(x) of a real variable x This is simply

the sum of an infinite series of terms each of which contains a non-negative integral power

n of x, namely x n , multiplied by a factor that depends upon n in a specific way A general function of this kind is known as a power series in x; such series are discussed in much

more detail in Section6.5 Written both as a formal sum and as an explicit series, theparticular function we need is

For integers m that are ≥1, the symbol m! stands for the multiple product 1 × 2 ×

3× · · · × m; it is read either as ‘factorial m’ or as ‘m factorial’ For example, factorial 4

is written as 4! and has the value 1× 2 × 3 × 4 = 24 The factorial function clearly hasthe elementary property

The first term in the explicit series for exp(x), which is given as 1, corresponds to the

n = 0 term in the sum; it is therefore x0/0! By (1.7), the numerator has the value 1,

whatever the value of x The value of the denominator, 0!, is also 1, though this will not

be obvious The general definition of m! for m real, but not necessarily a positive integer,8involves the gamma function, (n), which is defined in Problem4.13 There it is shownthat 0!= (1) = 1 Thus, though it appears to involve x and looks as if it might also involve dividing by zero, x0/ 0! has, in fact, the simple value 1 for all x.

It can be shown (see Chapter6) that, despite the fact that whenever x > 1 the quantity

x n grows as n increases, because of the rapidly increasing factors n! in the denominators, the series always ‘converges’ That is, as more and more terms are added, they become

vanishingly small and the total sum becomes arbitrarily close to a definite value (dependent

on x); that value is denoted by exp(x).

It should be noted that definition (1.22) is valid for all real values of x in therange −∞ < x < +∞ At the extremes of the range, exp(x) → 0 as x → −∞, and exp(x) → +∞ as x → +∞ In between, it is a monotonically increasing function that has the value 1 when x= 0, as is obvious from its definition:

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8 It is defined for negative values of m, so long as they are not negative integers A couple of, possibly intriguing,

values are ( − 1 )! =√πand ( − 3 )! = −2√π.

11 1.2 Exponential and logarithmic functions

The value of exp(x) that is of particular relevance to the choice of a base9for logarithms

is exp(1) This is the quantity that is referred to as e and given numerically by the sum of

the infinite series obtained by setting x = 1 in definition (1.22):

If, in our definitions leading to identity (1.16), we set a equal to e (which is positive

and >1), then we have that the natural logarithm ln x of x satisfies the statement

if e y = x then y = ln x and vice versa. (1.26)

In order to study the properties of ln x, we make a slightly different initial definition of a

natural logarithm, namely

if exp(y) = x then y = ln x and vice versa. (1.27)Then, by proving that

we show that the validity (by definition) of (1.27) implies that of (1.26) as well The proof,

which is given in AppendixA, uses only the information implied by the definition (1.27)

to establish (1.28) and hence (1.26) In the course of the proof, the following important

calculus-based properties of the functions exp(x) and ln x are established as by-products:

1

These three simple, but powerful, properties of logarithms to base e are major reasons for

this particular choice of base They are listed here for future use in later chapters of this

main text

1.2.3 The use of logarithms

Following our extensive discussion of the definition of a logarithm and the connection

between the power e x and the series defining the exponential function exp(x), we return to

the practical uses that can be made of logarithms Nowadays, these are mostly of historical

interest, since the invention of small but powerful hand calculators means that nearly all

numerical calculations can be carried out at the touch of a few buttons Nevertheless, it is

important that the practical scientist appreciates the mathematical basis of some of these

automated procedures

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9 Show that if a general base a is chosen, the graph of y = a x can be superimposed on that of y = e xby a simple

scaling of the x-axis, x → x ln a.

We first turn our attention to the multiplication and division of powers, as given inEquations (1.14) and (1.15) We have seen that, given any two positive quantities x and

y , there are two corresponding unique quantities ln x and ln y such that

with x/y equal to e u , where u is the RHS of (1.34).

These two results show how the multiplication or division of two positive numberscan be reduced to an addition or subtraction calculation, with no actual multiplication ordivision required If either or both of the numbers to be multiplied or divided are negative,then they have to be treated as positive so far as the use of logarithms is concerned, and aseparate, but simple, determination of the sign of the answer made

We next turn to the use of logarithms in connection with the analysis of experimentaldata Many experiments in both physics and engineering are aimed at establishing aformula that connects the values of two measured variables, or of verifying a proposedformula and then extracting values for some of its parameters For the graphical analysis

of the experimental data, it is very convenient, whenever possible, to re-cast the expectedrelationship between the measured quantities into a standard ‘straight-line’ form Thisboth helps to give a quick visual impression of whether the plotted data is compatible withthe expected relationship and makes the extraction of parameters a routine procedure If

we denote one, or a particular combination, of the physical variables by y say, and another such single or composite variable by x, then a straight-line plot of y against x takes the form

The slope m is equal to the ratio y/x, where y is the difference in y-values (positive

or negative) corresponding to any arbitrary difference x in x-values (again, positive

or negative); if x and/or y have physical dimensions (see Section 1.3) associated with

13 1.2 Exponential and logarithmic functions

them then so does m.10The intercept the line makes on the y-axis gives the value of c;

its dimensions are the same as those of y.

As a simple example, consider analysing data giving the distance v from a thin lens of

an image formed by the lens when the object is placed a distance u from it The relevant

where f is the focal length of the lens In terms of u and f , v is given by v = uf/(u − f )

and a plot of v against u is not very helpful (the reader may find it instructive to sketch it for

a fixed positive f ).11However, if y = 1/v is plotted against x = 1/u then a straight-line

plot, as given in (1.35), is obtained Its slope m should be−1; this can be used either

as a check on the accuracy of measurement or as a constraint when drawing the best

straight-line fit The intercept made on the y-axis by the fitted line gives a value for f−1,

and hence for f , the focal length of the lens.

There are no logarithms directly involved in this optical example, but if the actual or

expected form of the relationship between the two variables is a power law, i.e one of the

form y = Ax n, then it too can be cast into straight-line form by taking the logarithms of

both sides As previously noted, whilst it is normal in mathematical work to use natural

logarithms, for practical investigations logarithms to base 10 are often employed In either

case the form is the same, but it needs to be remembered which has been used when

recovering the value of A from fitted data In the mathematical form, the power law

relationship becomes

So, a plot of ln y against ln x has a slope of n, whilst the intercept on the ln y axis is ln A,

from which A can be found by exponentiation.

Of course, for practical applications, some means of converting x to its logarithm, and

of recovering x from its logarithm, has to be available Historically, these two procedures

were carried out using tables of logarithms and anti-logarithms, respectively

Since numbers are usually presented in decimal form, the logarithms and anti-logarithms

given in published tables use base 10, i.e they are log x rather than ln x Only the

non-integral part of the logarithm (the mantissa) needs to be provided, as the non-integral part n

can be determined by imagining x written as x = ξ × 10 n where n is chosen to make ξ

lie in the range 1≤ ξ < 10 As two examples:

log 365.25 = log 3.6525 × 102= 2 + log 3.6525 = 2 + 0.5626 = 2.5626,

log 0.003 652 5 = log 3.6525 × 10−3= −3 + log 3.6525 = −3 + 0.5626 = −2.4374.

As noted earlier, even the most basic scientific calculator provides these values at the touch

of a few buttons, and multiplication and division can be equally easily effected; further,

the signs of x, y and the answer are handled automatically.

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10 If x and y are dimensionless and equal scales are employed, then m is equal to the tangent of the angle the line

makes with the x-axis See also Section2.2.1

11 And, to make matters worse, the objective of such an experiment is usually that of finding the actual value of f ,

which is therefore unknown!

As illustrated at the end of Section1.1, logarithms can also be used to evaluate

expres-sions of the form a m where a is positive and m is a general real number, and not necessarily

positive or an integer As given in Equation (1.11),

Thus, for example, 17−0.2is found by first determining that ln 17= 2.833 21 ,

multi-plying this by−0.2 to give −0.566 64 , and then evaluating e −0.566 64 as 0.567 43.

When a = 1, and consequently ln a = 0, (1.37) reads

to give the expected result that unity raised to any power is still unity

An equally expected result is that

a0 = e 0 ln a = e0 = 1, (1.39)

i.e the zeroth power of any positive quantity is unity; this is also true when a is negative,

though we have not proved it here

Further, we expect that for a = 0 and m = 0, a mwill have the value zero This is in

accord with a natural extension of the prescription, discussed in Section 1.1, that, for

integral n, a n is the result of multiplying unity n times by a factor of a.

Less obvious is the value to be assigned to a m when both a = 0 and m = 0 However,

the same prescription indicates that the value should be 1, since not multiplying unity by

anything must leave it unchanged The same conclusion can be reached more

mathemat-ically by starting from (1.37), taking a= m = x to give x x = e x ln x and examining the

behaviour of x ln x as x → 0 By comparing the representation of ln x as the integral of

t−1with the corresponding integral of t −1+β for any positive β, it can be shown that x ln x

tends to zero as x tends to zero, and so x x tends to unity in the same limit To summarise:

0m = 0 for m = 0, but 0 m = 1 if m = 0. (1.40)

E X E R C I S E S 1.2

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1 Arrange the following expressions into distinct sets, each of which consists of members

with a common value:

(a) e ln a , (b) aloga b , (c) a−2, (d) alogb1, (e) (1/a2)−1/2 ,

(f) exp(2 log a), (g) 10 −2 log a , (h) e −2 ln a /a−1, (i) aloga1,

(j) blogb a , (k) ln[exp(a)], (l) 2log2 2/10log 2, (m) logb b.

2 Using only the numeric keypad and the+, −, =, ln, exp, x−1and ‘answer’ keys on a

hand calculator, evaluate the following to 4 s.f.:

(a) (2.25) −2.25 , (b) (0.3) 0.3 , (c) (0.3) −2.25 , (d) (0.3) 1/2.25

15 1.3 Physical dimensions

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In Section1.1we saw how quantities or algebraic expressions that have positive numerical

values can be raised to any finite power So far as arithmetic and algebra are concerned,

that is all that is needed However, when we come to use equations that describe physical

situations, and therefore contain symbols that represent physical quantities, we also need

to take into account the units in which the quantities are measured This additional

consideration has, in general, two distinct consequences

The first and obvious one is that all the quantities involved must be expressed in

the same system of units The almost universal choice for scientific purposes is the SI

system, though some branches of engineering still use other systems and several areas of

physics use derived units that make the values to be manipulated more manageable Other

derived units have less scientific origins.12In the SI system the main base units and their

abbreviations are the metre (m), the kilogram (kg), the second (s), the ampere (A) and the

kelvin (K); they are augmented by the mole for measuring the amount of a substance and

the candela for measuring luminous intensity

In addition, there are many derived units that have specific names of their own, for

example the joule (J) However, if need be, the derived units can always be expressed in

terms of the base units; in the case of the joule, which is the unit of energy, the equivalence

is that 1 J is equal to 1 kg m2s−2 As another example, 1 V (volt), being the electric

potential difference between two points when 1 J of energy is needed to make a current

of 1 A flow for 1 s between the points, can be represented as 1 kg m2s−3A−1

A second, and more fundamental, consequence of the implied presence of appropriate

units in equations relating to physical systems is the need to ensure that the ‘dimensions’

in the various terms in an equation or formula are consistent This is more fundamental

in the sense that it does not depend upon the particular system of units in use – the mean

Sun–Earth distance is always a length, whether it is measured in metres, astronomical

units or feet and inches Similarly, a velocity always consists of a length divided by a

time, whether it is measured in metres per second or miles per hour These properties are

described by saying that the Earth’s distance from the Sun has the dimension of length,

whilst its speed has the dimensions of length divided by time

There is one dimension associated with each of the base units of any system Purely

numerical quantities such as 2,13, π2, etc have no dimensions and affect only the numerical

value of an expression; from the point of view of checking the consistency of dimensions,

they are to be ignored For our discussion we will use only the SI system, though references

to other systems appear in some examples and problems We denote the dimensions of a

physical quantity X by [X], with those of the five main base units being denoted by the

symbols L, M, T , I and as follows:

[length]= L, [mass] = M, [time] = T , [current] = I, [temperature] =

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12 For interest or amusement, the reader may like to identify, and determine SI values for, the following derived

units: (a) barn, (b) denier, (c) hand, (d) hefner, (e) jar, (f) noggin, (g) rod, pole or perch, (h) shake, (i) shed, (j) slug,

(k) tog If help is needed, see G Woan, The Cambridge Handbook of Physics Formulas (Cambridge: Cambridge

University Press, 2000).

The dimensions of derived quantities are formally obtained by expressing the quantity in

terms of its base SI units and then replacing kg by M, etc Purely numerical quantities, such

as those mentioned above, are formally treated as if their dimensions were L0M0T0I0 0;this means they can be ignored without appearing to multiply the dimensions of the rest

of an expression by zero

More substantial examples are provided by the two quantities specifically mentioned

in an earlier paragraph, energy and voltage They have dimensions as follows:

[E] = M L2T−2 and [V ] = M L2T−3I−1.

It should be emphasised that the dimensions of a physical quantity do not depend on themagnitude of that quantity, nor upon the units in which it is measured Thus, for example,

energy has the same dimensions, whether it represents 1 erg or 7.93 MJ.

We now turn to the role of dimensions in the construction of derived quantities and use

as a simple example the expression for energy We have already given the dimensions of

energy as [E] = M L2T−2, and the dimensions of any formula that is supposed to be one for energy must have this same form It is almost immediately obvious that the expression for the kinetic energy T of a body of mass m moving with a speed v, namely T = 1

2mv2,satisfies this requirement;13the formal calculation is as follows:

[T ]= [1

2mv2]= [1

2][m][v2]= [m][v]2 = M(L T−1)2 = M L2T−2.

It will be noticed that the dimensions of a physical quantity obey the same algebraic rules

as the symbols that represent that quantity Thus, in the above illustration, the fact that the

velocity appears squared in the expression for the kinetic energy means that the L T−1

giving the dimensions of a velocity also appears squared in the dimensions of the energy,

i.e as L2T−2.The dimensions of any one physical quantity can contain only integer powers (posi-tive or negative) of the base dimensions, although fractional powers of basic or deriveddimensions may appear in formulae; when they do, they again follow the same rules as

‘ordinary’ powers For example, the period of oscillation τ of a simple pendulum of length

is given by τ = 2π(/g) 1/2 , where g is the acceleration due to gravity The dimensional

equation reads as follows:

Examination of the dimensions of quantities and combinations of quantities appearing

in quoted or derived formulae or equations can be used both positively and negatively.The constructive use takes the form of dimensional analysis, in which all of the physicalvariables the investigator thinks might influence a particular phenomenon are formed intocombinations that are dimensionless, i.e for each combination the net index of each ofthe base units is zero For the pendulum just considered, such a combination would be

(gτ2)/, as the reader should check If only one such combination can be formed, then it

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13 Check that the formula 12kx2for the energy stored in a stretched spring of spring constant k has the correct

dimensional form.

17 1.3 Physical dimensions

must be equal to a constant, but one whose value has to be determined in some other way;

for the pendulum it is 4π2

If more than one dimensionless combination can be formed, the best that can be said is

that some function of these combinations (but not of the individual variables with non-zero

dimensions that make up these combinations) is equal to a constant In some complicated

areas of physics and engineering, particularly those involving fluids in motion, this type

of analysis is an essential research tool The following worked example gives some idea

of the basic method involved – but hardly produces a previously unknown result!

Example One system of units, proposed by Max Planck and known as natural units, is based on five physical

constants of nature which are defined to have unit value when expressed in those units The five

constants are: c, the speed of light in a vacuum; G, the gravitational constant; k, the Boltzmann

We start by determining the dimensions of each of the five constants; after that we will aim

to construct a combination of them that has the dimensions of a temperature, i.e has just the

would be involved, and hence so would some power of I , the base dimension of current But, as it

happens, none of the four other ‘natural’ physical constants includes current in its dimensions, and,

can only contain k as an overall factor of 1/k Thus we assume a combination of the form

The -dimension has already been arranged to be correct, but equating the powers of L, M and T

on the two sides gives the three simultaneous equations

These have solution α= 5

We now turn to the more negative and humdrum topic of checking possible equationsand formulae for internal dimensional consistency Suppose that you have derived, or beenpresented with, an equation purporting to describe a certain physical situation Beforerisking examination credit-loss by submitting it for marking, or your academic reputation

by publishing, it is well worth checking the equation’s dimensional plausibility; findingconsistency does not guarantee that the equation is correct, but finding inconsistency

guarantees that it is wrong, and could save a lot of embarrassment Dimensional aspects

that should be checked are

r Both sides of the equation must have exactly the same dimensions.

r Any two items that are added or subtracted must have the same dimensions as eachother

r The arguments of any mathematical functions that can be written as a power series withmore than one term must be dimensionless Examples include the exponential function,the sinusoidal functions, and polynomials

You should also check that the equation has the expected behaviour for extreme values

of the variables and parameters, within the range of validity claimed, as well as for anyparticularly simple set of values for which the solution can be found by other means Weillustrate some of these checks by means of the following example

Example It is claimed that the speed v of waves of wavelength λ travelling on the surface of a liquid, under

the influence of both gravity and surface tension, is given by

ρλ ,

where ρ and σ are the density and surface tension, respectively, of the liquid The

units of joules per square metre; a and b are dimensionless constants Is the claimed formula

plausible?

We first note that, as we are not given any experimental data, and the values of a and b are unknown

in any case, the numerical values provided for g and σ are of no help when the possible validity of

19 1.3 Physical dimensions

the formula is being examined However, we can use the data to establish the dimensions of g and

RHS of the formula consists of two combinations of variables that are added, we must next check

that they each have the same overall dimensions Recalling that a and b are dimensionless, we

have



bσ ρλ



M L−3L = L2T−2.

As we can see, they do have the same dimensions and therefore can be added together Furthermore,

The problems at the end of this chapter further illustrate the uses of, and the

con-straints imposed by, the notion of dimensions, and at the same time introduce equations

and formulae from some of the more intriguing areas of quantum and cosmological

physics

E X E R C I S E S 1.3

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1 Demonstrate that each of the formulae given below is dimensionally acceptable.

(a) Bernoulli’s equation for the speed v and pressure p at height z in an incompressible

ideal fluid of density ρ is

1

2ρv2+ p + ρgz = constant.

(b) The speed v of a wave of wavelength λ travelling through a thin plate of thickness

t (in the direction of travel) is

Here E is the Young modulus, ρ the density, and σ the (dimensionless) Poisson

ratio for the material of the plate The Young modulus is defined as the ratio ofthe longitudinal stress (force per unit area) to the longitudinal strain (fractionalincrease in length) in a thin wire made of the material

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15 In fact, the formula is a valid one for surface waves whose wavelength is much less than the depth of the liquid;

a has the value 1/2π and b = 2π.

(c) The probability density pr(c) of particle speeds c in a classical gas at temperature

2 Below are the names and formulae for three physical constants, together with a set of

quoted values Using the values given in AppendixE, check the formulae and quotedvalues for numerical and dimensional consistency, and so determine which, if any,have been wrongly quoted (beyond rounding errors)

Fine structure constant α = µ0ce2

[Note that the force F on a conductor of length carrying a current i perpendicular to

a magnetic field of flux density B is F = Bi The unit of magnetic flux density is the

tesla (with symbol T).]

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Earlier in this chapter we considered powers of a single quantity or variable, such as a n , e n

or x n We now extend our discussion to functions that are powers of the sum or difference

of two terms, e.g (x − α) m Later in this book we will find numerous occasions on which

we wish to write such a product of repeated factors as a polynomial in x or, more generally,

as a sum of terms each of which is a power of x multiplied by a power of α, as opposed

to a power of their sum or difference

To make the discussion general and the result applicable to a wide variety of situations,

we will consider the general expansion of f (x, y) = (x + y) n , where x and y may stand for constants, variables or functions but, for the time being, n is a positive integer It may

not be obvious what form the general expansion takes, but some idea can be obtained by

carrying out the multiplication explicitly for small values of n Thus we obtain successively (x + y)1 = x + y,

(x + y)2 = (x + y)(x + y) = x2+ 2xy + y2,

(x + y)3 = (x + y)(x2+ 2xy + y2)= x3+ 3x2y + 3xy2+ y3,

(x + y)4 = (x + y)(x3+ 3x2y + 3xy2+ y3)= x4+ 4x3y + 6x2y2+ 4xy3+ y4.

This does not establish a general formula, but the regularity of the terms in the

expan-sions and the suggestion of a pattern in the coefficients indicate that a general formula for

the nth power will have n + 1 terms, that the powers of x and y in every term will add up

21 1.4 The binomial expansion

to n, and that the coefficients of the first and last terms will be unity, whilst those of the

second and penultimate terms will be n.16,17

In fact, the general expression, the binomial expansion for power n, is given by

wheren C k is called the binomial coefficient When it is expressed in terms of the factorial

functions introduced in Section1.2it takes the form n!/[k!(n − k)!] with 0! = 1 Clearly,

simply to make such a statement does not constitute proof of its validity, but, as we will see

in Section1.4.2, Equation (1.41) can be proved using a method called induction Before

turning to that proof, we investigate some of the elementary properties of the binomial

We note that, for any given n, the largest coefficient in the binomial expansion is the

middle one (k = n/2) if n is even; the middle two coefficients [k = 1

16 Write down your prediction for the expansion of (x + y)5 and then check it by direct calculation.

17 One examination paper question read: ‘Expand (x + y)5’ The submitted response was, (x + y)5= (x + y)5 =

(x + y)5= (x + y)5= (x + y) 5= (x + y) 5=

1.4.2 Proof of the binomial expansion

We are now in a position to prove the binomial expansion (1.41) In doing so, we introduce the reader to a procedure applicable to certain types of problems and known as the method

of induction The method is discussed much more fully in Section2.4.1

We start by assuming that (1.41) is true for some positive integer n = N, and then

proceed to show that, given the assumption, it follows that (1.41) also holds for n= N + 1:

where in the first line we have used the initial assumption and in the third line have moved

the second summation index by unity, by writing k + 1 = j We now separate off the

first term of the first sum,N C0x N+1, and write it asN+1C0x N+1; we can do this since, asnoted in (i) following (1.42),n C0= 1 for every n Similarly, the last term of the second

summation can be replaced byN+1C N+1y N+1.The remaining terms of each of the two summations are now written together, with the

summation index denoted by k in both terms.18Thus

it can be proved to be true for n = N + 1 But it holds trivially for n = 1, and therefore for n = 2 also By the same token it is valid for n = 3, 4, , and hence is established for all positive integers n.

1.4.3 Negative and non-integral values of n

Up till now we have restricted n in the binomial expansion to be a positive integer Negative

values can be accommodated, but only at the cost of an infinite series of terms rather than

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18 Note that the first summation, having lost its first term, now has an index that runs from 1 to N , and that the

second summation, having lost its last term, also has an index that now runs from 1 to N

23 1.4 The binomial expansion

the finite one represented by (1.41) For reasons that are intuitively sensible and will bediscussed in more detail in Chapter6, very often we require an expansion in which, at leastultimately, successive terms in the infinite series decrease in magnitude For this reason,

if|x| > |y| and we need to consider (x + y) −m , where m itself is a positive integer, then

Since the ratio|y/x| is less than unity, terms containing higher powers of it will be small

in magnitude, whilst raising the unit term to any power will not affect its magnitude If

|y| > |x| the roles of the two must be interchanged.

We can now state, but will not explicitly prove, the form of the binomial expansion

appropriate to negative values of n (n equal to −m):

is well defined since m + k − 1 ≥ k.

Thus we have a definition of binomial coefficients for negative integer values of n in terms of those for positive n The connection between the two may not be obvious, but

they are both formed in the same way in terms of recurrence relations Whatever the sign

of n, or its integral or non-integral nature, the series of coefficients n C k can be generated

by starting withn C0 = 1 and using the recurrence relation

Finally, to summarise, Equation (1.47) generates the appropriate coefficients for all

values of n, positive or negative, integer or non-integer, with the obvious exception of the case in which x = −y and n is negative.

1.4.4 Relationship with the exponential function

Before we leave the binomial expansion, we use it to establish an alternative representation

of the exponential function The representation takes the form of a limit and is

The formal definition of a limit is not discussed until Chapter 6, but for our present

purposes an intuitive notion of one will suffice We start by expanding the nth power

of 1+ (a/n) using the binomial theorem, and remembering that n C k can be written as

We now take the limit of both sides as n → ∞; n−1→ 0 and all the factors containing it

on the RHS tend to unity, leaving

and thus establishing (1.48)

The most practical example of this result is the way that compound interest on capital

A, borrowed or lent, leads to ‘exponential growth’ of that capital If the annual rate of

interest is a and the interest is paid only at the end of the year, the capital then stands at

A(1+ a) However, if it is paid monthly, then the corresponding figure is A[1 + (a/12)]12,

and if it is paid daily the capital stands at A[1 + (a/365)]365 at the end of the year Asthe interval between payments becomes shorter the end-of-year capital amount becomeslarger However, it does not increase indefinitely and ‘continuous interest payment’ results

in a capital of Ae a at the end of the year, and Ae na at the end of n years.19

E X E R C I S E S 1.4

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1 Evaluate the binomial coefficients (a)−3C1, (b)−5C7, (c)−1C k

2 Evaluate the binomial coefficients (a)1/2 C3, (b)−1/2 C3, (c)5/3 C3

3 Demonstrate explicitly the validity of (1.44) for n= 4 and k = 0, 1, 2 Using only

the general simple properties of binomial coefficients, and the simplest of arithmetic,deduce the validity of (1.44) for k= 3

19 Show that capital that attracts an annual interest rate of 5% will exceed twice its initial value more than 400 days

earlier if interest is paid continuously rather than yearly (in arrears).