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MATHEMATICAL

METHODS FOR

PHYSICISTS

SEVENTH EDITION

MATHEMATICAL

METHODS FOR

PHYSICISTS

A Comprehensive Guide

SEVENTH EDITION

George B. Arfken

Miami University

Oxford, OH

Hans J. Weber

University of Virginia

Charlottesville, VA

Frank E. Harris

University of Utah, Salt Lake City, UT

and

University of Florida, Gainesville, FL

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v

CONTENTS

• 1. MATHEMATICAL PRELIMINARIES PREFACE XI
  • 1.1. Infinite Series..................................................................................................................
  • 1.2. Series of Functions
  • 1.3. Binomial Theorem
  • 1.4. Mathematical Induction...............................................................................................
  • 1.5. Operations of Series Expansions of Functions
  • 1.6. Some Important Series
  • 1.7. Vectors
  • 1.8. Complex Numbers and Functions
  • 1.9. Derivatives and Extrema
  • 1.10. Evaluation of Integrals
  • 1.11. Dirac Delta Functions
    • Additional Readings
• 2. D ETERMINANTS AND MATRICES
  • 2.1 Determinants
  • 2.2 Matrices
    • Additional Readings
• 3. V ECTOR A NALYSIS
  • 3.1 Review of Basics Properties........................................................................................
  • 3.2 Vector in 3 ‐ D Spaces
  • 3.3 Coordinate Transformations
  • 3.4 Rotations in  vi
  • 3.5 Differential Vector Operators
  • 3.6 Differential Vector Operators: Further Properties
  • 3.7 Vector Integrations
  • 3.8 Integral Theorems
  • 3.9 Potential Theory
  • 3.10 Curvilinear Coordinates
    • Additional Readings
• 4. TENSOR AND D IFFERENTIAL F ORMS
  • 4.1 Tensor Analysis
  • 4.2 Pseudotensors, Dual Tensors
  • 4.3 Tensor in General Coordinates
  • 4.4 Jacobians....................................................................................................................
  • 4.5 Differential Forms
  • 4.6 Differentiating Forms
  • 4.7 Integrating Forms
    • Additional Readings
• 5. V ECTOR S PACES
  • 5.1 Vector in Function Spaces
  • 5.2 Gram ‐ Schmidt Orthogonalization
  • 5.3 Operators
  • 5.4 Self ‐ Adjoint Operators
  • 5.5 Unitary Operators
  • 5.6 Transformations of Operators....................................................................................
  • 5.7 Invariants
  • 5.8 Summary – Vector Space Notations
    • Additional Readings
• 6. EIGENVALUE P ROBLEMS
  • 6.1 Eigenvalue Equations
  • 6.2 Matrix Eigenvalue Problems
  • 6.3 Hermitian Eigenvalue Problems
  • 6.4 Hermitian Matrix Diagonalization
  • 6.5 Normal Matrices
    • Additional Readings
• 7. ORDINARY D IFFERENTIAL EQUATIONS
  • 7.1 Introduction
  • 7.2 First ‐ Order Equations
  • 7.3 ODEs with Constant Coefficients
  • 7.4 Second ‐ Order Linear ODEs
  • 7.5 Series Solutions ‐ Frobenius‘ Method
  • 7.6 Other Solutions
  • 7.7 Inhomogeneous Linear ODEs vii
  • 7.8 Nonlinear Differential Equations
    • Additional Readings
• 8. S TURM – LIOUVILLE THEORY
  • 8.1 Introduction
  • 8.2 Hermitian Operators
  • 8.3 ODE Eigenvalue Problems
  • 8.4 Variation Methods
  • 8.5 Summary, Eigenvalue Problems
    • Additional Readings
• 9. P ARTIAL D IFFERENTIAL EQUATIONS
  • 9.1 Introduction
  • 9.2 First ‐ Order Equations
  • 9.3 Second – Order Equations
  • 9.4 Separation of Variables
  • 9.5 Laplace and Poisson Equations
  • 9.6 Wave Equations
  • 9.7 Heat – Flow, or Diffution PDE.....................................................................................
  • 9.8 Summary
    • Additional Readings
• 10. G REEN’ F UNCTIONS
  • 10.1 One – Dimensional Problems
  • 10.2 Problems in Two and Three Dimensions
    • Additional Readings
• 11. COMPLEX V ARIABLE THEORY
  • 11.1 Complex Variables and Functions
  • 11.2 Cauchy – Riemann Conditions
  • 11.3 Cauchy’s Integral Theorem
  • 11.4 Cauchy’s Integral Formula
  • 11.5 Laurent Expansion
  • 11.6 Singularities
  • 11.7 Calculus of Residues
  • 11.8 Evaluation of Definite Integrals..................................................................................
  • 11.9 Evaluation of Sums.....................................................................................................
  • 11.10 Miscellaneous Topics
    • Additional Readings
• 12. F URTHER TOPICS IN A NALYSIS
  • 12.1 Orthogonal Polynomials.............................................................................................
  • 12.2 Bernoulli Numbers
  • 12.3 Euler – Maclaurin Integration Formula
  • 12.4 Dirichlet Series
  • 12.5 Infinite Products viii
  • 12.6 Asymptotic Series
  • 12.7 Method of Steepest Descents.....................................................................................
  • 12.8 Dispertion Relations
    • Additional Readings
• 13. G AMMA F UNCTION
  • 13.1 Definitions, Properties
  • 13.2 Digamma and Polygamma Functions
  • 13.3 The Beta Function
  • 13.4 Stirling’s Series
  • 13.5 Riemann Zeta Function
  • 13.6 Other Ralated Function
    • Additional Readings
• 14. BESSEL F UNCTIONS
  • 14.1 Bessel Functions of the First kind, J ν (x)
  • 14.2 Orthogonality
  • 14.3 Neumann Functions, Bessel Functions of the Second kind
  • 14.4 Hankel Functions
  • 14.5 Modified Bessel Functions, I ν (x) and K ν (x)
  • 14.6 Asymptotic Expansions
  • 14.7 Spherical Bessel Functions
    • Additional Readings
• 15. LEGENDRE F UNCTIONS
  • 15.1 Legendre Polynomials
  • 15.2 Orthogonality
  • 15.3 Physical Interpretation of Generating Function
  • 15.4 Associated Legendre Equation
  • 15.5 Spherical Harmonics...................................................................................................
  • 15.6 Legendre Functions of the Second Kind
    • Additional Readings
• 16. A NGULAR MOMENTUM
  • 16.1 Angular Momentum Operators..................................................................................
  • 16.2 Angular Momentum Coupling
  • 16.3 Spherical Tensors
  • 16.4 Vector Spherical Harmonics
    • Additional Readings
• 17. G ROUP THEORY
  • 17.1 Introduction to Group Theory
  • 17.2 Representation of Groups
  • 17.3 Symmetry and Physics
  • 17.4 Discrete Groups
  • 17.5 Direct Products ix
  • 17.6 Simmetric Group
  • 17.7 Continous Groups
  • 17.8 Lorentz Group
  • 17.9 Lorentz Covariance of Maxwell’s Equantions
  • 17.10 Space Groups
    • Additional Readings
• 18. MORE S PECIAL F UNCTIONS
  • 18.1 Hermite Functions
  • 18.2 Applications of Hermite Functions
  • 18.3 Laguerre Functions.....................................................................................................
  • 18.4 Chebyshev Polynomials
  • 18.5 Hypergeometric Functions
  • 18.6 Confluent Hypergeometric Functions
  • 18.7 Dilogarithm
  • 18.8 Elliptic Integrals..........................................................................................................
    • Additional Readings
• 19. F OURIER S ERIES
  • 19.1 General Properties
  • 19.2 Application of Fourier Series
  • 19.3 Gibbs Phenomenon
    • Additional Readings
• 20. INTEGRAL TRANSFORMS
  • 20.1 Introduction
  • 20.2 Fourier Transforms
  • 20.3 Properties of Fourier Transforms
  • 20.4 Fourier Convolution Theorem.....................................................................................
  • 20.5 Signal – Proccesing Applications
  • 20.6 Discrete Fourier Transforms
  • 20.7 Laplace Transforms
  • 20.8 Properties of Laplace Transforms.............................................................................
  • 20.9 Laplace Convolution Transforms
  • 20.10 Inverse Laplace Transforms
    • Additional Readings
• 21. INTEGRAL EQUATIONS
  • 21.1 Introduction
  • 21.2 Some Special Methods
  • 21.3 Neumann Series
  • 21.4 Hilbert – Schmidt Theory
    • Additional Readings
• 22. C ALCULUS OF V ARIATIONS x
  • 22.1 Euler Equation
  • 22.2 More General Variations
  • 22.3 Constrained Minima/Maxima
  • 22.4 Variation with Constraints
    • Additional Readings
• 23. P ROBABILITY AND STATISTICS
  • 23.1 Probability: Definitions, Simple Properties
  • 23.2 Random Variables
  • 23.3 Binomial Distribution
  • 23.4 Poisson Distribution
  • 23.5 Gauss’ Nomal Distribution
  • 23.6 Transformation of Random Variables
  • 23.7 Statistics
    • Additional Readings
• INDEX

PREFACE

This, the seventh edition of Mathematical Methods for Physicists , maintains the tradition set by the six previous editions and continues to have as its objective the presentation of all the mathematical methods that aspiring scientists and engineers are likely to encounter as students and beginning researchers. While the organization of this edition differs in some respects from that of its predecessors, the presentation style remains the same: Proofs are sketched for almost all the mathematical relations introduced in the book, and they are accompanied by examples that illustrate how the mathematics applies to real-world physics problems. Large numbers of exercises provide opportunities for the student to develop skill in the use of the mathematical concepts and also show a wide variety of contexts in which the mathematics is of practical use in physics. As in the previous editions, the mathematical proofs are not what a mathematician would consider rigorous, but they nevertheless convey the essence of the ideas involved, and also provide some understanding of the conditions and limitations associated with the rela- tionships under study. No attempt has been made to maximize generality or minimize the conditions necessary to establish the mathematical formulas, but in general the reader is warned of limitations that are likely to be relevant to use of the mathematics in physics contexts.

TO THE STUDENT

The mathematics presented in this book is of no use if it cannot be applied with some skill, and the development of that skill cannot be acquired passively, e.g., by simply reading the text and understanding what is written, or even by listening attentively to presentations by your instructor. Your passive understanding needs to be supplemented by experience in using the concepts, in deciding how to convert expressions into useful forms, and in developing strategies for solving problems. A considerable body of background knowledge

xi

xii Preface

needs to be built up so as to have relevant mathematical tools at hand and to gain experi- ence in their use. This can only happen through the solving of problems, and it is for this reason that the text includes nearly 1400 exercises, many with answers (but not methods of solution). If you are using this book for self-study, or if your instructor does not assign a considerable number of problems, you would be well advised to work on the exercises until you are able to solve a reasonable fraction of them. This book can help you to learn about mathematical methods that are important in physics, as well as serve as a reference throughout and beyond your time as a student. It has been updated to make it relevant for many years to come.

WHAT’S NEW

This seventh edition is a substantial and detailed revision of its predecessor; every word of the text has been examined and its appropriacy and that of its placement has been consid- ered. The main features of the revision are: (1) An improved order of topics so as to reduce the need to use concepts before they have been presented and discussed. (2) An introduc- tory chapter containing material that well-prepared students might be presumed to know and which will be relied on (without much comment) in later chapters, thereby reducing redundancy in the text; this organizational feature also permits students with weaker back- grounds to get themselves ready for the rest of the book. (3) A strengthened presentation of topics whose importance and relevance has increased in recent years; in this category are the chapters on vector spaces, Green’s functions, and angular momentum, and the inclu- sion of the dilogarithm among the special functions treated. (4) More detailed discussion of complex integration to enable the development of increased skill in using this extremely important tool. (5) Improvement in the correlation of exercises with the exposition in the text, and the addition of 271 new exercises where they were deemed needed. (6) Addition of a few steps to derivations that students found difficult to follow. We do not subscribe to the precept that “advanced” means “compressed” or “difficult.” Wherever the need has been recognized, material has been rewritten to enhance clarity and ease of understanding. In order to accommodate new and expanded features, it was necessary to remove or reduce in emphasis some topics with significant constituencies. For the most part, the material thereby deleted remains available to instructors and their students by virtue of its inclusion in the on-line supplementary material for this text. On-line only are chapters on Mathieu functions, on nonlinear methods and chaos, and a new chapter on periodic sys- tems. These are complete and newly revised chapters, with examples and exercises, and are fully ready for use by students and their instuctors. Because there seems to be a sig- nificant population of instructors who wish to use material on infinite series in much the same organizational pattern as in the sixth edition, that material (largely the same as in the print edition, but not all in one place) has been collected into an on-line infinite series chapter that provides this material in a single unit. The on-line material can be accessed at www.elsevierdirect.com.

Preface xiii

PATHWAYS THROUGH THE MATERIAL

This book contains more material than an instructor can expect to cover, even in a two-semester course. The material not used for instruction remains available for reference purposes or when needed for specific projects. For use with less fully prepared students, a typical semester course might use Chapters 1 to 3, maybe part of Chapter 4, certainly Chapters 5 to 7, and at least part of Chapter 11. A standard graduate one-semester course might have the material in Chapters 1 to 3 as prerequisite, would cover at least part of Chapter 4, all of Chapters 5 through 9, Chapter 11, and as much of Chapters 12 through 16 and/or 18 as time permits. A full-year course at the graduate level might supplement the foregoing with several additional chapters, almost certainly including Chapter 20 (and Chapter 19 if not already familiar to the students), with the actual choice dependent on the institution’s overall graduate curriculum. Once Chapters 1 to 3, 5 to 9, and 11 have been covered or their contents are known to the students, most selections from the remain- ing chapters should be reasonably accessible to students. It would be wise, however, to include Chapters 15 and 16 if Chapter 17 is selected.

ACKNOWLEDGMENTS

This seventh edition has benefited from the advice and help of many people; valuable advice was provided both by anonymous reviewers and from interaction with students at the University of Utah. At Elsevier, we received substantial assistance from our Acqui- sitions Editor Patricia Osborn and from Editorial Project Manager Kathryn Morrissey; production was overseen skillfully by Publishing Services Manager Jeff Freeland. FEH gratefully acknowledges the support and encouragement of his friend and partner Sharon Carlson. Without her, he might not have had the energy and sense of purpose needed to help bring this project to a timely fruition.

CHAPTER 1

MATHEMATICAL

PRELIMINARIES

This introductory chapter surveys a number of mathematical techniques that are needed throughout the book. Some of the topics (e.g., complex variables) are treated in more detail in later chapters, and the short survey of special functions in this chapter is supplemented by extensive later discussion of those of particular importance in physics (e.g., Bessel func- tions). A later chapter on miscellaneous mathematical topics deals with material requiring more background than is assumed at this point. The reader may note that the Additional Readings at the end of this chapter include a number of general references on mathemati- cal methods, some of which are more advanced or comprehensive than the material to be found in this book.

1.1 INFINITE SERIES

Perhaps the most widely used technique in the physicist’s toolbox is the use of infinite series (i.e., sums consisting formally of an infinite number of terms) to represent functions, to bring them to forms facilitating further analysis, or even as a prelude to numerical eval- uation. The acquisition of skill in creating and manipulating series expansions is therefore an absolutely essential part of the training of one who seeks competence in the mathemat- ical methods of physics, and it is therefore the first topic in this text. An important part of this skill set is the ability to recognize the functions represented by commonly encountered expansions, and it is also of importance to understand issues related to the convergence of infinite series.

Mathematical Methods for Physicists. © 2013 Elsevier Inc. All rights reserved.

2 Chapter 1 Mathematical Preliminaries

Fundamental Concepts

The usual way of assigning a meaning to the sum of an infinite number of terms is by introducing the notion of partial sums. If we have an infinite sequence of terms u 1 , u 2 , u 3 , u 4 , u 5 ,... , we define the i th partial sum as

si =

∑^ i

n = 1

un. (1.1)

This is a finite summation and offers no difficulties. If the partial sums si converge to a finite limit as i → ∞, lim i →∞

si = S , (1.2)

the infinite series

∑∞

n = 1 un^ is said to be^ convergent^ and to have the value^ S. Note that we define the infinite series as equal to S and that a necessary condition for convergence to a limit is that lim n →∞ un = 0. This condition, however, is not sufficient to guarantee convergence. Sometimes it is convenient to apply the condition in Eq. (1.2) in a form called the Cauchy criterion , namely that for each ε > 0 there is a fixed number N such that | s (^) j − si | < ε for all i and j greater than N. This means that the partial sums must cluster together as we move far out in the sequence. Some series diverge , meaning that the sequence of partial sums approaches ±∞; others may have partial sums that oscillate between two values, as for example, ∑^ ∞

n = 1

un = 1 − 1 + 1 − 1 + 1 − · · · − (− 1 ) n^ + · · ·.

This series does not converge to a limit, and can be called oscillatory. Often the term divergent is extended to include oscillatory series as well. It is important to be able to determine whether, or under what conditions, a series we would like to use is convergent.

Example 1.1.1 THE GEOMETRIC SERIES

The geometric series, starting with u 0 = 1 and with a ratio of successive terms r = un + 1 / un , has the form 1 + r + r^2 + r^3 + · · · + rn −^1 + · · ·. Its n th partial sum sn (that of the first n terms) is^1

sn =

1 − rn 1 − r

. (1.3)

Restricting attention to | r | < 1 , so that for large n , rn^ approaches zero, and sn possesses the limit

lim n →∞ sn =

1 − r

, (1.4)

(^1) Multiply and divide sn = ∑ n − 1 m = 0 r m (^) by 1 − r.

1.1 Infinite Series 3

showing that for | r | < 1 , the geometric series converges. It clearly diverges (or is oscilla- tory) for | r | ≥ 1 , as the individual terms do not then approach zero at large n. 

Example 1.1.2 THE HARMONIC SERIES

As a second and more involved example, we consider the harmonic series

∑^ ∞

n = 1

n

= 1 +

+

+

+ · · · +

n

+ · · ·. (1.5)

The terms approach zero for large n , i.e., lim n →∞ 1 / n = 0 , but this is not sufficient to guarantee convergence. If we group the terms (without changing their order) as

1 +

+

(

+

)

+

(

+

+

+

)

+

(

+ · · · +

)

+ · · · ,

each pair of parentheses encloses p terms of the form

1 p + 1

+

p + 2

+ · · · +

p + p

>

p 2 p

=

.

Forming partial sums by adding the parenthetical groups one by one, we obtain

s 1 = 1 , s 2 =

, s 3 >

, s 4 >

,... , sn >

n + 1 2

,

and we are forced to the conclusion that the harmonic series diverges. Although the harmonic series diverges, its partial sums have relevance among other places in number theory, where Hn =

∑ n m = 1 m − (^1) are sometimes referred to as harmonic numbers. 

We now turn to a more detailed study of the convergence and divergence of series, considering here series of positive terms. Series with terms of both signs are treated later.

Comparison Test

If term by term a series of terms un satisfies 0 ≤ un ≤ an , where the an form a convergent series, then the series

∑

n un^ is also convergent. Letting^ si^ and^ s^ j^ be partial sums of the u series, with j > i , the difference s (^) j − si is

∑ (^) j n = i + 1 un^ , and this is smaller than the corresponding quantity for the a series, thereby proving convergence. A similar argument shows that if term by term a series of terms v n satisfies 0 ≤ bn ≤ v n , where the bn form a divergent series, then

∑

n v n^ is also divergent. For the convergent series an we already have the geometric series, whereas the harmonic series will serve as the divergent comparison series bn. As other series are identified as either convergent or divergent, they may also be used as the known series for comparison tests.

4 Chapter 1 Mathematical Preliminaries

Example 1.1.3 A DIVERGENT SERIES

Test

∑∞

n = 1 n − p , p = 0. 999 , for convergence. Since n − 0. (^999) > n − (^1) and bn = n − (^1) forms the divergent harmonic series, the comparison test shows that

∑

n n − 0. (^999) is divergent. Generalizing,

∑

n n − p (^) is seen to be divergent for all p ≤ 1. 

Cauchy Root Test

If ( an )^1 / n^ ≤ r < 1 for all sufficiently large n , with r independent of n , then

∑

n an^ is convergent. If ( an )^1 / n^ ≥ 1 for all sufficiently large n , then

∑

n an^ is divergent. The language of this test emphasizes an important point: The convergence or divergence of a series depends entirely on what happens for large n. Relative to convergence, it is the behavior in the large- n limit that matters. The first part of this test is verified easily by raising ( an )^1 / n^ to the n th power. We get an ≤ rn^ < 1. Since rn^ is just the n th term in a convergent geometric series,

∑

n an^ is convergent by the comparison test. Conversely, if ( an )^1 / n^ ≥ 1 , then an ≥ 1 and the series must diverge. This root test is particularly useful in establishing the properties of power series (Section 1.2).

D’Alembert (or Cauchy) Ratio Test

If an + 1 / an ≤ r < 1 for all sufficiently large n and r is independent of n , then

∑

n an^ is convergent. If an + 1 / an ≥ 1 for all sufficiently large n , then

∑

n an^ is divergent. This test is established by direct comparison with the geometric series ( 1 + r + r^2 + · · · ). In the second part, an + 1 ≥ an and divergence should be reasonably obvious. Although not quite as sensitive as the Cauchy root test, this D’Alembert ratio test is one of the easiest to apply and is widely used. An alternate statement of the ratio test is in the form of a limit: If

lim n →∞

an + 1 an







< 1 , convergence,

1 , divergence, = 1 , indeterminate.

(1.6)

Because of this final indeterminate possibility, the ratio test is likely to fail at crucial points, and more delicate, sensitive tests then become necessary. The alert reader may wonder how this indeterminacy arose. Actually it was concealed in the first statement, an + 1 / an ≤ r <

1. We might encounter an + 1 / an < 1 for all finite n but be unable to choose an r < 1 and independent of n such that an + 1 / an ≤ r for all sufficiently large n. An example is provided by the harmonic series, for which an + 1 an

=

n n + 1

< 1.

Since lim n →∞

an + 1 an

= 1 ,

no fixed ratio r < 1 exists and the test fails.

1.1 Infinite Series 5

Example 1.1.4 D’ALEMBERT RATIO TEST

Test

∑

n n /^2 n (^) for convergence. Applying the ratio test,

an + 1 an

=

( n + 1 )/ 2 n +^1 n / 2 n^

=

n + 1 n

.

Since an + 1 an

≤

for n ≥ 2 ,

we have convergence. 

Cauchy (or Maclaurin) Integral Test

This is another sort of comparison test, in which we compare a series with an integral. Geometrically, we compare the area of a series of unit-width rectangles with the area under a curve. ∑Let^ f^ ( x )^ be a continuous,^ monotonic decreasing function^ in which^ f^ ( n )^ =^ an^. Then n an^ converges if^

∫ ∞

1 f^ ( x ) d x^ is finite and diverges if the integral is infinite. The^ i^ th partial sum is

si =

∑^ i

n = 1

an =

∑^ i

n = 1

f ( n ).

But, because f ( x ) is monotonic decreasing, see Fig. 1.1(a),

si ≥

∫^ i +^1

1

f ( x ) d x.

On the other hand, as shown in Fig. 1.1(b),

si − a 1 ≤

∫^ i

1

f ( x ) d x.

Taking the limit as i → ∞, we have ∫^ ∞

1

f ( x ) d x ≤

∑^ ∞

n = 1

an ≤

∫^ ∞

1

f ( x ) d x + a 1. (1.7)

Hence the infinite series converges or diverges as the corresponding integral converges or diverges. This integral test is particularly useful in setting upper and lower bounds on the remain- der of a series after some number of initial terms have been summed. That is, ∑^ ∞

n = 1

an =

∑^ N

n = 1

an +

∑^ ∞

n = N + 1

an , (1.8)

6 Chapter 1 Mathematical Preliminaries

1 2 3 4

f ( x ) f ( x )

(a)

x

f (1) = a 1 f^ (1)^ =^ a 1 f (2) = a 2

1 2 3 4 (b)

x i =

FIGURE 1.1 (a) Comparison of integral and sum-blocks leading. (b) Comparison of integral and sum-blocks lagging.

and ∫^ ∞

N + 1

f ( x ) d x ≤

∑^ ∞

n = N + 1

an ≤

∫^ ∞

N + 1

f ( x ) d x + aN + 1. (1.9)

To free the integral test from the quite restrictive requirement that the interpolating func- tion f ( x ) be positive and monotonic, we shall show that for any function f ( x ) with a continuous derivative, the infinite series is exactly represented as a sum of two integrals:

∑^ N^2

n = N 1 + 1

f ( n ) =

∫^ N^2

N 1

f ( x ) d x +

∫^ N^2

N 1

( x − [ x ]) f ′( x ) d x. (1.10)

Here [ x ] is the integral part of x , i.e., the largest integer ≤ x , so x − [ x ] varies sawtoothlike between 0 and 1. Equation (1.10) is useful because if both integrals in Eq. (1.10) converge, the infinite series also converges, while if one integral converges and the other does not, the infinite series diverges. If both integrals diverge, the test fails unless it can be shown whether the divergences of the integrals cancel against each other. We need now to establish Eq. (1.10). We manipulate the contributions to the second integral as follows:

1. Using integration by parts, we observe that ∫^ N^2

N 1

x f ′( x ) d x = N 2 f ( N 2 ) − N 1 f ( N 1 ) −

∫^ N^2

N 1

f ( x ) d x.

2. We evaluate ∫^ N^2

N 1

[ x ] f ′( x ) dx =

N ∑ 2 − 1

n = N 1

n

n ∫+ 1

n

f ′( x ) d x =

N ∑ 2 − 1

n = N 1

n

[

f ( n + 1 ) − f ( n )

]

= −

∑^ N^2

n = N 1 + 1

f ( n ) − N 1 f ( N 1 ) + N 2 f ( N 2 ).

Subtracting the second of these equations from the first, we arrive at Eq. (1.10).

1.1 Infinite Series 7

An alternative to Eq. (1.10) in which the second integral has its sawtooth shifted to be symmetrical about zero (and therefore perhaps smaller) can be derived by methods similar to those used above. The resulting formula is

∑^ N^2

n = N 1 + 1

f ( n ) =

∫^ N^2

N 1

f ( x ) d x +

∫^ N^2

N 1

( x − [ x ] − 12 ) f ′( x ) d x

[

f ( N 2 ) − f ( N 1 )

]

.

(1.11)

Because they do not use a monotonicity requirement, Eqs. (1.10) and (1.11) can be applied to alternating series, and even those with irregular sign sequences.

Example 1.1.5 RIEMANN ZETA FUNCTION

The Riemann zeta function is defined by

ζ ( p ) =

∑^ ∞

n = 1

n −^ p , (1.12)

providing the series converges. We may take f ( x ) = x −^ p^ , and then ∫^ ∞

1

x −^ p^ d x =

x −^ p +^1 − p + 1

∣

∣

∣

∣

∞

x = 1

, p 6 = 1 ,

= ln x

∣

∣

∣

∞ x = 1

, p = 1.

The integral and therefore the series are divergent for p ≤ 1 , and convergent for p > 1. Hence Eq. (1.12) should carry the condition p > 1. This, incidentally, is an independent proof that the harmonic series ( p = 1 ) diverges logarithmically. The sum of the first million terms

∑ 1 , 000 , 000

n = 1 n

− (^1) is only 14 .392 726 · · ·. 

While the harmonic series diverges, the combination

γ = lim n →∞

( (^) n ∑

m = 1

m −^1 − ln n

)

(1.13)

converges, approaching a limit known as the Euler-Mascheroni constant.

Example 1.1.6 A SLOWLY DIVERGING SERIES

Consider now the series

S =

∑^ ∞

n = 2

n ln n

.

8 Chapter 1 Mathematical Preliminaries

We form the integral

∫^ ∞

2

x ln x

d x =

∫^ ∞

x = 2

d ln x ln x

= ln ln x

∣

∣

∣

∞ x = 2

,

which diverges, indicating that S is divergent. Note that the lower limit of the integral is in fact unimportant so long as it does not introduce any spurious singularities, as it is the large- x behavior that determines the convergence. Because n ln n > n , the divergence is slower than that of the harmonic series. But because ln n increases more slowly than n ε^ , where ε can have an arbitrarily small positive value, we have divergence even though the series

∑

n n

−( 1 +ε) (^) converges. 

More Sensitive Tests

Several tests more sensitive than those already examined are consequences of a theorem by Kummer. Kummer’s theorem, which deals with two series of finite positive terms, un and an , states:

1. The series

∑

n un^ converges if

lim n →∞

(

an

un un + 1

− an + 1

)

≥ C > 0 , (1.14)

where C is a constant. This statement is equivalent to a simple comparison test if the series

∑

n a − 1 n converges, and imparts new information only if that sum diverges. The more weakly

∑

n a − 1 n diverges, the more powerful the Kummer test will be.

2. If

∑

n a − 1 n diverges and

lim n →∞

(

an

un un + 1

− an + 1

)

≤ 0 , (1.15)

then

∑

n un^ diverges.

The proof of this powerful test is remarkably simple. Part 2 follows immediately from the comparison test. To prove Part 1, write cases of Eq. (1.14) for n = N + 1 through any larger n , in the following form:

u (^) N + 1 ≤ ( aN u (^) N − aN + 1 u (^) N + 1 )/ C ,

u (^) N + 2 ≤ ( aN + 1 u (^) N + 1 − aN + 2 u (^) N + 2 )/ C ,

... ≤........................ ,

un ≤ ( an − 1 un − 1 − an un )/ C.

1.1 Infinite Series 9

Adding, we get ∑^ n

i = N + 1

ui ≤

aN u (^) N C

−

an un C

(1.16)

<

aN u (^) N C

. (1.17)

This shows that the tail of the series

∑

n un^ is bounded, and that series is therefore proved convergent when Eq. (1.14) is satisfied for all sufficiently large n. Gauss’ test is an application of Kummer’s theorem to series un > 0 when the ratios of successive un approach unity and the tests previously discussed yield indeterminate results. If for large n un un + 1

= 1 +

h n

+

B ( n ) n^2

, (1.18)

where B ( n ) is bounded for n sufficiently large, then the Gauss test states that

∑

n un^ con- verges for h > 1 and diverges for h ≤ 1 : There is no indeterminate case here. The Gauss test is extremely sensitive, and will work for all troublesome series the physi- cist is likely to encounter. To confirm it using Kummer’s theorem, we take an = n ln n. The series

∑

n a

− 1 n is weakly divergent, as already established in Example 1.1.6. Taking the limit on the left side of Eq. (1.14), we have

lim n →∞

[

n ln n

(

1 +

h n

+

B ( n ) n^2

)

− ( n + 1 ) ln( n + 1 )

]

= lim n →∞

[

( n + 1 ) ln n + ( h − 1 ) ln n +

B ( n ) ln n n

− ( n + 1 ) ln( n + 1 )

]

= lim n →∞

[

−( n + 1 ) ln

(

n + 1 n

)

• ( h − 1 ) ln n

]

. (1.19)

For h < 1 , both terms of Eq. (1.19) are negative, thereby signaling a divergent case of Kummer’s theorem; for h > 1 , the second term of Eq. (1.19) dominates the first and is pos- itive, indicating convergence. At h = 1 , the second term vanishes, and the first is inherently negative, thereby indicating divergence.

Example 1.1.7 LEGENDRE SERIES

The series solution for the Legendre equation (encountered in Chapter 7) has successive terms whose ratio under certain conditions is a 2 j + 2 a 2 j

=

2 j ( 2 j + 1 ) − λ ( 2 j + 1 )( 2 j + 2 )

.

To place this in the form now being used, we define u (^) j = a 2 j and write u (^) j u (^) j + 1

=

( 2 j + 1 )( 2 j + 2 ) 2 j ( 2 j + 1 ) − λ

.

10 Chapter 1 Mathematical Preliminaries

In the limit of large j , the constant λ becomes negligible (in the language of the Gauss test, it contributes to an extent B ( j )/ j^2 , where B ( j ) is bounded). We therefore have u (^) j u (^) j + 1

→

2 j + 2 2 j

+

B ( j ) j^2

= 1 +

j

+

B ( j ) j^2

. (1.20)

The Gauss test tells us that this series is divergent. 

Exercises

1.1.1 (a) Prove that if lim n →∞ n pun = A < ∞, p > 1 , the series

∑∞

n = 1 un^ converges. (b) Prove that if lim n →∞ nun = A > 0 , the series diverges. (The test fails for A = 0 .) These two tests, known as limit tests , are often convenient for establishing the convergence of a series. They may be treated as comparison tests, comparing with ∑

n

n − q^ , 1 ≤ q < p.

1.1.2 If lim n →∞ b ann = K , a constant with 0 < K < ∞, show that (^6) n bn converges or diverges with 6 an.

Hint. If 6 an converges, rescale bn to b n ′ =

bn 2 K

. If (^6) n an diverges, rescale to b ′′ n =

2 bn K

.

1.1.3 (a) Show that the series

∑∞

n = 2 1 n (ln n )^2 converges.

(b) By direct addition

∑ 100 , 000

n = 2 [ n (ln^ n )

(^2) ]− (^1) = 2. 02288. Use Eq. (1.9) to make a five- significant-figure estimate of the sum of this series. 1.1.4 Gauss’ test is often given in the form of a test of the ratio

un un + 1

=

n^2 + a 1 n + a 0 n^2 + b 1 n + b 0

.

For what values of the parameters a 1 and b 1 is there convergence? divergence? ANS. Convergent for a 1 − b 1 > 1 , divergent for a 1 − b 1 ≤ 1. 1.1.5 Test for convergence

(a)

∑^ ∞

n = 2

(ln n )−^1 (d)

∑^ ∞

n = 1

[ n ( n + 1 )]−^1 /^2

(b)

∑^ ∞

n = 1

n! 10 n^

(e)

∑^ ∞

n = 0

2 n + 1

(c)

∑^ ∞

n = 1

2 n ( 2 n + 1 )

1.1 Infinite Series 11

1.1.6 Test for convergence

(a)

∑^ ∞

n = 1

n ( n + 1 )

(d)

∑^ ∞

n = 1

ln

(

1 +

n

)

(b)

∑^ ∞

n = 2

n ln n

(e)

∑^ ∞

n = 1

n · n^1 / n

(c)

∑^ ∞

n = 1

n 2 n

1.1.7 For what values of p and q will

∑∞

n = 2

1 n p^ (ln n ) q^ converge?

ANS. Convergent for

{

p > 1 , all q , p = 1 , q > 1 ,

divergent for

{

p < 1 , all q , p = 1 , q ≤ 1.

1.1.8 Given

∑ 1 , 000

n = 1 n

− (^1) = 7 .485 470... set upper and lower bounds on the Euler-Mascheroni constant. ANS. 0. 5767 < γ < 0. 5778.

1.1.9 (From Olbers’ paradox .) Assume a static universe in which the stars are uniformly distributed. Divide all space into shells of constant thickness; the stars in any one shell by themselves subtend a solid angle of ω 0. Allowing for the blocking out of distant stars by nearer stars , show that the total net solid angle subtended by all stars, shells extending to infinity, is exactly 4 π. [Therefore the night sky should be ablaze with light. For more details, see E. Harrison, Darkness at Night: A Riddle of the Universe. Cambridge, MA: Harvard University Press (1987).]

1.1.10 Test for convergence

∑^ ∞

n = 1

[

1 · 3 · 5 · · · ( 2 n − 1 ) 2 · 4 · 6 · · · ( 2 n )

] 2

=

+

+

+ · · ·.

Alternating Series

In previous subsections we limited ourselves to series of positive terms. Now, in contrast, we consider infinite series in which the signs alternate. The partial cancellation due to alternating signs makes convergence more rapid and much easier to identify. We shall prove the Leibniz criterion, a general condition for the convergence of an alternating series. For series with more irregular sign changes, the integral test of Eq. (1.10) is often helpful. The Leibniz criterion applies to series of the form

∑∞

n = 1 (−^1 )

n + (^1) an with an > 0 , and states that if an is monotonically decreasing (for sufficiently large n ) and lim n →∞ an = 0 , then the series converges. To prove this theorem, note that the remainder R 2 n of the series beyond s 2 n , the partial sum after 2 n terms, can be written in two alternate ways: R 2 n = ( a 2 n + 1 − a 2 n + 2 ) + ( a 2 n + 3 − a 2 n + 4 ) + · · · = a 2 n + 1 − ( a 2 n + 2 − a 2 n + 3 ) − ( a 2 n + 4 − a 2 n + 5 ) − · · ·.

12 Chapter 1 Mathematical Preliminaries

Since the an are decreasing, the first of these equations implies R 2 n > 0 , while the second implies R 2 n < a 2 n + 1 , so 0 < R 2 n < a 2 n + 1.

Thus, R 2 n is positive but bounded, and the bound can be made arbitrarily small by taking larger values of n. This demonstration also shows that the error from truncating an alter- nating series after a 2 n results in an error that is negative (the omitted terms were shown to combine to a positive result) and bounded in magnitude by a 2 n + 1. An argument similar to that made above for the remainder after an odd number of terms, R 2 n + 1 , would show that the error from truncation after a 2 n + 1 is positive and bounded by a 2 n + 2. Thus, it is generally true that the error in truncating an alternating series with monotonically decreasing terms is of the same sign as the last term kept and smaller than the first term dropped. The Leibniz criterion depends for its applicability on the presence of strict sign alternation. Less regular sign changes present more challenging problems for convergence determination.

Example 1.1.8 SERIES WITH IRREGULAR SIGN CHANGES

For 0 < x < 2 π , the series

S =

∑^ ∞

n = 1

cos( nx ) n

= − ln

(

2 sin

x 2

)

(1.21)

converges, having coefficients that change sign often, but not so that the Leibniz criterion applies easily. To verify the convergence, we apply the integral test of Eq. (1.10), inserting the explicit form for the derivative of cos( nx )/ n (with respect to n ) in the second integral:

S =

∫^ ∞

1

cos( nx ) n

dn +

∫^ ∞

1

(

n − [ n ]

) [

−

x n

sin( nx ) −

cos( nx ) n^2

]

dn. (1.22)

Using integration by parts, the first integral in Eq. (1.22) is rearranged to ∫^ ∞

1

cos( nx ) n

dn =

[

sin( nx ) nx

]∞

1

+

x

∫^ ∞

1

sin( nx ) n^2

dn ,

and this integral converges because ∣ ∣ ∣ ∣ ∣ ∣ ∫^ ∞

1

sin( nx ) n^2

dn

∣ ∣ ∣ ∣ ∣ ∣

<

∫^ ∞

1

dn n^2

= 1.

Looking now at the second integral in Eq. (1.22), we note that its term cos( nx )/ n^2 also leads to a convergent integral, so we need only to examine the convergence of ∫^ ∞

1

(

n − [ n ]

) (^) sin( nx ) n

dn.