A Course of Modern Analysis 5th Edition by E. T. Whittaker, ISBN-13: 978-1316518939

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A Course of Modern Analysis 5th Edition by E. T. Whittaker, ISBN-13: 978-1316518939

[PDF eBook eTextbook]

• Publisher: ‎ Cambridge University Press; 5th edition (November 4, 2021)
• Language: ‎ English
• ‎718 pages
• ISBN-10: ‎ 1316518930
• ISBN-13: ‎ 978-1316518939

New edition of a true classic in mathematics. Preserves the flavour of the original, updating where appropriate and improving usability.

This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902. Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge. The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis. This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate. All the formulas have been checked and many corrections made. A complete bibliographical search has been conducted to present the references in modern form for ease of use. A new foreword by Professor S.J. Patterson sketches the circumstances of the book’s genesis and explains the reasons for its longevity. A welcome addition to any mathematician’s bookshelf, this will allow a whole new generation to experience the beauty contained in this text.

Table of Contents:

Half-title

Frontispiece

Title page

Copyright information

Contents

Foreword

Preface to the Fifth Edition

Preface to the Fourth Edition

Preface to the Third Edition

Preface to the Second Edition

Preface to the First Edition

Introduction

Part I The Process of Analysis

1 Complex Numbers

1.1 Rational numbers

1.2 Dedekind’s theory of irrational numbers

1.3 Complex numbers

1.4 The modulus of a complex number

1.5 The Argand diagram

1.6 Miscellaneous examples

2 The Theory of Convergence

2.1 The definition of the limit of a sequence

2.11 Definition of the phrase ‘of the order of’

2.2 The limit of an increasing sequence

2.21 Limit-points and the Bolzano–Weierstrass theorem

2.22 Cauchy’s theorem on the necessary and sufficient condition for the existence of a limit [120,

2.3 Convergence of an infinite series

2.31 Dirichlet’s test for convergence

2.32 Absolute and conditional convergence

2.33 The geometric series, and the series [sum[sup(infty)sub(n=1)]frac[sup(1)sub(n[sup(x)])]]

2.34 The comparison theorem

2.35 Cauchy’s test for absolute convergence

2.36 D’Alembert’s ratio test for absolute convergence

2.37 A general theorem on series for which [lim[sub(n[rightarrow]infty)]|frac[u[sub(n)+1]][u[sub(n)]

2.38 Convergence of the hypergeometric series

2.4 Effect of changing the order of the terms in a series

2.41 The fundamental property of absolutely convergent series

2.5 Double series

2.51 Methods of summing a double series

2.52 Absolutely convergent double series

2.53 Cauchy’s theorem on the multiplication of absolutely convergent series

2.6 Power series

2.61 Convergence of series derived from a power series

2.7 Infinite products

2.71 Some examples of infinite products

2.8 Infinite determinants

2.81 Convergence of an infinite determinant

2.82 The rearrangement theorem for convergent infinite determinants

2.9 Miscellaneous examples

3 Continuous Functions and Uniform Convergence

3.1 The dependence of one complex number on another

3.2 Continuity of functions of real variables

3.21 Simple curves. Continua

3.22 Continuous functions of complex variables

3.3 Series of variable terms. Uniformity of convergence

3.31 On the condition for uniformity of convergence

3.32 Connexion of discontinuity with non-uniform convergence

3.33 The distinction between absolute and uniform convergence

3.34 A condition, due to Weierstrass, for uniform convergence

3.35 Hardy’s tests for uniform convergence

3.4 Discussion of a particular double series

3.5 The concept of uniformity

3.6 The modified Heine–Borel theorem

3.61 Uniformity of continuity

3.62 A real function, of a real variable, continuous in a closed interval, attains its upper bound

3.63 A real function, of a real variable, continuous in a closed interval, attains all values betwee

3.64 The fluctuation of a function of a real variable

3.7 Uniformity of convergence of power series

3.71 Abel’s theorem

3.72 Abel’s theorem on multiplication of convergent series

3.73 Power series which vanish identically

3.8 Miscellaneous examples

4 The Theory of Riemann Integration

4.1 The concept of integration

4.11 Upper and lower integrals

4.12 Riemann’s condition of integrability

4.13 A general theorem on integration

4.14 Mean-value theorems

4.2 Differentiation of integrals containing a parameter

4.3 Double integrals and repeated integrals

4.4 Infinite integrals

4.41 Infinite integrals of continuous functions. Conditions for convergence

4.42 Uniformity of convergence of an infinite integral

4.43 Tests for the convergence of an infinite integral

4.44 Theorems concerning uniformly convergent infinite integrals

4.5 Improper integrals. Principal values

4.51 The inversion of the order of integration of a certain repeated integral

4.6 Complex integration

4.61 The fundamental theorem of complex integration

4.62 An upper limit to the value of a complex integral

4.7 Integration of infinite series

4.8 Miscellaneous examples

5 The Fundamental Properties of Analytic Functions; Taylor’s, Laurent’s and Liouville’s Theore

5.1 Property of the elementary functions

5.11 Occasional failure of the property

5.12 Cauchy’s definition of an analytic function of a complex variable

5.13 An application of the modified Heine–Borel theorem

5.2 Cauchy’s theorem on the integral of a function round a contour

5.21 The value of an analytic function at a point, expressed as an integral taken round a contour en

5.22 The derivatives of an analytic function f(z)

5.23 Cauchy’s inequality for f[sup((n))](a)

5.3 Analytic functions represented by uniformly convergent series

5.31 Analytic functions represented by integrals

5.32 Analytic functions represented by infinite integrals

5.4 Taylor’s theorem

5.41 Forms of the remainder in Taylor’s series

5.5 The process of continuation

5.51 The identity of two functions

5.6 Laurent’s theorem

5.61 The nature of the singularities of one-valued functions

5.62 The ‘point at infinity’

5.63 Liouvillle’s theorem

5.64 Functions with no essential singularities

5.7 Many-valued functions

5.8 Miscellaneous examples

6 The Theory of Residues; Application to the Evaluation of Definite Integrals

6.1 Residues

6.2 The evaluation of definite integrals

6.21 The evaluation of the integrals of certain periodic functions taken between the limits 0 and 2

6.22 The evaluation of certain types of integrals taken between the limits -[infty] and +[infty]

6.23 Principal values of integrals

6.24 Evaluation of integrals of the form [int[sup(infty)sub(0)]x[sup(a-1)]Q(x)dx]

6.3 Cauchy’s integral

6.31 The number of roots of an equation contained within a contour

6.4 Connexion between the zeros of a function and the zeros of its derivative

6.5 Miscellaneous examples

7 The Expansion of Functions in Infinite Series

7.1 A formula due to Darboux

7.2 The Bernoullian numbers and the Bernoullian polynomials

7.21 The Euler–Maclaurin expansion

7.3 Bürmann’s theorem

7.31 Teixeira’s extended form of Bürmann’s theorem

7.32 Lagrange’s theorem

7.4 The expansion of a class of functions in rational fractions

7.5 The expansion of a class of functions as infinite products

7.6 The factor theorem of Weierstrass

7.7 The expansion of a class of periodic functions in a series of cotangents

7.8 Borel’s theorem

7.81 Borel’s integral and analytic continuation

7.82 Expansions in series of inverse factorials

7.9 Miscellaneous examples

8 Asymptotic Expansions and Summable Series

8.1 Simple example of an asymptotic expansion

8.2 Definition of an asymptotic expansion

8.21 Another example of an asymptotic expansion

8.3 Multiplication of asymptotic expansions

8.31 Integration of asymptotic expansions

8.32 Uniqueness of an asymptotic expansion

8.4 Methods of summing series

8.41 Borel’s method of summation [85, p. 97–115]

8.42 Euler’s method of summation [85, 201]

8.43 Cesàro’s method of summation [141]

8.44 The method of summation of Riesz [559]

8.5 Hardy’s convergence theorem

8.6 Miscellaneous examples

9 Fourier Series and Trigonometric Series

9.1 Definition of Fourier series

9.11 Nature of the region within which a trigonometrical series converges

9.12 Values of the coefficients in terms of the sum of a trigonometrical series

9.2 On Dirichlet’s conditions and Fourier’s theorem

9.21 The representation of a function by Fourier series for ranges other than (-[pi],[pi])

9.22 The cosine series and the sine series

9.3 The nature of the coefficients in a Fourier series

9.31 Differentiation of Fourier series

9.32 Determination of points of discontinuity

9.4 Fejér’s theorem

9.41 The Riemann–Lebesgue lemmas

9.42 The proof of Fourier’s theorem

9.43 The Dirichlet–Bonnet proof of Fourier’s theorem

9.44 The uniformity of the convergence of Fourier series

9.5 The Hurwitz–Liapounoff theorem concerning Fourier constants

9.6 Riemann’s theory of trigonometrical series

9.61 Riemann’s associated function

9.62 Properties of Riemann’s associated function; Riemann’s first lemma

9.63 Riemann’s theorem on trigonometrical series

9.7 Fourier’s representation of a function by an integral

9.8 Miscellaneous examples

10 Linear Differential Equations

10.1 Linear differential equations

10.2 Solution of a differential equation valid in the vicinity of an ordinary point

10.21 Uniqueness of the solution

10.3 Points which are regular for a differential equation

10.31 Convergence of the expansion of §10.3

10.32 Derivation of a second solution in the case when the difference of the exponents is an integer

10.4 Solutions valid for large values of |z|

10.5 Irregular singularities and confluence

10.6 The differential equations of mathematical physics

10.7 Linear differential equations with three singularities

10.71 Transformations of Riemann’s P-equation

10.72 The connexion of Riemann’s P-equation with the hypergeometric equation

10.8 Linear differential equations with two singularities

10.9 Miscellaneous examples

11 Integral Equations

11.1 Definition of an integral equation

11.11 An algebraical lemma

11.2 Fredholm’s equation and its tentative solution

11.21 Investigation of Fredholm’s solution

11.22 Volterra’s reciprocal functions

11.23 Homogeneous integral equations

11.3 Integral equations of the first and second kinds

11.31 Volterra’s equation

11.4 The Liouville–Neumann method of successive substitutions

11.5 Symmetric nuclei

11.51 Schmidt’s theorem that, if the nucleus is symmetric, the equation D(lambda) = 0 has at least

11.6 Orthogonal functions

11.61 The connexion of orthogonal functions with homogeneous integral equations

11.7 The development of a symmetric nucleus

11.71 The solution of Fredholm’s equation by a series

11.8 Solution of Abel’s integral equation

11.81 Schlömilch’s integral equation

11.9 Miscellaneous examples

Part II The Transcendental Functions

12 The Gamma-Function

12.1 Definitions of the Gamma-function. The Weierstrassian product

12.11 Euler’s formula for the Gamma-function

12.12 The difference equation satisfied by the Gamma-function

12.13 The evaluation of a general class of infinite products

12.14 Connexion between the Gamma-function and the circular functions

12.15 The multiplication-theorem of Gauss and Legendre

12.16 Expansion for the logarithmic derivates of the Gamma-function

12.2 Euler’s expression of [Gamma(z)] as an infinite integral

12.21 Extension of the infinite integral to the case in which the argument of the Gamma-function is

12.22 Hankel’s expression of [Gamma(z)] as a contour integral

12.3 Gauss’ expression for the logarithmic derivate of the Gamma-function as an infinite integral

12.31 Binet’s first expression for log [Gamma(z)] in terms of an infinite integral

12.32 Binet’s second expression for log [Gamma(z)] in terms of an infinite integral

12.33 The asymptotic expansion of the logarithms of the Gamma-function

12.4 The Eulerian integral of the first kind

12.41 Expression of the Eulerian integral of the first kind in terms of the Gamma-function

12.42 Evaluation of trigonometrical integrals in terms of the Gamma-function

12.43 Pochhammer’s extension of the Eulerian integral of the first kind

12.5 Dirichlet’s integral

12.6 Miscellaneous examples

13 The Zeta-Function of Riemann

13.1 Definition of the zeta-function

13.11 The generalised zeta-function

13.12 The expression of [zeta(s,a)] as an infinite integral

13.13 The expression of [zeta(s,a)] as a contour integral

13.14 Values of [zeta(s,a)] for special values of s

13.15 The formula of Hurwitz for [zeta(s,a)] when [sigma] < 0

13.2 Hermite’s formula for [zeta(s,a)]

13.21 Deductions from Hermite’s formula

13.3 Euler’s product for [zeta(s)]

13.31 Riemann’s hypothesis concerning the zeros of [zeta(s)]

13.4 Riemann’s integral for [zeta(s)]

13.5 Inequalities satisfied by [zeta(s,a)] when [sigma] > 0

13.51 Inequalities satisfied by [zeta(s,a)] when [zigma] [leq] 0

13.6 The asymptotic expansion of log [Gamma(z + a)]

13.7 Miscellaneous examples

14 The Hypergeometric Function

14.1 The hypergeometric series

14.11 The value of F(a, b; c; 1) when Re(c-a-b) > 0

14.2 The differential equation satisfied by F(a, b; c; z)

14.3 Solutions of Riemann’s P-equation by hypergeometric functions

14.4 Relations between particular solutions of the hypergeometric equation

14.5 Barnes’ contour integrals for the hypergeometric function

14.51 The continuation of the hypergeometric series

14.52 Barnes’ lemma

14.53 The connexion between hypergeometric functions of z and of 1-z

14.6 Solution of Riemann’s equation by a contour integral

14.61 Determination of an integral which represents P[sup(alpha)]

14.7 Relations between contiguous hypergeometric functions

14.8 Miscellaneous examples

15 Legendre Functions

15.1 Definition of Legendre polynomials

15.11 Rodrigues’ formula for the Legendre polynomials [561]

15.12 Schläfli’s integral for P[sub(n)](z) [579]

15.13 Legendre’s differential equation

15.14 The integral properties of the Legendre polynomials

15.2 Legendre functions

15.21 The recurrence formulae

15.22 Murphy’s expression of P[sub(n)](z) as a hypergeometric function

15.23 Laplace’s integrals for P[sub(n)](z)

15.3 Legendre functions of the second kind

15.31 Expansion of Q[sub(n)](z) as a power series

15.32 The recurrence formulae for Q[sub(n)](z)

15.33 The Laplacian integral for Legendre functions of the second kind

15.34 Neumann’s formula for Q[sub(n)](z), when n is an integer

15.4 Heine’s development of (t-z)[sup(-1)] as a series of Legendre polynomials in z

15.41 Neumann’s expansion of an arbitrary function in a series of Legendre polynomials

15.5 Ferrers’ associated Legendre functions P[sup(m)sub(n)](z) and Q[sup(m)sub(n)](z)

15.51 The integral properties of the associated Legendre functions

15.6 Hobson’s definition of the associated Legendre functions

15.61 Expression of P[sup(m)sub(n)](z) as an integral of Laplace’s type

15.7 The addition-theorem for the Legendre polynomials

15.71 The addition theorem for the Legendre functions

15.8 The function C[sup(v)sub(n)](z)

15.9 Miscellaneous examples

16 The Confluent Hypergeometric Function

16.1 The confluence of two singularities of Riemann’s equation

16.11 Kummer’s formulae

16.12 Definition of the function W[sub(k,m)](z)

16.2 Expression of various functions by functions of the type W[sub(k,m)](z)

16.3 The asymptotic expansion of W[sub(k,m)](z), when |z| is large

16.31 The second solution of the equation for W[sub(k,m)](z)

16.4 Contour integrals of the Mellin–Barnes type for W[sub(k,m)](z)

16.41 Relations between W[sub(k,m)](z) and Mk[sub(k, [pm]m)](z)

16.5 The parabolic cylinder functions. Weber’s equation

16.51 The second solution of Weber’s equation

16.52 The general asymptotic expansion of D[sub(n)](z)

16.6 A contour integral for D[sub(n)](z)

16.61 Recurrence formulae for D[sub(n)](z)

16.7 Properties of D[sub(n)](z) when n is an integer

16.8 Miscellaneous examples

17 Bessel Functions

17.1 The Bessel coefficients

17.11 Bessel’s differential equation

17.2 The solution of Bessel’s equation when n is not necessarily an integer

17.21 The recurrence formulae for the Bessel functions

17.22 The zeros of Bessel functions whose order n is real

17.23 Bessel’s integral for the Bessel coefficients

17.24 Bessel functions whose order is half an odd integer

17.3 Hankel’s contour integral for J[sub(n)](z)

17.4 Connexion between Bessel coefficients and Legendre functions

17.5 Asymptotic series for J[sub(n)](z) when |z| is large

17.6 The second solution of Bessel’s equation when the order is an integer

17.61 The ascending series for Y[sub(n)](z)

17.7 Bessel functions with purely imaginary argument

17.71 Modified Bessel functions of the second kind

17.8 Neumann’s expansion of an analytic function in a series of Bessel coefficients

17.81 Proof of Neumann’s expansion

17.82 Schlömilch’s expansion of an arbitrary function in a series of Bessel coefficients of order

17.9 Tabulation of Bessel functions

17.10 Miscellaneous examples

18 The Equations of Mathematical Physics

18.1 The differential equations of mathematical physics

18.2 Boundary conditions

18.3 A general solution of Laplace’s equation

18.31 Solutions of Laplace’s equation involving Legendre functions

18.4 The solution of Laplace’s equation which satisfies assigned boundary conditions at the surfac

18.5 Solutions of Laplace’s equation which involve Bessel coefficients

18.51 The periods of vibration of a uniform membrane

18.6 A general solution of the equation of wave motions

18.61 Solutions of the equation of wave motions which involve Bessel functions

18.7 Miscellaneous examples

19 Mathieu Functions

19.1 The differential equation of Mathieu

19.11 The form of the solution of Mathieu’s equation

19.12 Hill’s equation

19.2 Periodic solutions of Mathieu’s equation

19.21 An integral equation satisfied by even Mathieu functions

19.22 Proof that the even Mathieu functions satisfy the integral equation

19.3 The construction of Mathieu functions

19.31 The integral formulae for the Mathieu functions

19.4 The nature of the solution of Mathieu’s general equation; Floquet’s theory

19.41 Hill’s method of solution

19.42 The evaluation of Hill’s determinant

19.5 The Lindemann–Stieltjes theory of Mathieu’s general equation

19.51 Lindemann’s form of Floquet’s theorem

19.52 The determination of the integral function associated with Mathieu’s equation

19.53 The solution of Mathieu’s equation in terms of F(zeta)

19.6 A second method of constructing the Mathieu function

19.61 The convergence of the series defining Mathieu functions

19.7 The method of change of parameter

19.8 The asymptotic solution of Mathieu’s equation

19.9 Miscellaneous examples

20 Elliptic Functions. General Theorems and the Weierstrassian Functions

20.1 Doubly-periodic functions

20.11 Period-parallelograms

20.12 Simple properties of elliptic functions

20.13 The order of an elliptic function

20.14 Relation between the zeros and poles of an elliptic function

20.2 The construction of an elliptic function. Definition of [wp](z)

20.21 Periodicity and other properties of [wp](z)

20.22 The differential equation satisfied by [wp](z)

20.3 The addition-theorem for the function [wp](z)

20.31 Another form of the addition-theorem

20.32 The constants e[sub(1)], e[sub(2)], e[sub(3)]

20.33 The addition of a half-period to the argument of [wp](z)

20.4 Quasi-periodic functions. The function [zeta](z)

20.41 The quasi-periodicity of the function [zeta](z)

20.42 The function [sigma](z)

20.5 Formulae expressing any elliptic function in terms of Weierstrassian functions with the same pe

20.51 The expression of any elliptic function in terms of [wp](z) and [wp[sup(prime)]](z)

20.52 The expression of any elliptic function as a linear combination of zeta-functions and their de

20.53 The expression of any elliptic function as a quotient of sigma-functions

20.54 The connexion between any two elliptic functions with the same periods

20.6 On the integration of (a[sub(0)]x[sup(4)] + 4a[sub(1)]x[sup(3)] + 6a[sub(2)]x[sup(2)] + 4a[sub(

20.7 The uniformisation of curves of genus unity

20.8 Miscellaneous examples

21 The Theta-Functions

21.1 The definition of a theta-function

21.11 The four types of theta-functions

21.12 The zeros of the theta-functions

21.2 The relations between the squares of the theta-functions

21.21 The addition-formulae for the theta-functions

21.22 Jacobi’s fundamental formulae

21.3 Jacobi’s expressions for the theta-functions as infinite products

21.4 The differential equation satisfied by the theta-functions

21.41 A relation between theta-functions of zero argument

21.42 The value of the constant G

21.43 Connexion of the sigma-function with the theta-functions

21.5 The expression of elliptic functions by means of theta-functions

21.51 Jacobi’s imaginary transformation

21.52 Landen’s type of transformation

21.6 The differential equations satisfied by quotients of theta-functions

21.61 The genesis of the Jacobian elliptic function sn u

21.62 Jacobi’s earlier notation. The theta-function [Theta](u) and the eta-function H(u)

21.7 The problem of inversion

21.71 The problem of inversion for complex values of c. The modular functions f(tau), g(tau), h(tau)

21.72 The periods, regarded as functions of the modulus

21.73 The inversion-problem associated with Weierstrassian elliptic functions

21.8 The numerical computation of elliptic functions

21.9 The notations employed for the theta-functions

21.10 Miscellaneous examples

22 The Jacobian Elliptic Functions

22.1 Elliptic functions with two simple poles

22.11 The Jacobian elliptic functions, sn u, cn u, dn u

22.12 Simple properties of sn u, cn u, dn u

22.2 The addition-theorem for the function sn u

22.21 The addition-theorems for cn u and dn u

22.3 The constant K

22.31 The periodic properties (associated with K) of the Jacobian elliptic functions

22.32 The constant K[sup(prime)]

22.33 The periodic properties (associated with K + iK[sup(prime)]) of the Jacobian elliptic function

22.34 The periodic properties (associated with iK[sup(prime)]) of the Jacobian elliptic functions

22.35 General description of the functions sn u, cn u, dn u

22.4 Jacobi’s imaginary transformation

22.41 Proof of Jacobi’s imaginary transformation by the aid of theta-functions

22.42 Landen’s transformation

22.5 Infinite products for the Jacobian elliptic functions

22.6 Fourier series for the Jacobian elliptic functions

22.61 Fourier series for reciprocals of Jacobian elliptic functions

22.7 Elliptic integrals

22.71 The expression of a quartic as the product of sums of squares

22.72 The three kinds of elliptic integrals

22.73 The elliptic integral of the second kind. The function E(u)

22.74 The elliptic integral of the third kind

22.8 The lemniscate functions

22.81 The values of K and K[sup(prime)] for special values of k

22.82 A geometrical illustration of the functions sn u, cn u, dn u

22.9 Miscellaneous examples

23 Ellipsoidal Harmonics and Lamé’s Equation

23.1 The definition of ellipsoidal harmonics

23.2 The four species of ellipsoidal harmonics

23.21 The construction of ellipsoidal harmonics of the first species

23.22 Ellipsoidal harmonics of the second species

23.23 Ellipsoidal harmonics of the third species

23.24 Ellipsoidal harmonics of the fourth species

23.25 Niven’s expressions for ellipsoidal harmonics in terms of homogeneous harmonics

23.26 Ellipsoidal harmonics of degree n

23.3 Confocal coordinates

23.31 Uniformising variables associated with confocal coordinates

23.32 Laplace’s equation referred to confocal coordinates

23.33 Ellipsoidal harmonics referred to confocal coordinates

23.4 Various forms of Lamé’s differential equation

23.41 Solutions in series of Lamé’s equation

23.42 The definition of Lamé functions

23.43 The non-repetition of factors in Lamé functions

23.44 The linear independence of Lamé functions

23.45 The linear independence of ellipsoidal harmonics

23.46 Stieltjes’ theorem on the zeros of Lamé functions

23.47 Lamé functions of the second kind

23.5 Lamé’s equation in association with Jacobian elliptic functions

23.6 The integral equation satisfied by Lamé functions of the first and second species

23.61 The integral equation satisfied by Lamé functions of the third and fourth species

23.62 Integral formulae for ellipsoidal harmonics

23.63 Integral formulae for ellipsoidal harmonics of the third and fourth species

23.7 Generalisations of Lamé’s equation

23.71 The Jacobian form of the generalised Lamé equation

23.8 Miscellaneous examples

Appendix The Elementary Transcendental Functions

A.1 On certain results assumed in Chapters 1 to 4

A.11 Summary of the Appendix

A.12 A logical order of development of the elements of analysis

A.2 The exponential function exp z

A.21 The addition-theorem for the exponential function, and its consequences

A.22 Various properties of the exponential function

A.3 Logarithms of positive numbers

A.31 The continuity of the Logarithm

A.32 Differentiation of the Logarithm

A.33 The expansion of Log(1 + a) in powers of a

A.4 The definition of the sine and cosine

A.41 The fundamental properties of sin z and cos z

A.42 The addition-theorems for sin z and cos z

A.5 The periodicity of the exponential function

A.51 The solution of the equation exp [gamma] = 1

A.52 The solution of a pair of trigonometrical equations

A.6 Logarithms of complex numbers

A.7 The analytical definition of an angle

References

Author index

Subject index

E. T. Whittaker was Professor of Mathematics at the University of Edinburgh. He was awarded the Copley Medal in 1954, ‘for his distinguished contributions to both pure and applied mathematics and to theoretical physics’.

G. N. Watson was Professor of Pure Mathematics at the University of Birmingham. He is known, amongst other things, for the 1918 result now known as Watson’s lemma and was awarded the De Morgan Medal in 1947.

Victor H. Moll is Professor in the Department of Mathematics at Tulane University. He co-authored Elliptic Curves (Cambridge, 1997) and was awarded the Weiss Presidential Award in 2017 for his Graduate Teaching. He first received a copy of Whittaker and Watson during his own undergraduate studies at the Universidad Santa Maria in Chile.

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