A Course of Modern Analysis

Edmund Taylor Whittaker (1873–1956); George Neville Watson(1886–1965)
	Contents
	Foreword    S.J. Patterson
	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
		2.2 The limit of an increasing sequence
		2.3 Convergence of an infinite series
		2.4 Effect of changing the order of the terms in a series
		2.5 Double series
		2.6 Power series
		2.7 Infinite products
		2.8 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
		3.3 Series of variable terms. Uniformity of convergence
		3.4 Discussion of a particular double series
		3.5 The concept of uniformity
		3.6 The modified Heine–Borel theorem
		3.7 Uniformity of convergence of power series
		3.8 Miscellaneous examples
	4. The Theory of Riemann Integration
		4.1 The concept of integration
		4.2 Differentiation of integrals containing a parameter
		4.3 Double integrals and repeated integrals
		4.4 Infinite integrals
		4.5 Improper integrals. Principal values
		4.6 Complex integration
		4.7 Integration of infinite series
		4.8 Miscellaneous examples
	5. The Fundamental Properties of Analytic Functions; Taylor’s, Laurent’s and Liouville’s Theorems
		5.1 Property of the elementary functions
		5.2 Cauchy’s theorem on the integral of a function round a contour
		5.3 Analytic functions represented by uniformly convergent series
		5.4 Taylor’s theorem
		5.5 The process of continuation
		5.6 Laurent’s theorem
		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
		6.3 Cauchy’s integral
		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
		7.3 Bürmann’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
		7.9 Miscellaneous examples
	8. Asymptotic Expansions and Summable Series
		8.1 Simple example of an asymptotic expansion
		8.2 Definition of an asymptotic expansion
		8.3 Multiplication of asymptotic expansions
		8.4 Methods of summing series
		8.5 Hardy’s convergence theorem
		8.6 Miscellaneous examples
	9. Fourier Series and Trigonometric Series
		9.1 Definition of Fourier series
		9.2 On Dirichlet’s conditions and Fourier’s theorem
		9.3 The nature of the coefficients in a Fourier series
		9.4 Fejér’s theorem
		9.5 The Hurwitz–Liapounoff theorem concerning Fourier constants
		9.6 Riemann’s theory of 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
		10.3 Points which are regular for a differential equation
		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
		10.8 Linear differential equations with two singularities
		10.9 Miscellaneous examples
	11. Integral Equations
		11.1 Definition of an integral equation
		11.2 Fredholm’s equation and its tentative solution
		11.3 Integral equations of the first and second kinds
		11.4 The Liouville–Neumann method of successive substitutions
		11.5 Symmetric nuclei
		11.6 Orthogonal functions
		11.7 The development of a symmetric nucleus
		11.8 Solution of Abel’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
		12.2 Euler’s expression of Γ(z) as an infinite integral
		12.3 Gauss’ expression for the logarithmic derivate of the Gamma-function as an infinite integral
		12.4 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
		13.2 Hermite’s formula for ζ(s,a)
		13.3 Euler’s product for ζ(s)
		13.4 Riemann’s integral for ζ(s)
		13.5 Inequalities satisfied by ζ(s,a) when σ>0
		13.6 The asymptotic expansion of log Γ(z+a)
		13.7 Miscellaneous examples
	14. The Hypergeometric Function
		14.1 The hypergeometric series
		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
		14.6 Solution of Riemann’s equation by a contour integral
		14.7 Relations between contiguous hypergeometric functions
		14.8 Miscellaneous examples
	15. Legendre Functions
		15.1 Definition of Legendre polynomials
		15.2 Legendre functions
		15.3 Legendre functions of the second kind
		15.4 Heine’s development of (t-z)^(-1) as a series of Legendre polynomials in z
		15.5 Ferrers’ associated Legendre functions P^m_n (z) and Q^m_n (z)
		15.6 Hobson’s definition of the associated Legendre functions
		15.7 The addition-theorem for the Legendre polynomials
		15.8 The function C^v_n (z)
		15.9 Miscellaneous examples
	16. The Confluent Hypergeometric Function
		16.1 The confluence of two singularities of Riemann’s equation
		16.2 Expression of various functions by functions of the type W_{k,m}(z)
		16.3 The asymptotic expansion of W_{k,m} (z), when |z| is large
		16.4 Contour integrals of the Mellin–Barnes type for W_{k,m} (z)
		16.5 The parabolic cylinder functions. Weber’s equation
		16.6 A contour integral for D_n (z)
		16.7 Properties of D_n (z) when n is an integer
		16.8 Miscellaneous examples
	17. Bessel Functions
		17.1 The Bessel coefficients
		17.2 The solution of Bessel’s equation when n is not necessarily an integer
		17.3 Hankel’s contour integral for J_n (z)
		17.4 Connexion between Bessel coefficients and Legendre functions
		17.5 Asymptotic series for J_n (z) when |z| is large
		17.6 The second solution of Bessel’s equation when the order is an integer
		17.7 Bessel functions with purely imaginary argument
		17.8 Neumann’s expansion of an analytic function in a series of Bessel coefficients
		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
		18.4 The solution of Laplace’s equation which satisfies assigned boundary conditions at the surface of a sphere
		18.5 Solutions of Laplace’s equation which involve Bessel coefficients
		18.6 A general solution of the equation of wave motions
		18.7 Miscellaneous examples
	19. Mathieu Functions
		19.1 The differential equation of Mathieu
		19.2 Periodic solutions of Mathieu’s equation
		19.3 The construction of Mathieu functions
		19.4 The nature of the solution of Mathieu’s general equation; Floquet’s theory
		19.5 The Lindemann–Stieltjes theory of Mathieu’s general equation
		19.6 A second method of constructing the Mathieu function
		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
		20.2 The construction of an elliptic function. Definition of ℘(z)
		20.3 The addition-theorem for the function ℘(z)
		20.4 Quasi-periodic functions. The function ζ(z)
		20.5 Formulae expressing any elliptic function in terms of Weierstrassian functions with the same periods
		20.6 On the integration of (a₀x⁴ + 4a₁x³ + 6a₂x² + 4a₃x + a₄) ^{-1/2}
		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
		21.2 The relations between the squares of the theta-functions
		21.3 Jacobi’s expressions for the theta-functions as infinite products
		21.4 The differential equation satisfied by the theta-functions
		21.5 The expression of elliptic functions by means of theta-functions
		21.6 The differential equations satisfied by quotients of theta-functions
		21.7 The problem of inversion
		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
		22.2 The addition-theorem for the function sn u
		22.3 The constant K
		22.4 Jacobi’s imaginary transformation
		22.5 Infinite products for the Jacobian elliptic functions
		22.6 Fourier series for the Jacobian elliptic functions
		22.7 Elliptic integrals
		22.8 The lemniscate functions
		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
		23.3 Confocal coordinates
		23.4 Various forms of Lamé’s differential equation
		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
		23.7 Generalisations of Lamé’s equation
		23.8 Miscellaneous examples
Appendix. The Elementary Transcendental Functions
	A.1 On certain results assumed in Chapters 1 to 4
	A.2 The exponential function exp z
	A.3 Logarithms of positive numbers
	A.4 The definition of the sine and cosine
	A.5 The periodicity of the exponential function
	A.6 Logarithms of complex numbers
	A.7 The analytical definition of an angle
References
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Author index
Subject index