A Course of Modern Analysis 5th Edition by E. T. Whittaker 9781009008594 1009008595

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• ISBN 10: 1009008595
• ISBN 13: 9781009008594
• Author:  E. T. Whittaker

A Course of Modern Analysis

Table contents:

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

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