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Course of Modern Analysis 5th Revised edition [Kõva köide]

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  • Formaat: Hardback, 718 pages, kõrgus x laius x paksus: 260x185x62 mm, kaal: 1660 g, Worked examples or Exercises
  • Ilmumisaeg: 26-Aug-2021
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1316518930
  • ISBN-13: 9781316518939
Teised raamatud teemal:
Course of Modern Analysis 5th Revised editionSuurem pilt
  • Formaat: Hardback, 718 pages, kõrgus x laius x paksus: 260x185x62 mm, kaal: 1660 g, Worked examples or Exercises
  • Ilmumisaeg: 26-Aug-2021
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1316518930
  • ISBN-13: 9781316518939
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781316518939.html
Märksõnad:
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.

Arvustused

'Generations of mathematicians have referred to W&W, as it has been affectionately called, for information that is difficult to locate elsewhere, in particular, on special functions. This improved new edition will enable future generations to access and learn from one of the great classical texts in the mathematical literature. My personal references to W&W are legion; the cover of my worn copy has long been separated from the text because of constant use.' Bruce C. Berndt, University of Illinois at Urbana-Champaign 'Many of us who often use special functions revere the classics of complex analysis from the early 20th century. The names of Copson, MacRobert and Titchmarsh come to mind. However, the grandfather, indeed the overarching prototype, for most of these books is the one always referred to as "Whittaker and Watson." Fortunately for the world of mathematics, Victor Moll has presided over this wonderful fifth edition. Victor has provided an exceptionally valuable introduction that provides summaries of each chapter with ties to modern work. This new edition makes it easier for all to use the immense resources therein. Thank you, Victor! Thank you, Cambridge University Press.' George Andrews, The Pennsylvania State University 'In many cases the coverage here is still the best or one of the best available, and is concise and all in one volume.' Allen Stenger, Mathematical Association of America

Muu info

New edition of a true classic in mathematics. Preserves the flavour of the original, updating where appropriate and improving usability.
Foreword xvii
S. J. Patterson
Preface to the Fifth Edition xxi
Preface to the Fourth Edition xxiii
Preface to the Third Edition xxiv
Preface to the Second Edition xxv
Preface to the First Edition xxvi
Introduction xxvii
Part I The Process of Analysis
1(240)
1 Complex Numbers
3(7)
1.1 Rational numbers
3(1)
1.2 Dedekind's theory of irrational numbers
4(2)
1.3 Complex numbers
6(1)
1.4 The modulus of a complex number
7(1)
1.5 The Argand diagram
8(1)
1.6 Miscellaneous examples
9(1)
2 The Theory of Convergence
10(30)
2.1 The definition of the limit of a sequence
10(1)
2.11 Definition of the phrase `of the order of'
10(1)
2.2 The limit of an increasing sequence
10(1)
2.21 Limit-points and the Bolzano--Weierstrass theorem
11(1)
2.22 Cauchy's theorem on the necessary and sufficient condition for the existence of a limit
12(1)
2.3 Convergence of an infinite series
13(3)
2.31 Dirichlet's test for convergence
16(1)
2.32 Absolute and conditional convergence
17(1)
2.33 The geometric series, and the series Σ∞n=1 1/ns
17(1)
2.34 The comparison theorem
18(2)
2.35 Cauchy's test for absolute convergence
20(1)
2.36 D'Alembert's ratio test for absolute convergence
20(1)
2.37 A general theorem on series for which limn→∞ |un+1/un| = 1
21(1)
2.38 Convergence of the hypergeometric series
22(1)
2.4 Effect of changing the order of the terms in a series
23(1)
2.41 The fundamental property of absolutely convergent series
24(1)
2.5 Double series
24(1)
2.51 Methods of summing a double series
25(1)
2.52 Absolutely convergent double series
26(1)
2.53 Cauchy's theorem on the multiplication of absolutely convergent series
27(1)
2.6 Power series
28(1)
2.61 Convergence of series derived from a power series
29(1)
2.7 Infinite products
30(1)
2.71 Some examples of infinite products
31(3)
2.8 Infinite determinants
34(1)
2.81 Convergence of an infinite determinant
34(1)
2.82 The rearrangement theorem for convergent infinite determinants
35(1)
2.9 Miscellaneous examples
36(4)
3 Continuous Functions and Uniform Convergence
40(18)
3.1 The dependence of one complex number on another
40(1)
3.2 Continuity of functions of real variables
40(1)
3.21 Simple curves. Continua
41(1)
3.22 Continuous functions of complex variables
42(1)
3.3 Series of variable terms. Uniformity of convergence
43(1)
3.31 On the condition for uniformity of convergence
44(1)
3.32 Connexion of discontinuity with non-uniform convergence
45(1)
3.33 The distinction between absolute and uniform convergence
46(1)
3.34 A condition, due to Weierstrass, for uniform convergence
47(1)
3.35 Hardy's tests for uniform convergence
48(1)
3.4 Discussion of a particular double series
49(2)
3.5 The concept of uniformity
51(1)
3.6 The modified Heine--Borel theorem
51(1)
3.61 Uniformity of continuity
52(1)
3.62 A real function, of a real variable, continuous in a closed interval, attains its upper bound
53(1)
3.63 A real function, of a real variable, continuous in a closed interval, attains all values between its upper and lower bounds
54(1)
3.64 The fluctuation of a function of a real variable
54(1)
3.7 Uniformity of convergence of power series
55(1)
3.71 Abel's theorem
55(1)
3.72 Abel's theorem on multiplication of convergent series
55(1)
3.73 Power series which vanish identically
56(1)
3.8 Miscellaneous examples
56(2)
4 The Theory of Riemann Integration
58(23)
4.1 The concept of integration
58(1)
4.11 Upper and lower integrals
58(1)
4.12 Riemann's condition of integrability
59(1)
4.13 A general theorem on integration
60(2)
4.14 Mean-value theorems
62(2)
4.2 Differentiation of integrals containing a parameter
64(1)
4.3 Double integrals and repeated integrals
65(2)
4.4 Infinite integrals
67(1)
4.41 Infinite integrals of continuous functions. Conditions for convergence
67(1)
4.42 Uniformity of convergence of an infinite integral
68(1)
4.43 Tests for the convergence of an infinite integral
68(3)
4.44 Theorems concerning uniformly convergent infinite integrals
71(1)
4.5 Improper integrals. Principal values
72(1)
4.51 The inversion of the order of integration of a certain repeated integral
73(2)
4.6 Complex integration
75(1)
4.61 The fundamental theorem of complex integration
76(1)
4.62 An upper limit to the value of a complex integral
76(1)
4.7 Integration of infinite series
77(2)
4.8 Miscellaneous examples
79(2)
5 The Fundamental Properties of Analytic Functions; Taylor's, Laurent's and Liouville's Theorems
81(29)
5.1 Property of the elementary functions
81(1)
5.11 Occasional failure of the property
82(1)
5.12 Cauchy's definition of an analytic function of a complex variable
82(1)
5.13 An application of the modified Heine--Borel theorem
83(1)
5.2 Cauchy's theorem on the integral of a function round a contour
83(3)
5.21 The value of an analytic function at a point, expressed as an integral taken round a contour enclosing the point
86(2)
5.22 The derivatives of an analytic function f(z)
88(1)
5.23 Cauchy's inequality for f(n)(a)
89(1)
5.3 Analytic functions represented by uniformly convergent series
89(1)
5.31 Analytic functions represented by integrals
90(1)
5.32 Analytic functions represented by infinite integrals
91(1)
5.4 Taylor's theorem
91(3)
5.41 Forms of the remainder in Taylor's series
94(1)
5.5 The process of continuation
95(2)
5.51 The identity of two functions
97(1)
5.6 Laurent's theorem
98(2)
5.61 The nature of the singularities of one-valued functions
100(1)
5.62 The `point at infinity'
101(2)
5.63 Liouvillle's theorem
103(1)
5.64 Functions with no essential singularities
104(1)
5.7 Many-valued functions
105(1)
5.8 Miscellaneous examples
106(4)
6 The Theory of Residues; Application to the Evaluation of Definite Integrals
110(15)
6.1 Residues
110(1)
6.2 The evaluation of definite integrals
111(1)
6.21 The evaluation of the integrals of certain periodic functions taken between the limits 0 and 2π
111(1)
6.22 The evaluation of certain types of integrals taken between the limits -∞ and +∞
112(4)
6.23 Principal values of integrals
116(1)
6.24 Evaluation of integrals of the form ∫∞0 xa-1Q(x)dx
117(1)
6.3 Cauchy's integral
118(1)
6.31 The number of roots of an equation contained within a contour
119(1)
6.4 Connexion between the zeros of a function and the zeros of its derivative
120(1)
6.5 Miscellaneous examples
121(4)
7 The Expansion of Functions in Infinite Series
125(28)
7.1 A formula due to Darboux
125(1)
7.2 The Bernoullian numbers and the Bernoullian polynomials
125(2)
7.21 The Euler--Maclaurin expansion
127(2)
7.3 Burmann's theorem
129(2)
7.31 Teixeira's extended form of Burmann's theorem
131(2)
7.32 Lagrange's theorem
133(1)
7.4 The expansion of a class of functions in rational fractions
134(3)
7.5 The expansion of a class of functions as infinite products
137(1)
7.6 The factor theorem of Weierstrass
138(2)
7.7 Expansion in a series of cotangents
140(1)
7.8 Borel's theorem
141(1)
7.81 Borel's integral and analytic continuation
142(1)
7.82 Expansions in series of inverse factorials
143(2)
7.9 Miscellaneous examples
145(8)
8 Asymptotic Expansions and Summable Series
153(10)
8.1 Simple example of an asymptotic expansion
153(1)
8.2 Definition of an asymptotic expansion
154(1)
8.21 Another example of an asymptotic expansion
154(2)
8.3 Multiplication of asymptotic expansions
156(1)
8.31 Integration of asymptotic expansions
156(1)
8.32 Uniqueness of an asymptotic expansion
157(1)
8.4 Methods of summing series
157(1)
8.41 Borel's method of summation
158(1)
8.42 Euler's method of summation
158(1)
8.43 Cesaro's method of summation
158(1)
8.44 The method of summation of Riesz
159(1)
8.5 Hardy's convergence theorem
159(2)
8.6 Miscellaneous examples
161(2)
9 Fourier Series and Trigonometric Series
163(38)
9.1 Definition of Fourier series
163(1)
9.11 Nature of the region within which a trigonometrical series converges
164(3)
9.12 Values of the coefficients in terms of the sum of a trigonometrical series
167(1)
9.2 On Dirichlet's conditions and Fourier's theorem
167(1)
9.21 The representation of a function by Fourier series for ranges other than (-π, π)
168(1)
9.22 The cosine series and the sine series
169(2)
9.3 The nature of the coefficients in a Fourier series
171(1)
9.31 Differentiation of Fourier series
172(1)
9.32 Determination of points of discontinuity
173(1)
9.4 Fejer's theorem
174(3)
9.41 The Riemann--Lebesgue lemmas
177(2)
9.42 The proof of Fourier's theorem
179(2)
9.43 The Dirichlet--Bonnet proof of Fourier's theorem
181(2)
9.44 The uniformity of the convergence of Fourier series
183(2)
9.5 The Hurwitz--Liapounoff theorem concerning Fourier constants
185(2)
9.6 Riemann's theory of trigonometrical series
187(1)
9.61 Riemann's associated function
188(1)
9.62 Properties of Riemann's associated function; Riemann's first lemma
189(2)
9.63 Riemann's theorem on trigonometrical series
191(2)
9.7 Fourier's representation of a function by an integral
193(2)
9.8 Miscellaneous examples
195(6)
10 Linear Differential Equations
201(18)
10.1 Linear differential equations
201(1)
10.2 Solutions in the vicinity of an ordinary point
201(2)
10.21 Uniqueness of the solution
203(1)
10.3 Points which are regular for a differential equation
204(2)
10.31 Convergence of the expansion of §10.3
206(1)
10.32 Derivation of a second solution in the case when the difference of the exponents is an integer or zero
207(2)
10.4 Solutions valid for large values of \z\
209(1)
10.5 Irregular singularities and confluence
210(1)
10.6 The differential equations of mathematical physics
210(4)
10.7 Linear differential equations with three singularities
214(1)
10.71 Transformations of Riemann's P-equation
215(1)
10.72 The connexion of Riemann's P-equation with the hypergeometric equation
215(1)
10.8 Linear differential equations with two singularities
216(1)
10.9 Miscellaneous examples
216(3)
11 Integral Equations
219(22)
11.1 Definition of an integral equation
219(1)
11.11 An algebraical lemma
220(1)
11.2 Fredholm's equation and its tentative solution
221(2)
11.21 Investigation of Fredholm's solution
223(3)
11.22 Volterra's reciprocal functions
226(2)
11.23 Homogeneous integral equations
228(1)
11.3 Integral equations of the first and second kinds
229(1)
11.31 Volterra's equation
229(1)
11.4 The Liouville--Neumann method of successive substitutions
230(1)
11.5 Symmetric nuclei
231(1)
11.51 Schmidt's theorem that, if the nucleus is symmetric, the equation D(λ) = 0 has at least one root
232(1)
11.6 Orthogonal functions
233(1)
11.61 The connexion of orthogonal functions with homogeneous integral equations
234(2)
11.7 The development of a symmetric nucleus
236(1)
11.71 The solution of Fredholm's equation by a series
237(1)
11.8 Solution of Abel's integral equation
238(1)
11.81 Schlomilch's integral equation
238(1)
11.9 Miscellaneous examples
239(2)
Part II The Transcendental Functions
241(370)
12 The Gamma-Function
243(33)
12.1 Definitions of the Gamma-function
243(2)
12.11 Euler's formula for the Gamma-function
245(1)
12.12 The difference equation satisfied by the Gamma-function
245(1)
12.13 The evaluation of a general class of infinite products
246(2)
12.14 Connexion between the Gamma-function and the circular functions
248(1)
12.15 The multiplication-theorem of Gauss and Legendre
248(1)
12.16 Expansion for the logarithmic derivates of the Gamma-function
249(1)
12.2 Euler's expression of F(z) as an infinite integral
250(2)
12.21 Extension of the infinite integral to the case in which the argument of the Gamma-function is negative
252(1)
12.22 Hankel's expression of as a contour integral
253(2)
12.3 Gauss' infinite integral for Γ'(z)/Γ(z)
255(2)
12.31 Binet's first expression for log Γ(z) in terms of an infinite integral
257(2)
12.32 Binet's second expression for log Γ(z) in terms of an infinite integral
259(2)
12.33 The asymptotic expansion of the logarithms of the Gamma-function
261(2)
12.4 The Eulerian integral of the first kind
263(1)
12.41 Expression of the Eulerian integral of the first kind in terms of the Gamma-function
264(1)
12.42 Evaluation of trigonometrical integrals in terms of the Gamma-function
265(1)
12.43 Pochhammer's extension of the Eulerian integral of the first kind
266(1)
12.5 Dirichlet's integral
267(1)
12.6 Miscellaneous examples
268(8)
13 The Zeta-Function of Riemann
276(17)
13.1 Definition of the zeta-function
276(1)
13.11 The generalised zeta-function
276(1)
13.12 The expression of ζ(s, a) as an infinite integral
276(1)
13.13 The expression of ζ(s, a) as a contour integral
277(1)
13.14 Values of ζ(s, a) for special values of s
278(1)
13.15 The formula of Hurwitz for ζ(s, a) when σ < 0
279(1)
13.2 Hermite's formula for ζ(s, a)
280(2)
13.21 Deductions from Hermite's formula
282(1)
13.3 Euler's product for ζ(s)
282(1)
13.31 Riemann's hypothesis concerning the zeros of ζ(s)
283(1)
13.4 Riemann's integral for ζ(s)
283(2)
13.5 Inequalities satisfied by ζ(s, a) when σ > 0
285(1)
13.51 Inequalities satisfied by ζ(s, a) when σ ≤ 0
286(2)
13.6 The asymptotic expansion of log Γ(z + a)
288(2)
13.7 Miscellaneous examples
290(3)
14 The Hypergeometric Function
293(23)
14.1 The hypergeometric series
293(1)
14.11 The value of F(a, b c; 1)when Re(c - a - b) > 0
293(2)
14.2 The differential equation satisfied by F(a, b; c; z)
295(1)
14.3 Solutions of Riemann's P-equation
295(3)
14.4 Relations between particular solutions
298(1)
14.5 Barnes' contour integrals
299(1)
14.51 The continuation of the hypergeometric series
300(1)
14.52 Barnes' lemma
301(2)
14.53 The connexion between hypergeometric functions of z and of 1 -- z
303(1)
14.6 Solution of Riemann's equation by a contour integral
303(3)
14.61 Determination of an integral which represents P(α)
306(1)
14.7 Relations between contiguous hypergeometric functions
307(2)
14.8 Miscellaneous examples
309(7)
15 Legendre Functions
316(39)
15.1 Definition of Legendre polynomials
316(1)
15.11 Rodrigues' formula for the Legendre polynomials
317(1)
15.12 Schlafli's integral for Pn(z)
317(1)
15.13 Legendre's differential equation
318(1)
15.14 The integral properties of the Legendre polynomials
319(1)
15.2 Legendre functions
320(2)
15.21 The recurrence formulae
322(4)
15.22 Murphy's expression of Pn(z) as a hypergeometric function
326(1)
15.23 Laplace's integrals for Pn(z)
327(4)
15.3 Legendre functions of the second kind
331(1)
15.31 Expansion of Qn(z) as a power series
331(2)
15.32 The recurrence formulae for Qn{z)
333(1)
15.33 The Laplacian integral for Legendre functions of the second kind
334(1)
15.34 Neumann's formula for Qn(z), when n is an integer
335(2)
15.4 Heine's development of (t -- z)---1
337(1)
15.41 Neumann's expansion of an arbitrary function in a series of Legendre polynomials
338(1)
15.5 Ferrers' associated Legendre functions Pmn(z) and Qmn(z)
339(1)
15.51 The integral properties of the associated Legendre functions
340(1)
15.6 Hobson's definition of the associated Legendre functions
341(1)
15.61 Expression of Pmn(z) as an integral of Laplace's type
342(1)
15.7 The addition-theorem for the Legendre polynomials
342(2)
15.71 The addition theorem for the Legendre functions
344(2)
15.8 The function Cvn(z)
346(1)
15.9 Miscellaneous examples
347(8)
16 The Confluent Hypergeometric Function
355(18)
16.1 The confluence of two singularities of Riemann's equation
355(1)
16.11 Kummer's formulae
356(1)
16.12 Definition of the function Wk,m(z)
357(1)
16.2 Expression of various functions by functions of the type Wk,m{z)
358(2)
16.3 The asymptotic expansion of Wk,m(z), when |z| is large
360(1)
16.31 The second solution of the equation for Wk,m(z)
361(1)
16.4 Contour integrals of the Mellin-Barnes type for Wk,m(z)
361(2)
16.41 Relations between Wk,m(z) and Mk, ±m(z)
363(1)
16.5 The parabolic cylinder functions. Weber's equation
364(1)
16.51 The second solution of Weber's equation
365(1)
16.52 The general asymptotic expansion of Dn(z)
366(1)
16.6 A contour integral for Dn(z)
366(1)
16.61 Recurrence formulae for Dn(z)
367(1)
16.7 Properties of Dn(z) when n is an integer
367(2)
16.8 Miscellaneous examples
369(4)
17 Bessel Functions
373(34)
17.1 The Bessel coefficients
373(2)
17.11 Bessel's differential equation
375(1)
17.2 Bessel's equation when n is not necessarily an integer
376(1)
17.21 The recurrence formulae for the Bessel functions
377(2)
17.22 The zeros of Bessel functions whose order n is real
379(1)
17.23 Bessel's integral for the Bessel coefficients
380(2)
17.24 Bessel functions whose order is half an odd integer
382(1)
17.3 Hankel's contour integral for Jn(z)
383(2)
17.4 Connexion between Bessel coefficients and Legendre functions
385(1)
17.5 Asymptotic series for Jn(z) when |z| is large
386(2)
17.6 The second solution of Bessel's equation
388(2)
17.61 The ascending series for Yn(z)
390(1)
17.7 Bessel functions with purely imaginary argument
391(1)
17.71 Modified Bessel functions of the second kind
392(1)
17.8 Neumann's expansions
393(1)
17.81 Proof of Neumann's expansion
394(2)
17.82 Schlomilch's expansion of an arbitrary function in a series of Bessel coefficients of order zero
396(1)
17.9 Tabulation of Bessel functions
397(1)
17.10 Miscellaneous examples
397(10)
18 The Equations of Mathematical Physics
407(19)
18.1 The differential equations of mathematical physics
407(1)
18.2 Boundary conditions
408(1)
18.3 A general solution of Laplace's equation
409(3)
18.31 Solutions of Laplace's equation involving Legendre functions
412(2)
18.4 The solution of Laplace's equation
414(3)
18.5 Laplace's equation and Bessel coefficients
417(1)
18.51 The periods of vibration of a uniform membrane
417(1)
18.6 A general solution of the equation of wave motions
418(1)
18.61 Solutions of the equation of wave motions which involve Bessel functions
418(2)
18.7 Miscellaneous examples
420(6)
19 Mathieu Functions
426(25)
19.1 The differential equation of Mathieu
426(2)
19.11 The form of the solution of Mathieu's equation
428(1)
19.12 Hill's equation
428(1)
19.2 Periodic solutions of Mathieu's equation
428(1)
19.21 An integral equation satisfied by even Mathieu functions
429(1)
19.22 Proof that the even Mathieu functions satisfy the integral equation
430(1)
19.3 The construction of Mathieu functions
431(2)
19.31 The integral formulae for the Mathieu functions
433(1)
19.4 Floquet's theory
434(1)
19.41 Hill's method of solution
435(2)
19.42 The evaluation of Hill's determinant
437(1)
19.5 The Lindemann--Stieltjes theory of Mathieu's general equation
438(1)
19.51 Lindemann's form of Floquet's theorem
439(1)
19.52 The determination of the integral function associated with Mathieu's equation
439(2)
19.53 The solution of Mathieu's equation in terms of F(ζ)
441(1)
19.6 A second method of constructing the Mathieu function
442(2)
19.61 The convergence of the series denning Mathieu functions
444(2)
19.7 The method of change of parameter
446(1)
19.8 The asymptotic solution of Mathieu's equation
447(1)
19.9 Miscellaneous examples
448(3)
20 Elliptic Functions. General Theorems and the Weierstrassian Functions
451(35)
20.1 Doubly-periodic functions
451(1)
20.11 Period-parallelograms
452(1)
20.12 Simple properties of elliptic functions
452(1)
20.13 The order of an elliptic function
453(1)
20.14 Relation between the zeros and poles of an elliptic function
454(1)
20.2 The construction of an elliptic function. Definition of p(z)
455(1)
20.21 Periodicity and other properties of p(z)
456(2)
20.22 The differential equation satisfied by p(z)
458(4)
20.3 The addition-theorem for the function p(z)
462(1)
20.31 Another form of the addition-theorem
462(3)
20.32 The constants e1, e2, e3
465(1)
20.33 The addition of a half-period to the argument of p(z)
466(1)
20.4 Quasi-periodic functions. The function ζ(z)
467(1)
20.41 The quasi-periodicity of the function ζ(z)
468(1)
20.42 The function σ(z)
469(2)
20.5 Formulae in terms of Weierstrassian functions
471(1)
20.51 The expression of any elliptic function in terms of p(z) and p(z)
471(1)
20.52 The expression of any elliptic function as a linear combination of zeta-functions and their derivatives
472(1)
20.53 The expression of any elliptic function as a quotient of sigma-functions
473(1)
20.54 The connexion between any two elliptic functions with the same periods
474(1)
20.6 On the integration of (aox4 + 4a|x3 + 6a2x2 + 4a3x + a4)--1/2
475(2)
20.7 The uniformisation of curves of genus unity
477(1)
20.8 Miscellaneous examples
478(8)
21 The Theta-Functions
486(31)
21.1 The definition of a theta-function
486(1)
21.11 The four types of theta-functions
487(2)
21.12 The zeros of the theta-functions
489(1)
21.2 The relations between the squares of the theta-functions
490(1)
21.21 The addition-formulae for the theta-functions
491(1)
21.22 Jacobi's fundamental formulae
491(2)
21.3 Theta-functions as infinite products
493(1)
21.4 The differential equation satisfied by the theta-functions
494(1)
21.41 A relation between theta-functions of zero argument
495(1)
21.42 The value of the constant G
496(2)
21.43 Connexion of the sigma-function with the theta-functions
498(1)
21.5 Elliptic functions in terms of theta-functions
498(1)
21.51 Jacobi's imaginary transformation
499(2)
21.52 Landen's type of transformation
501(1)
21.6 Differential equations of theta quotients
502(1)
21.61 The genesis of the Jacobian elliptic function sn u
503(1)
21.62 Jacobi's earlier notation. The theta-function O(u) and the eta-function H(u)
504(1)
21.7 The problem of inversion
505(1)
21.71 The problem of inversion for complex values of c. The modular functions f(τ), g(τ), h(τ)
506(4)
21.72 The periods, regarded as functions of the modulus
510(1)
21.73 The inversion-problem associated with Weierstrassian elliptic functions
510(1)
21.8 The numerical computation of elliptic functions
511(1)
21.9 The notations employed for the theta-functions
512(1)
21.10 Miscellaneous examples
513(4)
22 The Jacobian Elliptic Functions
517(50)
22.1 Elliptic functions with two simple poles
517(1)
22.11 The Jacobian elliptic functions, sn u, cn u, dn u
517(2)
22.12 Simple properties of sn u, cn u, dn u
519(2)
22.2 The addition-theorem for the function sn u
521(2)
22.21 The addition-theorems for cn u and dn u
523(2)
22.3 The constant K
525(1)
22.31 The periodic properties (associated with K) of the Jacobian elliptic functions
526(1)
22.32 The constant K'
527(2)
22.33 The periodic properties (associated with K + iK') of the Jacobian elliptic functions
529(1)
22.34 The periodic properties (associated with iK') of the Jacobian elliptic functions
530(1)
22.35 General description of the functions sn u, cn u, dnu
531(1)
22.4 Jacobi's imaginary transformation
532(1)
22.41 Proof of Jacobi's imaginary transformation by the aid of theta-functions
533(1)
22.42 Landen's transformation
534(1)
22.5 Infinite products for the Jacobian elliptic functions
535(2)
22.6 Fourier series for the Jacobian elliptic functions
537(2)
22.61 Fourier series for reciprocals of Jacobian elliptic functions
539(1)
22.7 Elliptic integrals
540(1)
22.71 The expression of a quartic as the product of sums of squares
541(1)
22.72 The three kinds of elliptic integrals
542(3)
22.73 The elliptic integral of the second kind. The function E(u)
545(6)
22.74 The elliptic integral of the third kind
551(1)
22.8 The lemniscate functions
552(2)
22.81 The values of K and K' for special values of k
554(2)
22.82 A geometrical illustration of the functions sn u, cn u, dn u
556(1)
22.9 Miscellaneous examples
557(10)
23 Ellipsoidal Harmonics and Lame's Equation
567(44)
23.1 The definition of ellipsoidal harmonics
567(1)
23.2 The four species of ellipsoidal harmonics
568(1)
23.21 The construction of ellipsoidal harmonics of the first species
568(3)
23.22 Ellipsoidal harmonics of the second species
571(1)
23.23 Ellipsoidal harmonics of the third species
572(1)
23.24 Ellipsoidal harmonics of the fourth species
573(1)
23.25 Niven's expressions for ellipsoidal harmonics in terms of homogeneous harmonics
574(3)
23.26 Ellipsoidal harmonics of degree n
577(1)
23.3 Confocal coordinates
578(2)
23.31 Uniformising variables associated with confocal coordinates
580(2)
23.32 Laplace's equation referred to confocal coordinates
582(2)
23.33 Ellipsoidal harmonics referred to confocal coordinates
584(1)
23.4 Various forms of Lame's differential equation
585(2)
23.41 Solutions in series of Lame's equation
587(2)
23.42 The definition of Lame functions
589(1)
23.43 The non-repetition of factors in Lame functions
590(1)
23.44 The linear independence of Lame functions
590(1)
23.45 The linear independence of ellipsoidal harmonics
591(1)
23.46 Stieltjes' theorem on the zeros of Lame functions
591(2)
23.47 Lame functions of the second kind
593(1)
23.5 Lame's equation in association with Jacobian elliptic functions
594(1)
23.6 The integral equation for Lame functions
595(2)
23.61 The integral equation satisfied by Lame functions of the third and fourth species
597(1)
23.62 Integral formulae for ellipsoidal harmonics
598(2)
23.63 Integral formulae for ellipsoidal harmonics of the third and fourth species
600(1)
23.7 Generalisations of Lame's equation
601(3)
23.71 The Jacobian form of the generalised Lame equation
604(3)
23.8 Miscellaneous examples
607(4)
Appendix. The Elementary Transcendental Functions
611(14)
A.1 On certain results assumed in
Chapters 1 to 4
611(2)
A.11 Summary of the Appendix
612(1)
A.12 A logical order of development of the elements of analysis
612(1)
A.2 The exponential function exp z
613(2)
A.21 The addition-theorem for the exponential function, and its consequences
613(1)
A.22 Various properties of the exponential function
614(1)
A.3 Logarithms of positive numbers
615(2)
A.31 The continuity of the Logarithm
616(1)
A.32 Differentiation of the Logarithm
616(1)
A.33 The expansion of Log(1 + a) in powers of a
616(1)
A.4 The definition of the sine and cosine
617(2)
A.41 The fundamental properties of sin z and cos z
618(1)
A.42 The addition-theorems for sin z and cos z
618(1)
A.5 The periodicity of the exponential function
619(4)
A.51 The solution of the equation exp γ =1
619(2)
A.52 The solution of a pair of trigonometrical equations
621(2)
A.6 Logarithms of complex numbers
623(1)
A.7 The analytical definition of an angle
623(2)
References 625(23)
Author index 648(4)
Subject index 652
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.