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Whittaker E.T., Watson G.N. — A Course of Modern Analysis

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Название: A Course of Modern Analysis

Авторы: Whittaker E.T., Watson G.N.

Аннотация:

This classic text has entered and held the field as the standard book on the applications of analysis to the transcendental functions. The authors explain the methods of modern analysis in the first part of the book and then proceed to a detailed discussion of the transcendental function, unhampered by the necessity of continually proving new theorems for special applications. In this way the authors have succeeded in being rigorous without imposing on the reader the mass of detail that so often tends to make a rigorous demonstration tedious. Researchers and students will find this book as valuable as ever.

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Рубрика: Математика/Анализ/Продвинутый анализ/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: Fourth Edition

Год издания: 1927

Количество страниц: 608

Добавлена в каталог: 02.04.2005

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Предметный указатель

Schlaefli's integral, modified, for Gegenbauer's function      329
Schloemilch's expansion in series of Bessel coefficients      377
Schloemilch's function      352
Schloemilch's integral equation      229
Schmidt's theorem      223
Schwartz' lemma      186
Second kind, Bessel function of, (Hankel's)      370
Second kind, Bessel function of, (modified)      373
Second kind, Bessel function of, (Neumann's)      372
Second kind, Bessel function of, (Weber — Schlaefli)      370
Second kind, elliptic integral of [E(u), Z(u)]      517
Second kind, elliptic integral of [E(u), Z(u)], complete      518
Second kind, Eulerian integral of      241
Second kind, Eulerian integral of, extended      244
Second kind, integral equation of      213 221
Second kind, Lame functions of      562
Second kind, Legendre functions of      316—320 325 326
Second mean-value theorem      66
Second solution of Bessel's equation      370 372
Second solution of Bessel's equation, modified      373
Second solution of Legendre's equation      316
Second solution of Mathieu's equation      413 427
Second solution of the hypergeometric equation      286
Second solution of the hypergeometric equation, confluent form      343
Second solution of Weber's equation      347
Second species of ellipsoidal harmonics      537
Second species of ellipsoidal harmonics, construction of      540
Section      4
Selffert's spherical spiral      527
Sequences      11
Sequences, decreasing      12
Sequences, increasing      12
Series (infinite series)      15
Series of analytic functions      91
Series of cosines      165
Series of cotangents      139
Series of inverse factorials      142
Series of powers      see Power series
Series of rational functions      134
Series of sines      166
Series of variable terms      44. See also Uniformity of convergence
Series, absolutely convergent      18
Series, change of order of terms in      25
Series, conditionally convergent      18
Series, convergence of      15
Series, differentiation of      31 79 92
Series, divergence of      15
Series, geometric      19
Series, integration of      32 78
Series, methods of summing      154—156
Series, multiplication of      29 58 59
Series, order of terms in      25
Series, remainder of      15
Series, representing particular functions      see under the name of the function
Series, solutions of differential and integral equations in      194—202 228
Series, Taylor's      93. See also Asymptotic expansions Convergence Expansions Fourier Trigonometrical
Set, irreducible (of zeros or poles)      430
Sigma-functions of Weieratrass [ $\sigma(z)$ , $\sigma_1(z)$ , $\sigma_2(z)$ , $\sigma_3(z)$ ]      447 448
Sigma-functions, addition formula for      451 458 460
Sigma-functions, analogy with circular functions      447
Sigma-functions, duplication formulae      459 460
Sigma-functions, expression of elliptic functions by      450
Sigma-functions, four types of      448
Sigma-functions, quasi-periodic properties      447
Sigma-functions, singly infinite product for      448
Sigma-functions, Theta-functions connected with      448 473 487
Sigma-functions, three-term equation involving      4o"i 461
Sigma-functions, triplication formula      459
Signless integers      3
Simple curve      43
Simple pole      102
Simple zero      94
Simply-connected region      455
Sine, product for      137. See also Circular functions
Sine-integral [Si(z)]      352
Sine-series (Fourier series)      166
Singly-periodic functions      429. See also Circular functions
Singular points (singularities) of linear differential equations      194 202
Singularities      83 84 102 194 197 202
Singularities at infinity      104
Singularities, confluence of      203 337
Singularities, equations with five      203
Singularities, equations with r      209
Singularities, equations with three      206 S10
Singularities, equations with two      208
Singularities, essential      102 104
Singularities, irregular      197 202
Singularities, regular      197
Solid harmonics      392
Solution of Riemann's P-equation by hypergeometric functions      283 288
Solutions of differential equations      see Chapters X XVIII XXIII and
Solutions of integral equations      see Chapter XI
Sonine's polynomial []      352
Species (various) of ellipsoidal harmonics      537
Spherical harmonics      see Harmonics
Spherical spiral, Seiffert's      527
Spheroidal harmonics      403
Squares of Bessel fractions      379 380
Squares of hypergeometric functions      298
Squares of Jacobian elliptic functions (relations between)      492
Squares of Theta-functions (relations between)      488
Statement of Fourier's theorem, Dirichlets's      181 168 194 176
Steadily tending to zero      17
Stieltjes' theorem on zeros ol Lame functions      560
Stieltjes' theorem on zeros ol Lame functions, generalised      562
Stieltjes' theory of Mattieu's equation      417
Stirling's series lor the Gamma-function      361
Stokes' equation      304
Stolz' condition for convergence of double series      47
Strings, vibrations of      160
Successive substitutions, method of      221
Sum-formula of Euler and Maclaurin      127
Summability of Fourier series      169
Summability, methods of      154—158
Summability, uniform      156
Surface harmonic      392
Surface, modular      41
Surfaces, nearly spherical      332
Sylvester's theorem concerning integrals of harmonics      400
Symmetric nucleus      223 228
Tabulation of Bessel functions      378
Tabulation of complete elliptic integrals      518
Tabulation of Gamma functions      253
Taylor's series      93
Taylor's series, failure of      100 104 110
Taylor's series, remainder in      95
Teaseral harmonics      392
Teaseral harmonics, factorisation of      536
Teixeira's extension of Buermann's theorem      131
Telegraphy, equation of      387
Tests for convergence      see Infinite integrals Infinite
Thermometric parameter      405
Theta-function [ $\mathcal J_1$ , $\mathcal J_2$ , $\mathcal J_3$ , $\mathcal J_4,$ \Theta(u)$]      462—490 (Chapter XXI)
Theta-function, abridged notation for products      468 469
Theta-function, addition formulas      487
Theta-function, connexion with Sigma-functions      448 473 487
Theta-function, duplication formulae      488
Theta-function, expression of elliptic functions by      418
Theta-function, four types of      463
Theta-function, fundamental formulae (Jacobi's)      467 488
Theta-function, infinite products for      469 473 488
Theta-function, Jacobi's first notation, $\Theta(u)$ and       479
Theta-function, multipliers      463
Theta-function, notations      464 479 487
Theta-function, parameters q, $\tau$       463
Theta-function, partial differential equation satisfied by      470
Theta-function, periodicity factors      463
Theta-function, periods      463
Theta-function, quotients of      477
Theta-function, quotients yielding Jacobian elliptic functions      478

Theta-function, relation $\mathcal J_1'=\mathcal J_2\mathcal J_3\mathcal J_4$       470
Theta-function, squares of (relations between)      466
Theta-function, transformations of (Jacobi's imaginary)      124 474
Theta-function, transformations of (Landen's)      476
Theta-function, triplication formulae for      490
Theta-function, with zero argument      464
Theta-function, zeros of      465
Third kind of elliptic integral, $\Pi(u, a)$       522
Third kind, a dynamical application of      52.3
Third order, linear differential equations of      210 298 418 428
Third species of ellipsoidal harmonics      537
Third species of ellipsoidal harmonics, construction of      541
Three kinds of elliptic integrals      514
Three-term equation involving Sigma-functions      451 461
Total fluctuation      57
Transcendental functions      see under the names of special functions
Transformations of elliptic functions and Theta-functions      508
Transformations, Jacobi's imaginary      474 504 506 519
Transformations, Landen's      476 507
Transformations, of Riemann's P-equation      207
Trigonometrical equations      587 588
Trigonometrical integrals      112 263
Trigonometrical integrals and Gamma-functions      388
Trigonometrical series      160—193 (Chapter IX)
Trigonometrical series, convergence of      181
Trigonometrical series, Riemann's theory of      182—188
Trigonometrical series, values of coefficients in      163
Trigonometrical series, which are not Fourier series      160 168.
Triplication formulae for Jaoobian elliptic functions and E(u)      530 534
Triplication formulae for Sigma-functions      459
Triplication formulae for Theta-functions      490
Triplication formulae for Zeta-funetiona      459
Twenty-four solutions of the hypergeometric equation      284
Twenty-four solutions, relations between      285 288 290
Two variables, continuous functions of      67
Two variables, hypergeometric function (Appell's) of      300
Two-dimensional continuum      43
Types of ellipsoidal harmonics      537
Unicursal      455
Uniformisation      454
Uniformising variables      455
Uniformising variables, associated with confocal coordinates      549
Uniformity of continuity      54
Uniformity of convergence      41—60 (Chapter III) defined 44
Uniformity of Fourier series      172 179 180
Uniformity of infinite integrals      70 72 73
Uniformity of infinite products      49
Uniformity of power series      57
Uniformity of series      44
Uniformity of series, condition for      45
Uniformity of series, Hardy's test for      50
Uniformity of series, Weierstrass's M-test for      49
Uniformity of summability      156
Uniformity, concept of      52
Uniformly convergent infinite integrals, properties of      73
Uniformly convergent series of analytic functions      91
Uniformly convergent series of analytic functions, differentiation of      92
Uniqueness of an asymptotic expansion      153
Uniqueness of solutions of linear differential equations      196
Upper bound      55
Upper integral      61
Upper limit, integral equation with variable      213 221
Upper to the value of a complex integral      78 91
Value of $\zeta(z)$ for special values of s      267 269
Value of Jacobian elliptic functions of $\frac12K$ , $\frac12iK'$ , $\frac12(K+iK')$       500 506 507
Value of K, K' for special values of k      521 524 525
Value of particular hypergeometric functions      281 293 298 301
Value of the argument of a complex number      9 588
Value of the coefficients in Fourier series and trigonometrical series      163 165 167 174
Value, absolute      see Modulus
Vanishing of power series      58
Variable, terms (series of)      see Uniformity of convergence
Variable, uniformising      455
Variable, upper limit, integral equation with      213 221
Vibrations of air in a sphere      399
Vibrations of circular membranes      396
Vibrations of elliptic membranes      404 405
Vibrations of strings      160
Volterra's integral equation      221
Volterra's reciprocal functions      218
Wave motions, equation of      386
Wave motions, general solution      397 402
Wave motions, solution involving Bessel functions      397
Weber's Bessel function of the second kind []      370
Weber's equation      204 209 342 347.
Weierstrass' factor theorem      137
Weierstrass' if-test for uniform convergence      49
Weierstrass' product for the Gamma-function      235
Weierstrass' theorem on limit points      12
Weierstrassian elliptic function      429—461 (Chapter XX)
Weierstrassian elliptic function, addition theorem for      440
Weierstrassian elliptic function, addition theorem for, Abel's method      442
Weierstrassian elliptic function, analogy with circular functions      438
Weierstrassian elliptic function, defined and constructed      432 433
Weierstrassian elliptic function, definition of $\{W(z)-e_r\}^{\frac12}$       451
Weierstrassian elliptic function, differential equation for      436
Weierstrassian elliptic function, discriminant of      444
Weierstrassian elliptic function, duplication formula      441
Weierstrassian elliptic function, expression of by Sigma-functions      451
Weierstrassian elliptic function, expression of elliptic functions by      448
Weierstrassian elliptic function, half-periods      444
Weierstrassian elliptic function, homogeneity properties      439
Weierstrassian elliptic function, integral formula for      437
Weierstrassian elliptic function, integration of irrational functions by      452
Weierstrassian elliptic function, invariants of      437
Weierstrassian elliptic function, inversion problem for      484
Weierstrassian elliptic function, Jacobian elliptic functions and      505
Weierstrassian elliptic function, periodicity      434
Weierstrassian elliptic function, roots , ,       443. See also Sigma-functions and Zeta-function (of Weierstrass)
Whittaker's function $W_{k, m}(z)$       see Confluent hypergeometric functions
Wronski's expansion      147
Zero argument, Theta-functions with      464
Zero argument, Theta-functions with, relation between      470
Zero of a function      94
Zero of a function, at infinity      104
Zero of a function, simple      94
Zeros of a function and poles (relation between)      433
Zeros of a function, connected with zeros of its derivate      121 123
Zeros of a function, irreducible set of      430
Zeros of a function, number of, in a cell      431
Zeros of a function, order of      94
Zeros of functions, Bessel's      361 367 378 381
Zeros of functions, Lame's      557 558 560 578
Zeros of functions, Legendre's      303 316 335
Zeros of functions, parabolic cylinder      354
Zeros of functions, Riemann's Zeta-      268 269 272 280
Zeros of functions, Theta-      465
Zeta-function (of Riemann), Euler's product for      271
Zeta-function (of Riemann), generalised by Hurwitz      265
Zeta-function (of Riemann), Hermite's integral for      269
Zeta-function (of Riemann), Hurwitz' integral for      268
Zeta-function (of Riemann), inequalities satisfied by      274 275
Zeta-function (of Riemann), logarithmic derivate of      279
Zeta-function (of Riemann), Riemann's hypothesis concerning      272 280
Zeta-function (of Riemann), Riemann's integrals for      266 273
Zeta-function (of Riemann), Riemann's relation connecting $\zeta(s)$ and $\zeta(1-s)$       269
Zeta-function (of Riemann), values of, for special values of s      267 269
Zeta-function (of Riemann), zeros of      268 269 272 280
Zeta-function (of Weierstrass), addition formula      446
Zeta-function (of Weierstrass), analogy with circular functions      446
Zeta-function (of Weierstrass), constants $\eta_1$ , $\eta_2$ connected with      446
Zeta-function (of Weierstrass), duplication formulae for      459
Zeta-function (of Weierstrass), expression of elliptic functions by      449
Zeta-function (of Weierstrass), quasi-periodicity      445
Zeta-function (of Weierstrass), triplication formulae      459. See also Weierstrassian elliptic functions
Zeta-function, $\zeta(s)$ ), (of Weierstrass)      445
Zeta-function, $\zeta(s)$ , $\zeta(s, a)$ , (of Riemann)      265—280 (Chapter XIII)
Zeta-function, Z(u), (of Jacobi)      518
Zeta-function, Z(u), (of Jacobi), addition formula for      518
Zeta-function, Z(u), (of Jacobi), connexion with E(u)      518

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