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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.

Язык:

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

Abel's discovery of elliptic functions      429 512
Abel's inequality      16
Abel's integral equation      211 229 230
Abel's method of establishing addition theorems      442 496 497 530 534
Abel's special form, $\phi(m)$ , of the confluent hypergeometric function      353
Abel's test for convergence      17
Abel's theorem on continuity of power series      57
Abel's theorem on multiplication of convergent series      58 59
Abridged notation for products of Theta-functions      468 469
Abridged notation for quotients and reciprocals of elliptic functions      494 498
Absolute convergence      18 28
Absolute convergence, Cauchy's test for      21
Absolute convergence, D'Alembert's ratio test for      22
Absolute convergence, De Morgan's test for      23
Absolute value      see Modulus
Absolutely convergent double series      28
Absolutely convergent infinite products      32
Absolutely convergent series      18
Absolutely convergent, fundamental property of      25
Absolutely convergent, multiplication of      29
Addition formula for Bessel functions      357 380
Addition formula for Gegenbauer's function      335
Addition formula for Legendre functions      328
Addition formula for Legendre polynomials      326 395
Addition formula for the Jacobian Zeta-function and for E(u)      518 534
Addition formula for the Sigma-function      451
Addition formula for the third kind of elliptic integral      523
Addition formula for the Weierstrassian Zeta-function      446
Addition formula for Theta-functions      467
Addition formulae, distinguished from addition theorems      519
Addition theorem for circular functions      535
Addition theorem for Jacobian elliptic functions      494 497 530
Addition theorem for the exponential function      531
Addition theorem for the Weierstrassian elliptic function      440 457
Addition theorem, proofs of, by Abel's method      442 496 497 530 534
Affix      9
Air in a sphere, vibrations of      399
Amplitude      9
Analytic continuation      96
Analytic continuation and Borel's integral      141
Analytic continuation, not always possible      98
Analytic continuation, of the hypergeometric function      288.See also Asymptotic expansions
Analytic functions      82—110 (Chapter V)
Analytic functions, defined      83
Analytic functions, derivates of      89
Analytic functions, distinguished from monogenic functions      99
Analytic functions, inequality satisfied by      91
Analytic functions, represented by integrals      92
Analytic functions, Riemann's equations connected with      84
Analytic functions, uniformly convergent series of      91
Analytic functions, values of, at points inside a contour      88
Angle and popular conception of an angle      589 590
Angle, analytical definition of      589
Angle, modular      492
Area represented by an integral      61 589
Argand diagram      9
Argument      9 588
Argument, continuity of      588
Argument, principal value of      9 588
Associated function of Borel      141
Associated function of Legendre and       323—326
Associated function of Riemann      183
Asymptotic expansions      150—159 (Chapter VIII)
Asymptotic expansions of Bessel functions      368 369 371 373 374
Asymptotic expansions of confluent hypergeometric functions      342 343
Asymptotic expansions of Gamma-functions      251 276
Asymptotic expansions of parabolic cylinder functions      347 348.
Asymptotic expansions, differentiation of      153
Asymptotic expansions, integration of      153
Asymptotic expansions, multiplication of      152
Asymptotic expansions, uniqueness of      153 154
Asymptotic inequality for parabolic cylinder functions of large order      354
Asymptotic solutions of Mathieu's equation      425
Auto-functions      226
Automorpblc functions      455
Axioms of arithmetic and geometry      579
Barnes' contour integrals for the confluent hypergeometric function      343—345
Barnes' contour integrals for the hypergeometric function      286 289
Barnes' G-function      264 279
Barnes' Lemma      289
Basic numbers      462
Bernoullian numbers      125
Bernoullian polynomials      126 127
Bertrand's test for convergence of infinite integrals      71
Bessel coefficients []      101 355
Bessel coefficients, addition formulae for      357
Bessel coefficients, Bessel's integral for      362
Bessel coefficients, by Schloemilch      377
Bessel coefficients, differential equation satisfied by      357
Bessel coefficients, expansion of $(t-z)^{-1}$ in series of      374 375 376
Bessel coefficients, expansion of functions in series of (by Neumann)      374 375 384
Bessel coefficients, expansion of, as power series      355
Bessel coefficients, expressible as a confluent form of Legendre functions      367
Bessel coefficients, expressible as confluent hypergeometrie functions      358
Bessel coefficients, inequality satisfied by      379
Bessel coefficients, Neumann's function connected with      see Neumann's function
Bessel coefficients, order of      356
Bessel coefficients, recurrence formulae for      359
Bessel coefficients, special case of confluent hypergeometrie functions      358. See also Bessel functions
Bessel functions      355—385 (Chapter XVII)
Bessel functions, defined      358—360
Bessel functions, $Y^{(n)}$ (Neumann)      372
Bessel functions, (Weber — Schlaefli)      370
Bessel functions, addition formulae for      380
Bessel functions, asymptotic expansion of      368 369 371 373 374
Bessel functions, expansion of functions in series of      374 375 377 381
Bessel functions, expansion of, as an ascending series      358 371
Bessel functions, first kind of      359
Bessel functions, Hankel's integral for      365
Bessel functions, integral connecting Legendre functions with      364 401
Bessel functions, integral properties of      380 381 384 385
Bessel functions, integrals involving products of      380 383 385
Bessel functions, notations for      356 372 373
Bessel functions, order of      356
Bessel functions, products of      379 380 383 385 428
Bessel functions, recurrence formulae for      359 373 374
Bessel functions, relation between Gegenbauer's function and      378
Bessel functions, relations between      360 371 372
Bessel functions, Schlaefli's form of Bessel's integral for      362 372
Bessel functions, second kind of modified,       373
Bessel functions, second kind of, (Hankel)      370
Bessel functions, solution of Laplace's equation by      396
Bessel functions, solution of the wave-motion equation by      397
Bessel functions, tabulation of      378
Bessel functions, whose order is half an odd integer      364
Bessel functions, whose order is large      368 383
Bessel functions, with imaginary argument, ,       372 373 384
Bessel functions, zeros of      361 367 378 381.
Bessel's equation      204 357 373
Bessel's equation, fundamental system of solutions of (when n is not an integer)      359 372
Bessel's equation, second solution when n is an integer      370 373.
Binomial theorem      95
Bluet's Integrals for $\log\Gamma(z)$       248—251
Bocher's theorem on linear differential equations with five singularities      203
Bolzano's theorem on limit points      12
Bonnet's form of the second mean value theorem      66
Borel's associated function      141
Borel's integral      140
Borel's integral and analytic continuation      141
Borel's method of “summing” series      154
Borel's theorem (the modified Heine — Borel theorem)      53
Boundary      44
boundary conditions      387
Boundary conditions and Laplace's equation      393
Bounds of continuous functions      55
Branch of a function      106
Branch-point      106
Buermann's theorem      128
Buermann's theorem, extended by Teixeira      131
Cantor's Lemma      183

Cauchy's condition for the existence of a limit      13
Cauchy's discontinuous factor      123
Cauchy's formula for the remainder in Taylor's series      96
Cauchy's inequality for derivatives of an analytic function      91
Cauchy's integral      119
Cauchy's integral representing $\Gamma(z)$       243
Cauchy's numbers      372
Cauchy's tests for convergence of series and integrals      21 71
Cauchy's theorem      85
Cauchy's theorem, extension to curves on a cone      87
Cauchy's theorem, Morera's converse of      87 110
cell      430
Cesaro's method of “summing” series      155
Cesaro's method of “summing” series, generalised      156
Change of order of terms in a series      25
Change of order of terms in an infinite determinant      37
Change of order of terms in an infinite product      33
Change of parameter (method of solution of Mathieu's equation)      424
Characteristic functions      226
Characteristic numbers      219
Characteristic numbers associated with symmetric nuclei are real      226
Chartier's test for convergence of infinite integrals      72
Circle of convergence      30
Circle, area of sector of      589
Circle, limiting      98
Circular functions      435 584
Circular functions, addition theorems for      585
Circular functions, continuity of      585
Circular functions, differentiation of      585
Circular functions, duplication formulae      585
Circular functions, periodicity of      587
Circular functions, relation with Gamma-functions      239
Circular membrane, vibrations of      356 396
Class, left (L)      4
Class, right (R)      4
Closed      44
Cluster-point      13
Coefficients in Fourier series, nature of      167 174
Coefficients in trigonometrical series, values of      163 165
Coefficients of Bessel      see Bessel coefficients
Coefficients, equating      59
Comparison theorem for convergence of integrals      71
Comparison theorem for convergence of series      20
Complementary moduli      479 493
Complementary moduli, elliptic integrals with      479 501 520
Complete elliptic integrals [E, K, E', K'] (first and second kinds)      498 499 518
Complete elliptic integrals, Legendre's relation between      520
Complete elliptic integrals, properties of (qua functions of the modulus)      484 498 499 501 521
Complete elliptic integrals, series for      299
Complete elliptic integrals, tables of      518
Complete elliptic integrals, the Gaussian transformation      533
Complete elliptic integrals, values (as Gamma-functions) for special values of k      524—527
Complete elliptic integrals, values for small values of |k|      521
Complete elliptic integrals, with complementary moduli      479 501 520
Complex Integrals      77
Complex Integrals, upper limit to value of      78
Complex Integration, fundamental theorem of      78
complex numbers      3—10 (Chapter I)
Complex numbers, amplitude of      9
Complex numbers, argument of      9 588
Complex numbers, defined      6
Complex numbers, dependence of one on another      41
Complex numbers, imaginary part of (I)      9
Complex numbers, logarithm of      589
Complex numbers, modulus of      8
Complex numbers, real part of (R)      9
Complex numbers, representative point of      9
Complex variable, continuous function of a      44
Computation of elliptic functions      485
Computation of solutions of integral equations      211
Condition of integrability (Riemann's)      63
Conditional convergence of infinite determinants      415. See also Convergence and Absolute convergence
Conditional convergence of series      18
Conditions, Dirichlet's      161 163 164 176
Conduction of Heat, equation of      387
Confluence      202 337
Confluent form      203 337
Confluent hypergeometric function [ $W_{k, m}(z)$ ]      337—354 (Chapter XV)
Confluent hypergeometric function, equation for      337
Confluent hypergeometric function, general asymptotic expansion of      342 345
Confluent hypergeometric function, integral defining      339
Confluent hypergeometric function, integrals of Barnes' type for      343—345
Confluent hypergeometric function, Kummer's formulae for      338
Confluent hypergeometric function, recurrence formulae for      352
Confluent hypergeometric function, relations with Bessel functions      360
Confluent hypergeometric function, the functions $W_{k, m}(z)$ and $M_{k, m}(z)$       337—339
Confluent hypergeometric function, the relations between functions of these types      346
Confluent hypergeometric function, various functions expressed in terms of $W_{k, m}(z)$       340 352 353 360.
Confocal coordinates      405 547
Confocal coordinates in association with ellipsoidal harmonics      552
Confocal coordinates, form a triply orthogonal system      648
Confocal coordinates, Laplace's equation referred to      551
Confocal coordinates, uniformising variables associated with      549
Congruence of points in the Argand diagram      430
Constant, Euler's or Mascheroni's, [ $\gamma$ ]      235 246 248
Constants , ,       443
Constants $\eta_1$ , $\eta_2$       171 172 446
Constants E, E'      518 520
Constants of Fourier      164
Constants, G      469 472
Constants, K      484 498 499
Constants, K'      484 501 503
Constants, relation between $\eta_1$ and $\eta_2$       446
Construction of elliptic functions      433 478 492
Construction of Mathieu functions      409
Construction of Mathieu functions, second method      420
Contiguous hypergeometric functions      294
Continua      43
Continuants      36
Continuation, analytic      96
Continuation, analytic and Borel's integral      141
Continuation, analytic of the hypergeometric function      288. See also Asymptotic expansions
Continuation, analytic, not always possible      98
Continuity      41
Continuity of power series      57
Continuity of power series, Abel's theorem      57
Continuity of the argument of a complex variable      588
Continuity of the circular functions      585
Continuity of the exponential function      581
Continuity of the logarithmic function      583 589
Continuity, uniformity of      54
Continuous functions      41—60 (Chapter III)
Continuous functions, bounds of      55
Continuous functions, defined      41
Continuous functions, integrability of      63
Continuous functions, of a complex variable      44
Continuous functions, of two variables      67
contour      85
Contour integrals      85
Contour integrals, evaluation of definite integrals by      112—124
Contour integrals, the Mellin — Barnes type of      286 343.
Contour, roots of an equation in the interior of a      119 123
Convergence      11—40 (Chapter II)
Convergence of a double series      27
Convergence of a series      15
Convergence of a series, Abel's test for      17
Convergence of a series, Dirichlet's test for      17
Convergence of an infinite determinant      36
Convergence of an infinite integral      70
Convergence of an infinite product      32
Convergence of Fourier series      174—179
Convergence of the geometric series      19
Convergence of the hypergeometric series      24
Convergence of the series $\sum n^{-2}$       19
Convergence of the series occurring in Mathieu functions      422
Convergence of trigonometrical series      161
Convergence, circle of      30
Convergence, conditional      18
Convergence, defined      13 15
Convergence, principle of      13

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