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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

Linear differential equations, irregular singularities of      197 202
Linear differential equations, ordinary point of      194
Linear differential equations, regular integral of      201
Linear differential equations, regular point of      197
Linear differential equations, singular points of      194 197
Linear differential equations, singular points of, confluence of      202
Linear differential equations, solution of      194 197
Linear differential equations, solution of, uniqueness of      196
Linear differential equations, special types of equations: Bessel's for circular cylinder functions      204 342 357 358 373
Linear differential equations, special types of equations: Gauss' for hypergeometric functions      202 207 283
Linear differential equations, special types of equations: Gegenbauer's      329
Linear differential equations, special types of equations: Heroiite's      204 209 342 347
Linear differential equations, special types of equations: Hill's      406 413
Linear differential equations, special types of equations: Jacobi's for Theta-functions      463
Linear differential equations, special types of equations: Lame's      204 540—543 554—558 570—575
Linear differential equations, special types of equations: Laplace's      386 388 536 551
Linear differential equations, special types of equations: Legendre's for zonal and surface harmonics      204 304 324
Linear differential equations, special types of equations: Mathieu's for elliptic cylinder functions      204 406
Linear differential equations, special types of equations: Neumann's      385
Linear differential equations, special types of equations: Riemann's for P-functions      206 283 291 294
Linear differential equations, special types of equations: Stokes'      204
Linear differential equations, special types of equations: Weber's for parabolic cylinder functions      204 209 342 347
Linear differential equations, special types of equations: Whittaker's for confluent hypergeometric functions      337
Lioiville's method of solving integral equations      221
Liouville's theorem      105 431
Logarithm      583
Logarithm of complex numbers      589
Logarithm, continuity of      583 589
Logarithm, differentiation of      586 589
Logarithm, expansion of      584 589
Logarithmic derivate of the Gamma-function [ $\Gamma(z)$ ]      240 241
Logarithmic derivate of the Riemann Zeta-function      379
Logarithmic derivate, Binet's integrals for      248—251
Logarithmic derivate, circular functions and      240
Logarithmic derivate, Dirichlet's integral for      247
Logarithmic derivate, Gauss' integral for      246
Logarithmic-Integral function [Li z]      341
Lower integral      61
Lunar perigee and node, motions of      406
M-test for uniformity of convergence      49
Maclaurin's (and Euler's) expansion      127
Maclaurin's series      94
Maclaurin's series, failure of      104 110
Maclaurin's test for convergence of infinite integrals      71
Many-valued functions      106
Mascheroni's constant [ $\gamma$ ]      235 246 248
Mathematical Physics, equations of      203 386—403
Mathieu functions [, , ]      404—428 (Chapter XIX)
Mathieu functions, construction of      409 420
Mathieu functions, convergence of series in      422
Mathieu functions, even and odd      407
Mathieu functions, expansions as Fourier series      409 411 420
Mathieu functions, integral equations satisfied by      407 409
Mathieu functions, integral formulae      411
Mathieu functions, order of      410
Mathieu functions, second kind of      427
Mathieu's equation      204 404—428
Mathieu's equation, by Lindemann and Stieltjes      417
Mathieu's equation, by the method of change of parameter      424
Mathieu's equation, general form, solutions by Floquet      412
Mathieu's equation, second solution of      413 420 427
Mathieu's equation, solutions in asymptotic series      425
Mathieu's equation, solutions which are periodic      see Mathieu functions
Mathieu's equation, the integral function associated with      418. See also Hill's equation
Mean-value theorems      65 66 96
Mehler's integral for Legendre functions      314
Mellin's (and Barnes') type of contour integral      286 343
Membranes, vibrations of      356 396 404 405
mesh      430
Methods of “summing” series      154—156
Minding's formula      119
Minimum value of $\Gamma(z)$       253
Modified Heine — Borel theorem      53
Modular angle      492
Modular angle, -surface      41
Modular angle, function      481
Modular angle, function, equation connected with      482
Modulus      430
Modulus of a complex number      8
Modulus of Jacobian elliptic functions      479 492
Modulus of Jacobian elliptic functions, complementary      479 493
Modulus, periods of elliptic functions regarded as functions of the      484 498 499 501 521
Monogenic      83
Monogenic distinguished from analytic      99
Monotonlc      57
Morera's theorem (converse of Cauchy's theorem)      87 110
Motions of lunar perigee and node      406
Multiplication formula for $\Gamma(z)$       240
Multiplication formula for the Sigma-function      460
Multiplication of absolutely convergent series      29
Multiplication of asymptotic expansions      152
Multiplication of convergent series (Abel's theorem)      58 59
Multipliers of Theta-functions      463
Murphy's formulae for Legendre functions and polynomials      311 312
Neumann's (F. E. Neumann's) integral for the Legendre function of the second kind      320
Neumann's definition of Bessel functions of the second kind      372
Neumann's expansions in series of Legendre and Bessel functions      322 374
Neumann's function []      374
Neumann's function, differential equation satisfied by      385
Neumann's function, expansion of      374
Neumann's function, expansion of functions in series of      376 384
Neumann's function, integral for      375
Neumann's function, integral properties of      385
Neumann's function, recurrence formulae for      375
Neumann's method of solving integral equations      221
Non-uniform convergence      44
Non-uniform convergence and discontinuity      47
Normal functions      224
Normal solutions of Laplace's equation      553
Notations for Bessel functions      356 372 373
Notations for Legendre functions      325 326
Notations for quotients and reciprocals of elliptic functions      494 498
Notations for Theta-functions      464 479 487
Nucleus of an integral equation      213
Nucleus, symmetric      223 228
Numbers      3—10 (Chapter I)
Numbers, basic      462
Numbers, Bernoulli's      125
Numbers, Cauchy's      379
Numbers, characteristic      219
Numbers, characteristic, complex      6
Numbers, characteristic, reality of      226
Numbers, irrational      6
Numbers, irrational-real      5
Numbers, pairs of      6
Numbers, rational      3 4
Numbers, rational-real      5
Numbers, real      5
Odd functions      115 166
Odd functions of Mathieu, []      407
open      44
Order (O and o)      11
Order of Bernoullian polynomials      126
Order of Bessel functions      356
Order of elliptic functions      432
Order of Legendre functions      324
Order of Mathieu functions      410
Order of poles of a function      102
Order of terms in a series      25
Order of the factors of a product      33
Order of zeros of a function      94
Ordinary discontinuity      42
Ordinary point of a linear differential equation      194
Orthogonal coordinates      394
Orthogonal functions      224
Oscillation      11
P-equation, Riemann's      206 337
P-equation, Riemann's, connexion with the hypergeometric equation      208 283
P-equation, Riemann's, solutions of      283 291
P-equation, Riemann's, solutions of, relations between      294

P-equation, Riemann's, transformations of      207
Parabolic cylinder functions []      347
Parabolic cylinder functions, contour integral for      349
Parabolic cylinder functions, differential equation for      204 209 347
Parabolic cylinder functions, expansion in a power series      347
Parabolic cylinder functions, expansion of a function as a series of      351
Parabolic cylinder functions, general asymptotic expansion of      348
Parabolic cylinder functions, inequalities satisfied by      354
Parabolic cylinder functions, integral equation satisfied by      231
Parabolic cylinder functions, integral properties      350
Parabolic cylinder functions, integrals involving      353
Parabolic cylinder functions, integrals representing      353
Parabolic cylinder functions, properties when n is an integer      350 353 354
Parabolic cylinder functions, recurrence formulae      350
Parabolic cylinder functions, relations between different kinds of [ and $D_{-n-1}(\pm iz)$ )]      348
Parabolic cylinder functions, zeros of      354. See also Weber's equation
Parallelogram of periods      430
Parameter of a point on a curve      442 496 497 527 530 533
Parameter of members of confocal systems of quadrics      547
Parameter of third kind of elliptic integral      522
Parameter, change of (method of solving Mathieu's equation)      424
Parameter, connected with Theta-functions      463 464
Parameter, thermometric      405
Parseval's theorem      182
Partial differential equations, property of      390 391.
Partition function      462
Parts, real and imaginary      9
Pearson's function [ $\omega_{n, m}(z)$ ]      353
Period-parallelogram      430
Period-parallelogram, fundamental      430
Periodic coefficients, equations with (Floquet's theory of)      412
Periodic functions, integrals involving      112 256.
Periodic solutions of Mathieu's equation      407
Periodicity factors      463
Periodicity of circular and exponential functions      585—587
Periodicity of elliptic functions      429 434 479 500 502 503
Periodicity of Theta-functions      463
Periods of elliptic functions      429
Periods of elliptic functions, qua functions of the modulus      484 498 499 501 521
Phase      9
Pincherle's functions (modified Legendre functions)      335
Plana's expansion      145
Pochhammer's extension of Eulerian integrals      256
Point, at infinity      103
Point, limit-      12
Point, representative      9
Point, singular      194 202
Poles of a function      102
Poles of a function, at infinity      104
Poles of a function, irreducible set of      430
Poles of a function, number in a cell      431
Poles of a function, relations between zeros of elliptic functions and      433
Poles of a function, residues at      432 504
Poles of a function, simple      102
Polygon, (fundamental) of automorphic functions      455
Polynomials of Abel      353
Polynomials of Bernoulli      126 127
Polynomials of Legendre      see Legendre polynomials
Polynomials of Sonine      352
Polynomials, expressed as series of Legendre polynomials      310
Popular conception of an angle      589
Popular conception of continuity      41
Positive integers      3
Power series      29
Power series, circle of convergence of      30
Power series, continuity of      57
Power series, continuity of, Abel's theorem      57
Power series, expansions of functions in      see under the names of special functions
Power series, identically vanishing      58
Power series, Maclaurin's expansion in      94
Power series, radius of convergence of      30 32
Power series, series derived from      31
Power series, Taylor's expansion in      93
Power series, uniformity of convergence of      57
Principal part of a function      102
Principal solution of a certain equation      482
Principal value of an integral      75 117
Principal value of the argument of a complex number      9 588
Principle of convergence      13
Pringsheim's theorem on summation of double series      28
Products of Bessel functions      379 380 383 385 428
Products of hypergeometric functions      298. See also Infinite products
Quarter periods K, iK'      479 498 499 501.
Quartic, canonical form of      513
Quartic, integration problem connected with      452 512
Quast-periodlcity      445 447 463
Quotients of elliptic functions (Glaisher's notation)      494 511
Quotients of Theta-functions      477
Raalschuetz' integral for the Gamma-function      243
Radius of convergence of power series      30 32
Rational functions      105
Rational functions, expansions in series of      134
Rational numbers      3 4
Rational-real numbers      5
Real functions of real variables      56
Real numbers, rational and irrational      5
Real part (R) of a complex number      9
Reality of characteristic numbers      226
Rearrangement of convergent series      25
Rearrangement of double series      28
Rearrangement of infinite determinants      37
Rearrangement of infinite products      33
Reciprocal functions, Volterra's      218
Reciprocals of elliptic functions (Glaisher's notation)      494 511
Recurrence formulae for Bessel functions      359 373 374
Recurrence formulae for confluent hypergeometric functions      352
Recurrence formulae for Gegenbauer's function      330
Recurrence formulae for Legendre functions      307 309 318
Recurrence formulae for Neumann's function      375
Recurrence formulae for parabolic cylinder functions      350. See also Contiguous hypergeometric functions
Region      44
Regular      83
Regular distribution of discontinuities      212
Regular integrals of linear differential equations      201
Regular integrals of linear differential equations of the third order      210
Regular points (singularities) of linear differential equations      197
Relations between Bessel functions      360 371
Relations between confluent hypergeometric functions $W_{\pm k, m}(\pm z)$ and $M_{k, \pm}(z)$       346
Relations between contiguous hypergeometric functions      294
Relations between elliptic functions      452
Relations between parabolic cylinder functions $D_n{\pm z}$ and $D_{-n-1}{\pm iz}$       348
Relations between poles and zeros of elliptic functions      433
Relations between Riemann Zeta-functions $\zeta(s)$ and $\zeta(1-s)$       269. See also Recurrence formulae
Remainder after n terms of a series      15
Remainder in Taylor's series      95
Removable discontinuity      42
Repeated integrals      68 75
Representative point      9
Residues      111—124 (Chapter VI) defined 111
Residues of elliptic functions      425 497
Riemann's associated function      183 184 185
Riemann's condition of integrability      63
Riemann's equations satisfied by analytic functions      84
Riemann's hypothesis concerning $\zeta(s)$       272 280
Riemann's lemmas      172 184 185
Riemann's P-equation      206 283 291 294
Riemann's P-equation and the hypergeometric equation      208. See also Hypergeometric functions
Riemann's P-equation, transformation of      207
Riemann's theory of trigonometrical series      182—188
Riemann's Zeta-function      see Zeta-function (of Riemann)
Riesz' method of “summing” series      156
Right (R-) class      4
Rodrigues' formula for Legendre polynomials      303
Rodrigues' formula, modified, for Gegenbaner's function      329
Roots of an equation, number of      120
Roots of an equation, number of inside a contour      119 123
Roots of Weierstrassian elliptic functions (, , )      443
Schlaefli's Bessel function of the second kind, []      370
Schlaefli's integral for Bessel functions      362 372
Schlaefli's integral for Legendre polynomials and functions      303 304 306

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