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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

Heine — Borel theorem (modified)      53
Heine's expansion of $(t-z)^{-1}$ in series of Legendre polynomials      321
Hermite's equation      204 209 342 347.
Hermite's formula for the generalised Zeta-function $\zeta(s, a)$       269
Hermite's solution of Lame's equation      573—575
Heun's equation      576 577
Hill's equation      406 413—417
Hill's infinite determinant      36 40 415
Hill's infinite determinant, evaluation of      415
Hill's method of solution      413
Hobson's associated Legendre functions      325
Holomorphic      83
Homogeneity of Weierstrassian elliptic functions      439
Homogeneous harmonics (associated with ellipsoid)      543 576
Homogeneous harmonics, ellipsoidal harmonics derived from (Niven's formula)      543
Homogeneous harmonics, linear independence of      560
Homogeneous integral equations      217 219
Hurwitz' definition of the generalised Zeta-function $\zeta(x, a)$       265
Hurwitz' formula for $\zeta(s, a)$       268
Hurwitz' theorem concerning Fourier constants      180
Hypergeometric equation      see Hypergeometric functions
Hypergeometric functions      281—301 (Chapter XIV)
Hypergeometric functions, Barnes' integrals      286 289
Hypergeometric functions, contiguous      294
Hypergeometric functions, continuation of      288
Hypergeometric functions, contour integrals for      291
Hypergeometric functions, differential equation for      202 207 283
Hypergeometric functions, functions expressed in terms of      281 311
Hypergeometric functions, of two variables (Appell's)      300
Hypergeometric functions, relations between twenty-four expressions involving      284 285 290
Hypergeometric functions, Riemann's P-equation and      208 283
Hypergeometric functions, series for (convergence of)      24 281
Hypergeometric functions, squares and products of      298
Hypergeometric functions, value of F(a, b, c)      281 293
Hypergeometric functions, values of special forms of hypergeometric functions      298 301. Confluent
Hypergeometric series      see Hypergeometric functions
Hypothesis of Rlemann on zeros of $\zeta(s)$       272 280
Identically vanishing power series      58
Identity of two functions      98
Imaginary argument, Bessel functions with [ and ]      372 373 384
Imaginary part (I) of a complex number      9
Imaginary transformation (Jacobl's) of E(u) and Z(u)      519
Imaginary transformation (Jacobl's) of elliptic functions      505 506 535
Imaginary transformation (Jacobl's) of Theta-functions      124 474
Improper integrals      75
Incomplete Gamma-functions [y (», x)]      341
Increasing sequence      12
Indicia! equation      198
Inequality (Abel's)      16
Inequality (Hadamard'g)      212
Inequality satisfied by $\zeta(s, a)$       274 275
Inequality satisfied by Bessel coefficients      379
Inequality satisfied by Legendre polynomials      303
Inequality satisfied by parabolic cylinder functions      354
Infinite determinants      see Determinants
Infinite integrals      69
Infinite integrals, convergence of      70 71 72
Infinite integrals, differentiation of      74
Infinite integrals, evaluation of      111—124
Infinite integrals, functions represented by      see under the names of special functions
Infinite integrals, representing analytic functions      92
Infinite integrals, theorems concerning      73
Infinite integrals, uniform convergence of      70 72 73.
Infinite products      32
Infinite products, absolute convergence of      32
Infinite products, convergence of      32
Infinite products, divergence to zero      33
Infinite products, expansions of functions as      136 137.
Infinite products, expressed by means of Theta-functions      473 488
Infinite products, uniform convergence of      49
Infinite series      see Series
infinity      11 103
Infinity, essential singularity at      104
Infinity, point at      103
Infinity, pole at      104
Infinity, zero at      104
Integers, positive      3
Integers, signless      3
Integrabillty of continuous functions      63
Integrabillty, Riemann's condition of      63
Integral equations      211—231 (Chapter XI)
Integral equations, Abel's      211 229 230
Integral equations, Fredholm's      213—217 228
Integral equations, homogeneous      217 219
Integral equations, kernel of      213
Integral equations, Liouville — Neumann method of solution of      221
Integral equations, nucleus of      213
Integral equations, numbers (characteristic) associated with      219
Integral equations, numerical solutions of      211
Integral equations, of the first and second kinds      213 221
Integral equations, satisfied by Lame functions      564—567
Integral equations, satisfied by Mathieu functions      407
Integral equations, satisfied by parabolic cylinder functions      231
Integral equations, Scloemilch's      229
Integral equations, solutions in series      228
Integral equations, Volterra's      221
Integral equations, with variable upper limit      213 221
Integral formulae for ellipsoidal harmonics      567
Integral formulae for the Jacobian elliptic functions      492 494
Integral formulae for the Weierstrassian elliptic function      437
Integral functions      106
Integral functions and Lamp's equation      571
Integral functions and Mathieu's equation      418
Integral properties of Bessel functions      380 381 385
Integral properties of Legendre functions      225 30—5 324
Integral properties of Mathieu functions      411
Integral properties of Neumann's function      385
Integral properties of parabolic cylinder functions      350
Integral theorem of Fourier — Bessel      385
Integral theorem, Fourier's      188 211
Integral, Borel's      140
Integral, Borel's, and analytic continuation      141
Integral, Cauchy's      119
Integral, Dlrichlet's      258
Integrals      61—81 (Chapter IV)
Integrals of harmonics (Sylvester's theorem)      400
Integrals of irrational functions      452 512Integrals
Integrals of periodic functions      112
Integrals, along curves (equivalence of)      87
Integrals, complex      77 78
Integrals, differentiation of      67
Integrals, double      68 255
Integrals, double-circuit      256 293
Integrals, evaluation of      111—124
Integrals, for derivates of an analytic function      89
Integrals, functions represented by      see under the names of the special functions
Integrals, improper      75
Integrals, lower      61
Integrals, principal values of      75 117
Integrals, regular      201
Integrals, repeated      68 75
Integrals, representing analytic functions      92
Integrals, representing areas      61 589
Integrals, round a contour      85
Integrals, upper      61. See also Elliptic integrals Infinite and
Integration      61
Integration of asymptotic expansions      153
Integration of integrals      68 74 75
Integration of series      78
Integration problem connected with cubics or quartics and elliptic functions      452 512.
Integration, complex      77
Integration, contour-      77
Integration, general theorem on      63
Integration, general theorem on complex      78
Interior      44
Internal spheroidal harmonics      403
Invariants of Weierstrassian elliptic functions      437
Inverse factorials, expansions in series of      142
Inversion of elliptic integrals      429 452 454 480 484 512 524
Irrational functions, integration of      452 512

Irrational-real numbers      5
Irreducible set of zeros or poles      430
Irregular points (singularities) of differential equations      197 202
Iterated functions      222
Jacobi's discovery of elliptic functions      429 512
Jacobi's earlier notation for Theta-functions      479
Jacobi's fundamental Theta-function formulae      467 488
Jacobi's imaginary transformations      124 474 505 506 519 535
Jacobi's Zeta-function      see under Zeta-function of J&cobi
Jacobian elliptic functions [sn u, cn u, dn u]      432 478 491—535
Jacobian elliptic functions, addition theorems for      494 497 530 535
Jacobian elliptic functions, connexion with Weierstrassian functions      505
Jacobian elliptic functions, definitions of am u, $\Delta\phi$ , sn u (sin am u), en u, dn u      478 492 494
Jacobian elliptic functions, differential equations satisfied by      477 492
Jacobian elliptic functions, differentiation of      493
Jacobian elliptic functions, duplication formulae for      498
Jacobian elliptic functions, Fourier series for      510 511 535
Jacobian elliptic functions, general description of      504
Jacobian elliptic functions, geometrical illustration of      524 527
Jacobian elliptic functions, Glaisher's notation for quotients and reciprocals of      494
Jacobian elliptic functions, infinite producte for      508 532
Jacobian elliptic functions, integral formulae for      492 494
Jacobian elliptic functions, Jacobi's imaginary transformation of      505 506
Jacobian elliptic functions, Lame functions expressed in terms of      564 573
Jacobian elliptic functions, Landen's transformation of      507
Jacobian elliptic functions, modular angle of      492
Jacobian elliptic functions, modulus of      479 492
Jacobian elliptic functions, modulus of (complementary)      479 493
Jacobian elliptic functions, parametric representation of points on curves by      524 527 527 533
Jacobian elliptic functions, periodicity of      479 500 502 503
Jacobian elliptic functions, poles of      432 503 504
Jacobian elliptic functions, quarter periods, K', iK', of      479 498 499 501
Jacobian elliptic functions, relations between      492
Jacobian elliptic functions, residues of      504
Jacobian elliptic functions, Seiffert's spherical spiral and      527
Jacobian elliptic functions, triplication formulae      530 534 535
Jacobian elliptic functions, values of, when the modulus is small      532. See also Elliptic functions Elliptic Lemniscate Theta-functions and
Jacobian elliptic functions, values of, when u is $\frac12K$ , $\frac12iK$ or $\frac12(K+IK)$       500 506 507
Jordan's lemma      115
Kernel      213
Klein's theorem on linear differential equations with five singularities      203
Kummer's formulae for confluent hypergeometric functions      338
Kummer's series for $\log\Gamma(z)$       250
Lacunary function      98
Lagrange's expansion      132 140
Lagrange's form for the remainder in Taylor's series      96
Lame functions, defined      558
Lame functions, expressed as algebraic functions      556 577
Lame functions, expressed by Jacobian elliptic functions      573—575
Lame functions, expressed by Weierstrassian elliptic functions      570—572
Lame functions, integral equations satisfied by      564—567
Lame functions, linear independence of      559
Lame functions, reality and distinctness of zeros of      557 558 578
Lame functions, second kind of      562
Lame functions, values of      558
Lame functions, zeros of (Stieltjes' theorem)      560. See also Lame's equation and Ellipsoidal harmonics
Lame's equation      204 536—578
Lame's equation, derived from theory of ellipsoidal harmonics      538—543 552—554
Lame's equation, different forms of      554 573
Lame's equation, generalised      204 570 573 576 577
Lame's equation, series solutions of      556 577 575
Lame's equation, solutions expressed in finite form      459 556 576 577 578
Lame's equation, solutions of a generalised equation in finite form      570 573.
Landen's transformation of Jacobian elliptic functions      476 507 533
Laplace's equation      386
Laplace's equation, its general solution      388
Laplace's equation, normal solutions of      553
Laplace's equation, solution with given boundary conditions      393
Laplace's equation, solutions involving functions of Legendre and Bessel      391 395
Laplace's equation, symmetrical solution of      399
Laplace's equation, transformations of      401 407 551 553
Laplace's integrals for Legendre polynomials and functions      312 313 314 319 326 337
Lapounoff's theorem concerning Fourier constants      180
Laurent's expansion      100
Least of limits      13
Lebesgue's lemma      172
Left (L-) class      4
Legendre functions      302—336 (Chapter XV)
Legendre functions, , , , defined      306 316 323 325
Legendre functions, addition formulae for      328 395
Legendre functions, Bessel functions and      364 367 401
Legendre functions, degree of      307 324
Legendre functions, differential equation for      204 306 324
Legendre functions, distinguished from Legendre polynomials      306
Legendre functions, expansion of a function as a series of      334
Legendre functions, expansions in ascending series      311 326
Legendre functions, expansions in descending series      302 317 326 334
Legendre functions, expressed by Murphy as hypergeometric functions      311 312
Legendre functions, expression of in terms of Legendre polynomials      319 320 333
Legendre functions, Ferrers' functions associated with      323 324
Legendre functions, first kind of      307
Legendre functions, Gegenbauer's function, $C_n^\nu(z)$ , associated with      see Gegenbauer's function
Legendre functions, Heine's expansion of $(t-z)^{-1}$ as a series of      321
Legendre functions, Hobson's functions associated with      325
Legendre functions, integral connecting Bessel functions with      364
Legendre functions, integral properties of      324
Legendre functions, Laplace's integrals for      312 313 319 326 334
Legendre functions, Mehler — Dirichlet integral for      314
Legendre functions, order of      326
Legendre functions, recurrence formulae for      307 318
Legendre functions, Schlaefli's integral for      304 306
Legendre functions, second kind of      316—320 325 326
Legendre functions, summation of $\sum h^nP_n(z)$ and $\sum h^nQ_n(z)$       302 321
Legendre functions, zeros of      303 316 335.
Legendre polynomials []      95 302
Legendre polynomials, addition formula for      326 387
Legendre polynomials, degree of      302
Legendre polynomials, differential equation for      204 304
Legendre polynomials, expansion in ascending series      311
Legendre polynomials, expansion in descending series      302 334
Legendre polynomials, expansion of a function as a series of      310 322 330 331 332 335
Legendre polynomials, expressed by Murphy as a hypergeometric function      311 312
Legendre polynomials, Heine's expansion of $(t-z)^{-1}$ as a series of      321
Legendre polynomials, integral connecting Bessel functions with      364
Legendre polynomials, integral properties of      225 305
Legendre polynomials, Laplace's equation and      391
Legendre polynomials, Laplace's integrals for      312 314
Legendre polynomials, Mehler — Dirichlet integral for      314
Legendre polynomials, Neumann's expansion in series of      322
Legendre polynomials, numerical inequality satisfied by      303
Legendre polynomials, recurrence formulae for      307 309
Legendre polynomials, Rodrigues' formula for      225 303
Legendre polynomials, Schlaefli's integral for      303 304
Legendre polynomials, summation of $\sum h^nP_n(z)$       302
Legendre polynomials, zeros of      303 316.
Legendre's equation      204 304
Legendre's equation for associated functions      324
Legendre's equation, second solution of      316. See also Legendre functions and Legendre polynomials
Legendre's relation between complete elliptic integrals      520
Lemniscate functions [ $\sin\, \mathrm{lemn}\, \phi$ and $\cos\, \mathrm{lemn}\, \phi$ ]      524
Limit of a function      42
Limit of a sequence      11 12
Limit to the value of a complex integral      78
Limit, condition for existence of      13
Limit-point (the Bolzano — Weierstrass theorem)      12
Limiting circle      98
Limits, greatest of and least of      13
Lindemann's theory of Mathieu's equation      417
Lindemann's theory, the similar theory of Lame's equation      570
Linear differential equations      194—210 (Chapter X) 386—403
Linear differential equations, equation for conduction of Heat      387
Linear differential equations, equation of Telegraphy      387
Linear differential equations, equation of the third order with regular integrals      210
Linear differential equations, equation of wave motions      386 397 402
Linear differential equations, equations with five singularities (the Klein-B&cher theorem)      203
Linear differential equations, equations with r singularities      209
Linear differential equations, equations with three singularities      206
Linear differential equations, equations with two singularities      208
Linear differential equations, exponents of      198
Linear differential equations, fundamental system of solutions of      197 200

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