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Apostol T. — Mathematical Analysis, Second Edition
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Название: Mathematical Analysis, Second EditionАвтор: Apostol T. Аннотация: It provides a transition from elementary calculus to advanced courses in real and complex function theory and introduces the reader to some of the abstract thinking that pervades modern analysis.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: 2Год издания: 1974Количество страниц: 492Добавлена в каталог: 17.03.2014Операции: Положить на полку |
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Exponential form, of Fourier series 323 Exponential function 7 19 Extended complex plane 25 Extended real-number system 14 Extension of a function 35 Exterior (or outer region) of a Jordan curve 447 Extremum problems 375 F(S), image of S under F 35 Fatou, Pierre (1878-1929) 299 Fatou’s Lemma 299 Fejer, Leopold (1880-1959) 179 312 320 Fejer’s theorem 179 320 Fekete, Michel 178 Field, of complex numbers 116 Field, of real numbers 2 Finite set 38 Fischer, Ernst (1875-1954) 297 311 Fixed point, of a function 92 Fixed-point theorem 92 Fourier coefficient 309 Fourier integral theorem 324 Fourier series 309 Fourier transform 326 Fourier, Joseph (1758-1830) 306 309 312 324 326 Fubini, Guido (1879-1943) 405 410 413 Fubini’s Theorem 410 413 Function, definition of 34 Fundamental theorem, of algebra 15 451 475 Fundamental theorem, of integral calculus 162 Gamma function, continuity of 282 Gamma function, definition of 277 Gamma function, derivative of 284 303 Gamma function, duplication formula for 341 Gamma function, functional equation for 278 Gamma function, series for 304 Gauss, Karl Friedrich (1777-1855) 17 464 Gaussian sum 464 Geometric series 190 195 Gibbs’ phenomenon 338 Global property 79 Goursat, Edouard (1858-1936) 434 Gradient 348 Gram — Schmidt process 335 Gram, Jurgen Pedersen (1850-1916) 335 Greatest lower bound 9 Hadamard determinant theorem 386 Hadamard, Jacques (1865-1963) 386 Half-open interval 4 Hardy, Godfrey Harold (1877-1947) 30 206 217 251 312 Harmonic series 186 Heine — Borel covering theorem 58 Heine's theorem 91 Heine, Eduard (1821-1881) 58 91 312 Hobson, Ernest William (1856-1933) 312 415 Homeomorphism 84 Homogeneous function 364 Homotopic paths 440 Hyperplane 394 Identity theorem for analytic functions 452 Image 35 Imaginary part 15 Imaginary unit 18 Implicit-function theorem 374 Improper Riemann integral 276 Increasing function 94 150 Increasing sequence, of functions 254 Increasing sequence, of numbers 71 185 Independent set of functions 335 Induction principle 4 Inductive set 4 Inequality, Bessel 309 Inequality, Cauchy — Schwarz 14 177 294 Inequality, Minkowski 27 Inequality, triangle 13 294 Infimum 9 Infinite, derivative 108 Infinite, product 206 Infinite, series 185 Infinite, set 38 Infinity, in 24 Infinity, in 14 Inner Jordan content 396 Inner product 48 294 int S, interior of S 49 61 Integers 4 Integrable function, Lebesgue 260 407 Integrable function, Riemann 141 389 Integral, equation 181 Integral, test 191 Integral, transform 326 Integration by parts 144 278 Integrator 142 Interior (or inner region) of a Jordan curve 447 Interior point 49 61 Interior, of a set 49 61 Intermediate-value theorem, for continuous functions 85 Intermediate-value theorem, for derivatives 112 Intersection of sets 41 Interval, in 50 52 Interval, in R 4 Inverse function 36 Inverse image 44 81 Inverse-function theorem 372 Inversion formula, for Fourier transforms 327 Inversion formula, for Laplace transforms 342 468 Irrational numbers 7 Isolated point 53 Isolated singularity 458 Isolated zero 452 Isometry 84 Iterated integral 167 287 Iterated limit 199 Iterated series 202 Jacobi, Carl Gustav Jacob (1804-1851) 351 368 Jacobian, determinant 368 Jacobian, matrix 351 Jordan, arc 435 Jordan, Camille (1838-1922) 312 319 396 435 447 Jordan, content 396 Jordan, curve 435 Jordan, curve theorem 447 Jordan, theorem on Fourier series 319 Jordan-measurable set 396 Jump, discontinuity 93 Jump, of a function 93 Kestelman, Hyman 165 182 Kronecker delta, 385 L(I), set of Lebesgue-integrable functions on an interval I 260 L(x,y), line segment joining x and y 355 Lagrange, identity 27 30 380 Lagrange, Joseph Louis (1736-1813) 27 30 380 Lagrange, multipliers 380 Landau, Edmund (1877-1938) 31 Laplace transform 326 342 468 Laplace, Pierre Simon (1749-1827) 326 342 468 Laurent expansion 455 Laurent, Pierre Alphonse (1813-1854) 455 Least upper bound 9 Lebesgue, bounded convergence theorem 273 Lebesgue, criterion for Riemann integrability 171 391 Lebesgue, dominated-convergence theorem 270 Lebesgue, Henri (1875-1941) 141 171 260 270 273 290 292 312 391 405 Lebesgue, integral of complex functions 292 Lebesgue, integral of real functions 260 407 Lebesgue, measure 290 408 Legendre polynomials 336 Legendre, Adrien-Marie (1752-1833) 336 Leibniz, Gottfried Wilhelm (1646-1716) 121 Leibniz’ formula 121 Length of a path 134 Levi monotone convergence theorem, for sequences 267 Levi monotone convergence theorem, for series 268 Levi monotone convergence theorem, for step functions 265 Levi, Beppo (1875-1961) 265 267 268 407 lim inf, limit inferor (lower limit) 184 lim sup, limit superior (upper limit) 184 Limit function 218 Limit theorem of Abel 245 Limit, in a metric space 71 Limit, inferior 184 Limit, superior 184 Lindeloef covering theorem 57 Lindeloef, Ernst (1870-1946) 56 Line segment in 88 Linear function 345 Linear space 48 Linear space of functions 137 Linearly dependent set of functions 122 Liouville, Joseph (1809-1882) 451 Liouville’s theorem 451 Lipschitz condition 121 137 316 Lipschitz, Rudolph (1831-1904) 121 137 312 316 Littlewood, John Edensor (1885-1977) 312 Local extremum 98 Local property 79 Localization theorem 318 Logarithm 23 Lower bound 8 Lower integral 152 Lower limit 184 M(I), set of measurable functions on an interval I 279 m(T), matrix of a linear function T 350 Mapping 35 Matrix 350 Matrix, product 351 max S, min S, largest (smallest) element of S 8 Maximum and minimum 83 375 Maximum-modulus principle 453 454 Mean convergence 232 Mean-Value Theorem for derivatives, of real-valued functions 110 Mean-Value Theorem for derivatives, of vector-valued functions 355 Mean-value theorem for integrals, multiple integrals 401 Mean-value theorem for integrals, Riemann integrals 160 165 Mean-value theorem for integrals, Riemann — Stieltjes integrals 160 Measurable function 279 407 Measurable set 290 408 Measure, of a set 290 408 Measure, zero 169 290 391 405 Mertens' theorem 204 Mertens, Franz (1840-1927) 204 Metric 60 Metric space 60 Minimum-modulus principle 454 Minkowski, Hermann (1864-1909) 27 Minkowski’s inequality 27 Mobius transformation 471 Modulus of a complex number 18 Moebius, Augustus Ferdinand (1790-1868) 471 Monotonic function 94 Monotonic sequence 185 Multiple integral 389 407 Multiplicative function 216 n-measure 408 Neighborhood 49 Neighborhood of infinity 15 25 Niven, Ivan M. (1915-1999) 180 Nonempty set 1 Nonmeasurable function 304 Nonmeasurable set 304 NonNegative 3 Norm, of a function 102 Norm, of a partition 141 Norm, of a vector 48 O, o, oh notation 192 One-to-one function 36 Onto 35 Open, covering 56 63 Open, interval in 50 Open, interval in R 4 Open, mapping 370 454 Open, mapping theorem 371 454 Open, set in 49 Open, set in a metric space 62 Operator 327 Order, of pole 458 Order, of zero 452 Order-pieserving function 38 Ordered 47 Ordered pair 33 Ordinate set 403 Orientation of a circuit 447 Orthogonal system of functions 306 Orthonormal set of functions 306 Oscillation of a function 98 170 Outer Jordan content 396 Parallelogram law 17 Parseval's formula 309 474 Parseval, Mark-Antoine (circa 1776-1836) 309 474 Partial derivative 115 Partial derivative of higher order 116 Partial sum 185 Partial summation formula 194 Partition of an interval 128 141 Path 88 133 435 Peano, Giuseppe (1858-1932) 224 Perfect set 67 Periodic function 224 317 Pi, 180 Piecewise-smooth path 435 Point, in 47 Point, in a metric space 60 Pointwise convergence 218 Poisson, integral formula 473 Poisson, Simeon Denis (1781-1840) 332 473 Poisson, summation formula 332 Polar coordinates 20 418 Polygonal curve 89 Polygonally connected set 89 Polynomial 80 Polynomial in two variables 462 Polynomial, zeros of 451 475 Power series 234 Powers of complex numbers 21 23 Prime number 5 Prime-number theorem 175 Principal part 456 Projection 394 Q, set of rational numbers 6 Quadratic form 378 Quadric surface 383 Quotient, of complex numbers 16 Quotient, of teal numbers 2 R, set of real numbers 1 Radius of convergence 234 Range of a function 34 Ratio Test 193 Rational function 81 462 Rational number 6 Real number 1 Real part 15 Rearrangement of series 196 Reciprocity law for Gauss sums 464 Rectifiable path 134 Reflexive relation 43 Region 89 Relation 34 Removable discontinuity 93 Removable singularity 458 Residue 459 Residue theorem 460
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