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Apostol T.M. — Mathematical Analysis
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Название: Mathematical AnalysisАвтор: Apostol T.M. Аннотация: A glance at the table of contents will reveal that this textbook treats topics in analysis at the "Advanced Calculus" level. The aim has been to provide a development of the subject which is honest, rigorous, up to date, and, at the same time, not too pedantic. The book provides a transition from elementary calculus to advanced courses in real and complex function theory, and it introduces the reader to some of the abstract thinking that pervades modem analysis.
Язык: Рубрика: Математика /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: 5th editionГод издания: 1981Количество страниц: 492Добавлена в каталог: 11.05.2005Операции: Положить на полку |
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293 295 Abel, limit theorem 245 Abel, limit theorem, partial summation formula 194 Abel, limit theorem, test for convergence of series 194 248 Abel, Neils Henrik, (1802—1829) 194 245 248 Absolute convergence of products 208 Absolute convergence of series 139 Absolute value 13 18 Absolutely continuous function 139 Accumulation point 52 62 Additive function 45 (Ex. 2.22) Additivity of Lebesgue measure 291 Adherent point 52 62 Algebraic number 45 (Ex. 2.15) Almost everywhere 172 391 Analytic function 434 Annulus 438 Approximation theorem of Weierstrass 322 ARC 88 435 Arc length 134 Archimedean property of real numbers 10 Arcwise connected set 8S Area (content) of a plane region 396 Argand, Jean-Robert (1768—1822) 17 Argument of complex number 21 Arithmetic mean 205 Arzela's theorem 228 273 Arzela, Cesare (1847—1912) 228 273 Associative law 2 16 Axioms for real numbers 1 2 9 Ball, in 49 Ball, in a metric space 61 Basis vectors 49 Bernoulli numbers 251 (Ex. 9.38) Bernoulli periodic functions 338 (Ex. 11.18) Bernoulli polynomials 251 (Ex. 9.38) 478 Bernoulli, James (1654—1705) 251 338 478 Bernstein's theorem 242 Bernstein, Sergei Natanovic 242 Bessel function 475 (Ex. 16.21) Bessel inequality 309 Bessel, Friedrich Wilhelm (1784—1846) 309 475 beta function 331 Binary system 225 Binomial series 244 Bolzano — Weierstrass theorem 54 Bolzano's theorem 85 Bolzano, Bernard (1781—1848) 54 85 Bonnet's theorem 165 Bonnet, Emile (1819—1892) 165 Borel, Emile (1871—1938) 58 Bound, greatest lower 9 Bound, least upper 9 Bound, lower 8 Bound, uniform 221 Bound, upper 8 Boundary, of a point 64 Boundary, of a set 64 Bounded convergence 227 273 Bounded function 83 Bounded set 54 63 Bounded variation 128 Bounded, away from zero 130 Cantor intersection theorem 56 Cantor set 180 (Ex. 7.32) Cantor — Bendixon theorem 67 (Ex. 3.25) Cantor, Georg (1845—1918) 8 32 56 67 180 312 cardinal number 38 Carleson, Lennart 312 Cartesian product 33 Casorati — Weierstrass theorem 475 (Ex. 16.23) Cauchy condition for products 207 Cauchy condition for sequences 73 183 Cauchy condition for series 186 Cauchy condition for uniform convergence 222 223 Cauchy — Riemann equations 118 Cauchy — Schwarz inequality, for inner products 294 Cauchy — Schwarz inequality, for integrals 177 (Ex. 7.16) 294 Cauchy — Schwarz inequality, for sums 14 27 30 Cauchy, Augustin-Louis (1789—1857) 14 73 118 177 183 207 222 Cauchy, inequalities 451 Cauchy, inequalities, integral formula 443 Cauchy, inequalities, integral theorem 439 Cauchy, inequalities, principal value 277 Cauchy, inequalities, product 204 Cauchy, inequalities, residue theorem 460 Cauchy, inequalities, sequence 73 Cesaro, Ernesto (1859—1906) 205 320 Cesaro, sum 205 Cesaro, summability of Fourier series 320 Chain rule, complex functions 117 Chain rule, matrix form of 353 Chain rule, real functions 107 Chain rule, vector-valued functions 114 Change of variables in a Lebesgue integral 262 Change of variables in a multiple Lebesgue integral 421 Change of variables in a Riemann integral 164 Change of variables in a Riemann — Stieltjes integral 144 Characteristic function 289 Circuit 435 Closed ball 67 (Ex. 3.31) Closed curve 435 Closed interval 4 52 Closed mapping 99 (Ex. 4.32) Closed region 90 Closed set 53 62 Closure of a set 53 Commutative law 2 16 Compact set 59 63 Comparison Test 190 Complement 41 Complete metric space 74 Complete orthonormal set 336 (Ex. 11.6) Completeness axiom 9 Complex number 15 Complex plane 17 Component, interval 51 Component, interval, of a metric space 87 Component, interval, of a vector 47 Composite function 37 Condensation point 67 (Ex. 3.23) Conditional convergent series 189 Conditional convergent series, rearrangement of 197 Conformal mapping 471 Conjugate complex number 28 (Ex. 1.29) Connected metric space 86 Connected set 86 Content 396 Continuity 78 Continuity, uniform 90 Continuously differentiable function 371 Contour integral 436 Contraction, constant 92 Contraction, fixed-point theorem 92 Contraction, mapping 92 Convergence in a metric space 70 Convergence of a product 207 Convergence of a sequence 183 Convergence of a series 185 Convergence, absolute 189 Convergence, bounded 227 Convergence, conditional 189 Convergence, mean 232 Convergence, pointwise 218 Convergence, uniform 221 Converse of a relation 36 Convex set 66 (Ex. 3.14) Convolution integral 328 Convolution theorem for Fourier transforms 329 Convolution theorem for Laplace transforms 342 (Ex. 11.36) Coordinate transformation 417 Countable additivity 291 Countable set 39 Covering of a set 56 Covering theorem, Heine — Borel 58 Covering theorem, Lindeloef 57 Cramer's rule 367 Curve, closed 435 Curve, Jordan 435 Curve, piecewise-smooth 435 Curve, rectifiable 134 Daniell, P. J. (1889—1946) 252 Darboux, Gaston (1842—1917) 152 De Moivre's theorem 29 (Ex. 1.44) De Moivre, Ham (1667—1754) 29 Decimals 11 12 27 Dedekind, Richard (1831—1916) 8 Deleted neighborhood 457 Dense set 68 (Ex. 3.32) Denumerable set 39 Derivative(s) of real-valued functions 104 Derivative(s) of vector-valued functions 114 Derivative(s), directional 344 Derivative(s), of complex functions 117 Derivative(s), partial 115 Derivative(s), total 347 Derived set 54 62 Determinant 367 Difference of two sets 41 Differentiation of integrals 162 167 Differentiation of sequences 229 Differentiation of series 230 Dim's theorem, on Fourier series 319 Dim's theorem, on uniform convergence 248 (Ex. 9.9) Dini, Ulisse (1845—1918) 248 312 319 Directional derivative 344 Dirichlet's test, for convergence of series 194 Dirichlet's test, for uniform convergence of series 230 Dirichlet, integrals 314 Dirichlet, kernel 317 Dirichlet, Peter Gustav Lejeune (1805—1859) 194 205 215 230 317 464 Dirichlet, product 205 Dirichlet, series 215 (Ex. 8.34) disconnected set 86 Discontinuity 93 Discrete metric space 61 Disjoint sets 41 Disjoint sets, collection of 42 Disk 49 Disk of convergence 234 Distance function (metric) 60 Distributive law 2 16 Divergent product 207 Divergent sequence 183 Divergent series 185 Divisor 4 Divisor, greatest common 5 Domain (open region) 90 Domain of a function 34 Dominated Convergence Theorem 270 Dot product 48 Double sequence 199 Double series 200 Double, integral 390 407 Du Bois-Reymond, Paul (1831—1889) 312 Duplication formula for the Gamma function 341 (Ex. 11.31) e, irrationality of 7 Element of a set 32 Empty set 33 Equivalence of paths 136 Equivalence relation 43 (Ex. 2.2) Essential singularity 458 Euclid's lemma 5 Euclidean metric 48 61 Euclidean space 47 Euler's constant 192 Euler's product for 209 Euler's summation formula 149 Euler's theorem on homogeneous functions 365 (Ex. 12.18) Euler, Leonard (1707—1783} 149 192 209 365 Exponential form of Fourier integral theorem 325 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 Fatou's lemma 299 (Ex. 10.8) Fatou, Pierre (1878—1929) 299 Fejer's theorem 179 (Ex. 7.23) 320 Fejer, Leopold (1880—1959) 179 312 320 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's theorem 410 413 Fubini, Guido (1879—1943) 405 410 413 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 (Ex. 11.31) Gamma function, functional equation for 278 Gamma function, series for 304 (Ex. 10.31) Gauss, Karl Friedrich (1777—1855) 17 464 Gaussian sum 464 Geometric series 190 195 Gibbs' phenomenon 338 (Ex. 11.19) Global property 79 Goursat, Edouard (1858—1936) 434 Gradient 348 Gram — Schmidt process 335 (Ex. 11.3) Gram, Jorgen Pedersen (1850—1916) 335 Greatest lower bound 9 Hadamard determinant theorem 386 (Ex. 13.16) 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 (Ex. 12.18) 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 (Ex. 11.2) Induction principle 4 Inductive set 4
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