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Krantz S.G. — Handbook of Real Variables
Krantz S.G. — Handbook of Real Variables

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Название: Handbook of Real Variables

Автор: Krantz S.G.


This concise, well-written handbook provides a distillation of the theory of real variables with a particular focus on the subject's significant applications to differential equations and Fourier analysis. Ideal for the working engineer or scientist, the book uses ample examples and brief explanations—-without a lot of proofs or axiomatic machinery—-to give the reader quick, easy access to all of the key concepts and touchstone results of real analysis. Topics are systematically developed, beginning with sequences and series, and proceeding to topology, limits, continuity, derivatives, and Riemann integration. In the second half of the work, Taylor series, the Weierstrass Approximation Theorem, Fourier series, the Baire Category Theorem, and the Ascoli—Arzela Theorem are carefully discussed. Picard iteration and differential equations are treated in detail in the final chapter.Key features: * Completely self-contained, methodical exposition for the mathematically-inclined researcher; also valuable as a study guide for students * Realistic, meaningful connections to ordinary differential equations, boundary value problems, and Fourier analysis * Example-driven, incisive explanations of every important idea, with suitable cross-references for ease of use * Illuminating applications of many theorems, along with specific how-to hints and suggestions * Extensive bibliography and index This unique handbook is a compilation of the major results, techniques, and applications of real analysis; it is a practical manual for applied mathematicians, physicists, engineers, economists, and others who use the fruits of real analysis but who do not necessarily have the time to appreciate all of the theory. Appropriate as a comprehensive reference or for a quick review, "A Handbook of Real Variables" will benefit a wide audience.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 201

Добавлена в каталог: 11.04.2008

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Предметный указатель
Abel's convergence test      29
Absolute convergence of series      31
Absolute maximum      63
Absolute minimum      63
Accumulation point of a set in a metric space      145
Addition of series      36
Alternating Series Test      29
Ascoli — Arzela theorem      151
Baire category theorem      149
Bessel's inequality      135
Boundary point      42
Bounded set      44
Bounded set in a metric space      145
Cantor set      48
Cauchy condensation test      24
Cauchy mean value theorem      78
Cauchy product of series      37
Cauchy sequences in a metric space      143
Chain rule      74
Change of variable      91
Characteristic curve      158
Characterization of connected subsets of $\mathbb{R}$      51
Closed ball in a metric space      144
Closure of a set in a metric space      148
Coefficients of a power series      120
Common refinement of partitions      88
Commuting limits      106
Compact set      45
Compact set in a metric space      146
Comparison of the root and ratio tests      26
Comparison Test      23
Completeness of a metric space      142
Conditional convergence of series      31
Connected set      50
Continuity      57
Continuity and closed sets      61
Continuity and open sets      60
Continuity and sequences      59
Continuity of a function on a metric space      143
Continuity under composition      60
Continuous functions are integrable      89
Continuous image(s) of a compact set      62
Continuous image(s) of connected sets      65
Continuously differentiable      82
Convergence in a metric space      140
Convergence of a sequence of functions      103
Cosine function      126
Counterexample to the convergence of taylor series      122
Darboux's theorem      75
Decomposition of a function of bounded variation      100
density      147
Derivative      71
Derivative of inverse function      80
Derived power series      118
Differentiable      71
Differential equations      153
differential equations first order      153
Dirichlet kernel      136
Dirichlet problem on the disc      171
disconnected set      50
Discontinuity of the first kind      66
Discontinuity of the second kind      66
Eigenfunction      175 176
Elementary operations on real analytic functions      115
Elementary properties of continuity      59
Elementary properties of sine and cosine      128
Elementary properties of the derivative      72
Elementary properties of the exponential function      124
Elementary properties of the integral      90
Equibounded family      151
Equicontinuous family      150
Euler's equidimensional equation      170
Euler's formula      127
Euler's number e      19 34
Exponential function(s)      123 131
Fourier coefficient      134
Fourier series      134
Function of bounded variation      99
Functional analysis      176
Fundamental theorem of calculus      92 93
Gamma function      132
Genericity of nowhere differentiable functions      150
Geometric series      24
Harmonic series      24
Heat distribution on the disc      171
Heine — Borel theorem      47
Image of a function      62
Initial condition      154
Initial curve      159
Integrable functions are bounded      89
Integral equation      154
Integration by parts      97
Interior point      43
Intermediate Value Theorem      65
Interval of convergence      114
Irrationality of e      34
Isolated point      43
l'Hopital's rule      79
Laplace equation      169
Least upper bound      7
Left limit      66
Legendre's equation      164
Length of a set      48
Limit of a function at a point      53
Limit of a function on a metric space      143
Limit of functions using sequences      57
Limit of riemann sums      87
Lipschitz condition      153
Local maximum      74
Local minimum      74
Lower integral      94
Lower riemann sum      93
Mean value theorem      76
Mesh of a partition      85
Method of bisection      45 147
Method of characteristics      158 159
Method of Frobenius      166
Metric space      139
Monotone decreasing function      67
Monotone decreasing sequences      13
Monotone function      67
Monotone increasing function      67
Monotone increasing sequences      13
Natural logarithm function      130
Nowhere differentiable function      73
Number $\pi$      129
Open ball in a metric space      144
Open covering      46
Open covering in a metric space      147
Open subcovering in a metric space      147
Orthogonality condition      176
Partial sum of a fourier series      136
Partition      85
Perfect set      51
Picard estimations of      157
Picard iterates      154
Picard iteration technique      154
Picard theorem      153
Pinching principle      14
Pointwise convergence of Fourier series      137
Poisson integral formula      172 173
Power sequences      17
Power series      113
Power series for solving a differential equation      160
Principle of superposition      175
Product of integrable functions      91
Pseudodifferential operators      176
Radius of convergence      117
Ratio Test      25 27
Rational and real exponents      17
Real analytic      114
Rearrangement of series      32
Refinement of a partition      94
Reversing the limits of integration      90
Riemann integral      87
Riemann lemma      96
Riemann sum      86
Riemann — Stieltjes integral      93 94
Riemann — stieltjes integral existence of      96
Right limit      66
Rolle's theorem      76
Root test      25 27
Scalar multiplication of series      36
Separation of variables      174
Separation of variables method      170
Sequence $j^{1/j}$      18
Sequence of functions      103
Series of functions      108
Set theory      176
Simple discontinuity      66
Sine function      126
Stirling's formula      133
Strictly monotonically decreasing      69
Strictly monotonically increasing      69
Subcovering      46
Summation by parts      29
Taylor's expansion      121
Term-by-term integration of power series      120
Total variation      99
Totally disconnected set      51
Uniform continuity      63
Uniform continuity and compact sets      64
Uniform convergence      104
Uniformly cauchy sequences of functions      107
Uniqueness of limits      54
Upper bound      7
Upper integral      94
Upper Riemann sum      93
Value of $\pi$      129
Vibrating string      174
Wave equation      174
Weierstrass approximation theorem      111
Weierstrass M-test      110
Weierstrass nowhere differentiable function      73
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