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Jones D.S. — Introduction to Asymptotics: A Treatment Using Nonstandard Analysis

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Название: Introduction to Asymptotics: A Treatment Using Nonstandard Analysis

Автор: Jones D.S.

Аннотация:

Many branches of science and engineering involve applications of mathematical analysis. An important part of applied analysis is asymptotic approximation which is, therefore, an active area of research with new methods and publications being found constantly. This book gives an introduction to the subject sufficient for scientists and engineers to grasp the fundamental techniques, both those which have been known for some time and those which have been discovered more recently. The asymptotic approximation of both integrals and differential equations is discussed and the discussion includes hyperasymptotics as well as uniform asymptotics. There are many numerical examples to illustrate the relation between theory and practice. Exercises in the chapters enable the book to be used as a text for an introductory course.

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Рубрика: Математика/Анализ/Асимптотические методы, Теория возмущений/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1996

Количество страниц: 165

Добавлена в каталог: 26.03.2005

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Предметный указатель

$J(\mu,z)$       66 67 79
$J(\mu,z)$ , analytic continuation      80
$V_{\mu}(x)$       90
$V_{\mu}(x)$ as parabolic cylinder function      90
$V_{\mu}(x)$ asymptotic behaviour      91
$\mathbb{C}$       142
$\mathbb{N}$       142
$\mathbb{R}$       142
(T)      143
Abramowitz, M.      90 91 159
Airy function      45 63
Airy function, analytic continuation      47
Airy function, asymptotic behaviour      46 63
Airy function, differential equation      47
Airy function, integral representation      45
Algorithm for nonstandard statement      3
Almost minimal term      74
Analytic continuation      24 47 51 66 76
Analytic continuation, singularities      51
Asymptotic expansion,      5
Asymptotic expansion, coefficient in      6
Asymptotic expansion, definition      5
Asymptotic expansion, division by      10 15
Asymptotic expansion, dominant      123
Asymptotic expansion, function of      9
Asymptotic expansion, generalised      12
Asymptotic expansion, integral of      7
Asymptotic expansion, inversion of      10
Asymptotic expansion, recessive      123
Asymptotic expansion, uniform in angle      12
Asymptotic expansion, uniqueness      6
Asymptotic expansions, addition of      7
Asymptotic expansions, multiplication of      8
Asymptotic sequence      3
Asymptotic sequence, definition      4
Asymptotic sequence, derivative      5
Asymptotic sequence, example      4
Asymptotic sequence, subsequence      4
Asymptotic series      43
Asymptotic series, general theorems      43
Asymptotics, uniform      79
Bernstein, A. R.      141 159
Berry, M. V.      99 159
Bessel function, modified      61 62 69 112
beta function      115
Bleistein, N.      2 159
Boyd, W. G. C      99 159
Chester, C      92 159
Coalescing saddle points      92
Coalescing saddle points, phase determination      93
Compact interval      150
Complex contours      27
Confluent hypergeometric function      21 61
Connection formulae      132 133
Continuity and S-continuity      149
Continuity and strong differentiability      153
Continuity on compact interval      150
Conventional quantities standard      143
Convergence of functions      148
Convergence, Maclaurin — Cauchy test      158
Critical point      39
Curve of steepest descent      28
Derivative, strong      153 155
Differential equation      119
Differential equation with parameter      123
Differential equation with pole      138
Differential equation with zero      136 137
Differential equation, analytic continuation of series      132
Differential equation, connection formulae      132 133
Differential equation, error bound for series      129 132
Differential equation, error bound variation with path      125
Differential equation, hyperasymptotics      135
Differential equation, solution in series      126
Differential equation, various cases      126
Differential equation, WKB approximation      119
Dingle, R. B.      99 159
Divergence      1
Dominant asymptotic expansion      123
Endpoint, near saddle point      89
Entire function      45
Entire function, asymptotic behaviour      59
Entire function, exponential type      50
Entire function, integral of      51
Expansion, asymptotic      5
Fourier integral      32
Fourier integral as limit      34
Fourier integral, integrand of bounded variation      34
Friedman, B.      92 159
Function, $J(\mu,z)$       66 67 79
Function, $V_{\mu}(x)$       90
Function, Airy      45 63
Function, Beta      115
Function, bounded variation      34
Function, confluent hypergeometric      21 61
Function, entire      45 59
Function, generalised      2
Function, incomplete gamma      61
Function, parabolic cylinder      90
Function, regular      11
Function, S-continuous      149
Function, S-differentiable      151
Function, standard      144
Function, strongly differentiable      153
Generalised function      2
Handelsman, R. A.      2 159
Howls, C. J.      99 159
Hyperasymptotics      99
Hyperasymptotics, accuracy      106
Hyperasymptotics, analytic continuation      109
Hyperasymptotics, differential equation      135
Hyperasymptotics, numerical examples      107 109 113 117 118
Hyperasymptotics, Stieltjes transform      110
Hyperasymptotics, technique      106
Idealisation rule      142
Incomplete gamma function      61
Infinite integral      156
Infinite integral of infinitesimal      156
Infinite series      158
Infinite series, sum of      158
Infinitesimal      3 20
Infinitesimal number      145
Infinitesimal, standard      145
Integer, nonstandard      142
Integral and series      158
Integral as standard part      154
Integral equation      121
Integral equation, iteration in      121
Integral of continuous function      154
Integral of entire function      51
Integral of infinitesimal      155 156
Integral, asymptotic behaviour      17
Integral, contour      27
Integral, Fourier      32
Integral, infinite      156
Integral, Laplace      23 44 100
Integral, Laplace-type      26
Integral, partial sum of      65
Integral, Pearcey      95

Integral, ray      25
Integral, singular      156
Integral, swallowtail      95
Interval, compact      150
Interval, standard      143
Iteration      121
Izumi, S.      56 159
Jones, D. S.      2 159
Laplace integral      23 44
Laplace integral, hyperasymptotics      100
Laplace-type integral      26
Lighthill, M. J.      2 159
Limited number      145
Maclaurin series      6 70 100
Maclaurin — Cauchy test      158
Maple      83 107
Mathematica      83 107
McLeod, J. B.      99 159
Method of stationary phase      39
Method of steepest descents      28
Modified Bessel function      61 62 69 112
Nelson, E.      3 141 159
Nonstandard analysis      141
Nonstandard integer      142
Nonstandard set      141
Number, infinitesimal      145
Number, limited      145
Number, unlimited      19 142
Olde Daalhuis, A. B.      99 134 159
Olver, F. W. J.      2 99 134 159 160
Optimal remainder      100
Parabolic cylinder function      90
Paris, R. B.      99 160
Part, standard      145 150
Partial sum      53 56 65
Partial sum of integral      65
Pearcey integral      95
Pole near saddle point      87
Power series      10 49
Power series, coefficients      49
Power series, derivative of      11 12
Power series, growth of coefficients      51
Power series, integral of      11 15
Power series, normalised      49
Ray integral      25 48 67
Recessive asymptotic expansion      123
Regular function      11
Remainder      1 2
Remainder, expansion of      99
Remainder, exponentially small      73
Remainder, hyperasymptotics      135
Remainder, minimising      102 103
Remainder, numerical examples      69
Remainder, optimal      73 100
Riemann surface      4
Riemann — Lebesgue lemma      32
Robert, A.      2 160
Robinson, A.      141 159 160
Robinson’s Continuous Lemma      13 149
Robinson’s Lemma      147
Rolle’s Theorem      152
S-continuity      149 150
S-continuity and continuity      149
S-continuity, examples      150
S-continuous      17
S-continuous function      149
S-derivative      151
S-derivative, higher      153
Saddle point      27
Saddle point, near endpoint      89
Saddle point, near pole      87
Saddle points, coalescing      92
SEQUENCE      146
Sequence of functions      148
Sequence, asymptotic      3
Sequence, standard      146 148
Series      43
Series and integral      158
Series, divergent      1
Series, infinite      158
Series, Maclaurin      6 70 100
Series, power      10 49
Set      141
Set of all subsets      143
Set of standard elements      144
Set theory      141
Set theory, Zermelo — Fraenkel system      141
Set, nonstandard      141
Set, standard      141
Singular integral      156
Standard function      144 147
Standard functions, equality of      144
Standard infinitesimal      145
Standard interval      143
Standard part      145 150
Standard sequence      146 148
Standard set      141
Standard sets, equality of      143
Standardisation rule      142 150
Stationary phase      38
Stationary phase, method of      39
Steepest descent, curve of      28
Steepest descent, method of      28
Stegun, I. A.      90 91 159
Stieltjes transform      110 135
Stieltjes transform, analytic continuation      113
Stieltjes transform, analytic continuation, hyperasymptotics      116
Stieltjes, T. J.      99 160
Stirling’s formula      17 19
Stokes line      79
Stokes’ phenomenon      79 113 135
Strong derivative      155
Strong differentiability      153
Strong differentiability and continuity      153
Sum of infinite series      158
Sum of infinitesimals      158
Sum, partial      53
Sum, partial of integral      65
Sum, partial, asymptotics      53 56 65
Surface, Riemann      4
Swallowtail integral      95
Taylor’s Theorem      153
Transfer rule      3 143
Truncation      1
Truncation, best      74
Truncation, numerical examples      69
Uniform asymptotic expansion      12
Uniform asymptotic representation      79
Uniform asymptotics      79
Unlimited number      19 142
Ursell, F.      92 159
van den Berg, L      2 160
Watson’s lemma      44 45
WKB approximation      119 120 126
WKB approximation, error bound      120
Wong, R.      2 160
Wood, A. D.      99 160
Zermelo — Fraenkel system      141

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