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Carslaw H.S. Introduction to the Theory of Fourier's Series and Integrals
Carslaw H.S.  Introduction to the Theory of Fourier's Series and Integrals







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: Introduction to the Theory of Fourier's Series and Integrals

: Carslaw H.S.

: en

: // /

:

ed2k: ed2k stats

: Second Edition

: 1921

: 323

: 09.04.2005

: | | ID
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$\mu$-test for convergence of integrals      104 116
Abel      124 146
Abels theorem on the power series      146
Abels theorem on the power series, extensions of      149
Absolute convergence of integrals      103
Absolute convergence of series      44
Absolute value      32
Aggregate, bounded      30
Aggregate, bounded above (or on the right)      29
Aggregate, bounded below (or on the left)      29
Aggregate, general notion of      29
Aggregate, limiting points of      31
Aggregate, upper and lower bounds of      30
Aggregate, Weierstrasss theorem on limiting points of      32
Approximation curves for a series      124 (see also Gibbss phenomenon)
Baker      122
bernoulli      1 3
BLISS      77
Bocher      8 234 243 263 268271 275 282
Bochers treatment of Gibbss phenomenon      268
Bonnet      97
Bounds (upper and lower) of an aggregate      30
Bounds (upper and lower) of f(x) in an interval      50
Bounds (upper and lower) of f(x, y) in a domain      72
Boussinesq      7
Bromwich      28 48 150 157 163 165 192 193
Bromwichs theorem      151
Brunei      119 193
Burkhardt      2 15
Byerly      198
Cantor      12 14 15 26 27 29
Carse and Shearer      298
Carslaw      155 264
Cauchy      8 9 77 124
Cesaro      12 151
Cesaros method of summing series (CI)      151 238240
Change of order of terms in a conditionally convergent series      47
Change of order of terms in an absolutely convergent series      45
Chapman      13
Chrystal      148
Clairaut      3 5
Closed interval, definition of      49
Conditional convergence of series, definition of      45
Continuity of $\int\limits_a^xf(x)dx$ when f(x) is bounded and integrable      93
Continuity of functions      59
Continuity of infinite integrals involving a single parameter      179 183
Continuity of ordinary integrals $\int\limits_a^xf(x)dx$, involving a single parameter      169
Continuity of the power series (Abels theorem)      146
Continuity of the sum of a uniformly convergent series of continuous functions      135
Continuous functions of two variables      73
Continuous functions, integrability of      84
Continuous functions, non-differentiable      77
Continuous functions, theorems on      60
Continuum, arithmetical      25
Continuum, linear      25
Convergence of functions      51
convergence of integrals      98 (see also Absolute convergence Conditional and
Convergence of sequences      33
Convergence of series      41
Cosine integral (Fouriers integral)      284 292
Cosine series (Fouriers series)      197 215
Darboux      4 6 7
Darbouxs theorem      79
De La Vallee Poussin      10 12 28 48 75 77 119 163 192 193 239 243 262 263
Dedekind      18 19 23 2528
Dedekinds axiom of continuity      24
Dedekinds sections      20
Dedekinds theorem on the system of real numbers      23
Dedekinds theory of irrational numbers      18
Definite integrals containing an arbitrary parameter (Chapter VI.), continuity, integration and differentiation of      169 179
Definite integrals containing an arbitrary parameter (Chapter VI.), infinite integrals      173
Definite integrals containing an arbitrary parameter (Chapter VI.), ordinary integrals      169
Definite integrals containing an arbitrary parameter (Chapter VI.), uniform convergence of      174
Definite integrals, ordinary (Chapter IV.), considered as functions of the upper limit      93
Definite integrals, ordinary (Chapter IV.), Darbouxs theorem      79
Definite integrals, ordinary (Chapter IV.), definition of      81
Definite integrals, ordinary (Chapter IV.), definition of upper and lower integrals      81
Definite integrals, ordinary (Chapter IV.), first theorem of mean value      92
Definite integrals, ordinary (Chapter IV.), necessary and sufficient conditions for existence      82
Definite integrals, ordinary (Chapter IV.), second theorem of mean value      94 (see also Dirichlets integrals Fouriers Infinite
Definite integrals, ordinary (Chapter IV.), some properties of      87
Definite integrals, ordinary (Chapter IV.), the sums S and s      77
Descartes      26
Differentiation of Fouriers series      261
Differentiation of infinite integrals      182
Differentiation of ordinary integrals      170
differentiation of power series      148
Differentiation of series      143
Dini      12 28 112 119 163 193 243
Dirichlet      4 8 9 11 77 200 208
Dirichlets conditions, definition of      206
Dirichlets integrals      200
Discontinuity of functions      64
Discontinuity, classification of      65 (see also Infinite discontinuity and Points of infinite discontinuity)
Divergence of functions      51
Divergence of integrals      98
Divergence of sequences      37
Divergence of series      41
Donkin      124
du Bois-Reymond      12 34 77 97 112
DAlembert      1 3
Euclid      25 26
Euler      15
Fejer      13 234 258
Fejers theorem      234
Fejers theorem and the convergence of Fouriers series      240 258
FOURIER      29 243 294
Fouriers constants, definition of      196
Fouriers integrals (Chapter X.), cosine and sine integrals      292
Fouriers integrals (Chapter X.), more general conditions for      287
Fouriers integrals (Chapter X.), simple treatment of      284
Fouriers integrals (Chapter X.), Sommerfelds discussion of      293
Fouriers series (Chapters VII. and VIII.) for even functions (the cosine series)      215
Fouriers series (Chapters VII. and VIII.) for intervals other than $(-\pi, \pi)      228
Fouriers series (Chapters VII. and VIII.) for odd functions (the sine series)      220
Fouriers series (Chapters VII. and VIII.), definition of      196
Fouriers series (Chapters VII. and VIII.), differentiation and integration of      261
Fouriers series (Chapters VII. and VIII.), Fejers theorem      234 240
Fouriers series (Chapters VII. and VIII.), Lagranges treatment of      198
Fouriers series (Chapters VII. and VIII.), more general theory of      262
Fouriers series (Chapters VII. and VIII.), order of the terms in      248
Fouriers series (Chapters VII. and VIII.), Poissons discussion of      230
Fouriers series (Chapters VII. and VIII.), proof of convergence of, under certain conditions      210
Fouriers series (Chapters VII. and VIII.), uniform convergence of      253
Functions of a single variable of bounded variation      207
Functions of a single variable, bounded in an interval      50
Functions of a single variable, continuous      59
Functions of a single variable, definition of      49
Functions of a single variable, discontinuous      64
Functions of a single variable, integrable      84
Functions of a single variable, inverse      68
Functions of a single variable, limits of      50
Functions of a single variable, monotonic      66
Functions of a single variable, oscillation in an interval      50
Functions of a single variable, upper and lower bounds of      50
Functions of several variables      71
General principle of convergence of functions      56
General principle of convergence of sequences      34
Gibb      301
Gibbs      268270 282
Gibbss phenomenon in Fouriers series (Chapter IX.)      264
Gibson      15 195 262
Gmeiner      see Stolz and Gmeiner
Goursat      28 48 59 75 77 94 119 153 163 165 193 243
Gronwall      270
Hardy      13 19 20 48 75 77 94 130 239
Hardys theorem      239
Harmonic analyser (Kelvins)      295
Harmonic analysis (Appendix I.)      295
Harnack      112
Heine      11 12 14 26
Hildebrandt      77
Hobson      12 17 28 69 89 94 138 163 193 234 243 263 275
Improper integrals, definition of      113
Infinite aggregate      see Aggregate
Infinite discontinuity      see Points of infinite discontinuity
Infinite integrals (integrand function of a single variable), $\mu$-test and other tests for convergence of      114
Infinite integrals (integrand function of a single variable), $\mu$-test for convergence of      104
Infinite integrals (integrand function of a single variable), absolute convergence of      103
Infinite integrals (integrand function of a single variable), integrand bounded and interval infinite      98
Infinite integrals (integrand function of a single variable), integrand infinite      111
Infinite integrals (integrand function of a single variable), mean value theorems for      109
Infinite integrals (integrand function of a single variable), necessary and sufficient condition for convergence of      100
Infinite integrals (integrand function of a single variable), other tests for convergence of      106
Infinite integrals (integrand function of a single variable), with positive integrand      101
Infinite integrals (integrand function of two variables), continuity, integration and differentiation of      179
Infinite integrals (integrand function of two variables), definition of uniform convergence of      174
Infinite integrals (integrand function of two variables), tests for uniform convergence of      174
Infinite sequences and series      see Sequences and Series
Infinity of a function, definition of      65
Integrable functions      84
Integration of integrals, infinite      180 183 190
Integration of integrals, ordinary      172
Integration of series (ordinary integrals)      140
Integration of series (ordinary integrals), (infinite integrals)      154
Integration of series (ordinary integrals), Fouriers series      261
Integration of series (ordinary integrals), power series      148
Interval: open, closed, open at one end and closed at the other      49
Inverse functions      68
Irrational numbers      see Numbers
Jackson      270
JORDAN      12 207 243 294
Jourdain      14
Kelvin, Lord      6
Kneser      14
Kowalewski      119 163
Lagrange      2 3 5 6 198
Laplace      6
Lebesgue      10 12 77 112 119 243 263 294
legendre      6
Leibnitz      26
Limit repeated      127
Limiting points of an aggregate      31
limits of functions      50
Limits of functions of two variables      72
Limits of sequences      33
Lipschitz      12
Littlewood      239
Lower integrals, definition of      81
M-test for convergence of series      134
Mean value theorems of the integral calculus, (infinite integrals)      109
Mean value theorems of the integral calculus, first theorem (ordinary integrals)      92
Mean value theorems of the integral calculus, infinite integrals      109
Mean value theorems of the integral calculus, second theorem (ordinary integrals)      94
Modulus      see Absolute value
Monotonic functions      66
Monotonic functions, admit only ordinary discontinuities      67
Monotonic functions, integrability of      84
Monotonic in the stricter sense, definition of      39
Moore      150
Neighbourhood of a point, definition of      52
Neumann      243 294
Newton      26
Numbers (Chapter I.), Dedekinds theorem on the system of real      23
Numbers (Chapter I.), Dedekinds theory of irrational      18
Numbers (Chapter I.), development of the system of real      25 (see also Dedekinds axiom of continuity and Dedekinds sections)
Numbers (Chapter I.), irrational      17
Numbers (Chapter I.), rational      16
Numbers (Chapter I.), real      21
Open interval, definition of      49
Ordinary or simple discontinuity, definition of      65
Oscillation of a function in an interval      50
Oscillation of a function in an interval of a function of two variables in a domain      72
Oscillatory, functions      52
Oscillatory, integrals      99
Oscillatory, sequences      38
Oscillatory, series      41
Osgood      45 58 75 119 129 163 193
Partial remainder $(_pR_n(x))$, definition of      123
Partial remainder $(_pR_n)$, definition of      42
Periodogram analysis      299
Picard      14 234
Pierpont      75 83 86 93 110 119 163 193
Poincare      7
Points of infinite discontinuity, definition of      66
Points of oscillatory discontinuity, definition of      66
poisson      7 8 230 231 293
Poissons discussion of Fouriers series      230
Poissons integral      231
Power series, Abels theorem on      146
Power series, integration and differentiation of      148
Power series, interval of convergence of      145
Power series, nature of convergence of      146
Pringsheim      16 25 28 34 48 75 163 287 291
Proper integrals, definition of      113
Raabe      165
Rational numbers and real numbers      see Numbers
Remainder after n terms, ($R_n$), definition of      43
Remainder after n terms, ($R_n(x)$)      123
Repeated integrals, (infinite)      180 183 190
Repeated integrals, (ordinary)      172
Repeated limits      127
Riemann      5 911 15 77 112 243
Runge      268 297
Russell      26 28
Sachse      15 124
Schuster      301
Sections      see Dedekinds sections
Seidel      11 130
Sequences, convergent      33
Sequences, divergent and oscillatory      37
Sequences, limit of      33
Sequences, monotonic      39
Sequences, necessary and sufficient condition for convergence of (general principle of convergence)      34
Series with positive terms      43
Series, absolute and conditional convergence of      44
Series, convergent      41
Series, definition of sum of an infinite      41
Series, definition of sum, when terms are functions of a single variable      122
Series, divergent and oscillatory      41
Series, necessary and sufficient condition for convergence of      42
Series, necessary and sufficient condition for uniform convergence of      132
Series, term by term differentiation and integration of      140 (see also Differentiation of series Fouriefs Integration Poiver
Series, uniform convergence and continuity of      135
Series, uniform convergence of      129
Series, Weierstrasss M-test for uniform convergence of      134
Shearer      see Carse and Shearer
Simple (or ordinary) discontinuity, definition of      65
Sine integral (Fouriers integral)      284 292
Sine series (Fouriers series)      197 220
Sommerfeld      293
Stekloff      14
Stokes      11 130
Stolz      119 163 193
Stolz and Gmeiner      16 28 48
Summable series (CI), definition of      151
Sums S and s, definition of      77
Tannery      28
Topler      14
Trigonometrical series      196
Turner      299
Uniform continuity of a function      62
uniform convergence of integrals      174
Uniform convergence of series      129
Upper integrals, definition of      81
Van Vleck      14
Watson      see Whittaker and Watson
Weber Riemann      243 294
Weierstrass      14 26 27 77 97 134
Weierstrasss M-test for uniform convergence      134
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