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Jordan C. — Calculus of Finite Differences
Jordan C. — Calculus of Finite Differences

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Название: Calculus of Finite Differences

Автор: Jordan C.

Аннотация:

This book, a result of nineteen years’ lectures on the Calculus of Finite Differences, Probability, and Mathematical Statistics in the Budapest University of Technical and Economical Sciences, and based on the venerable works of Stirling, Euler and Boole, has been written especially for practical use, with the object of shortening and facilitating the labours of the Computer. With this aim in view, some of the old and neglected, though useful, methods have been utilized and further developed: as for instance Stirling’s methods of summation, Boole’s symbolical methods, and Laplace’s method of Generating Functions, which last is especially helpful for the resolution of equations of partial differences.


Язык: en

Рубрика: Математика/Анализ/Учебники по элементарному анализу/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2-nd edition

Год издания: 1950

Количество страниц: 652

Добавлена в каталог: 09.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$(log(z+1))^{n}$ expansion into powers      146 202 204
$b_{m}$ numbers, table of      266
$C_{m\nu}$ numbers, table of      152
$C_{m}$, numbers, table of      172
$\beta$ functions      347
$\beta$ functions by a definite integral      351
$\beta$ functions into Boole series      438
$\beta$ functions into Euler polynomials      349
$\beta$ functions into Euler series      349
$\beta$ functions, derivatives of      352
$\beta$ functions, expansion by digamma function      351
$\beta$ functions, expansion into Boole polynomials      348
$\beta_{m}$ numbers, table of      449 450
$\Gamma( I’m)$ numbers, table of      155
$\Gamma(\int m)$ numbers, table of      154
$\gamma_{2m}$ numbers table of      429
Addition of differences, method of      76
Alternate functions      92—94
Alternate functions, sum of      320 324
Alternate functions, summation by parts      108
Alternate reciprocal powers am numbers, table of      232 251
Alternate reciprocal powers, expressed by sums      251
Alternate reciprocal powers, generating function of      251
Alternate reciprocal powers, sum of      251
Antilogarithmic tables      399 402
Approximation by Bernoulli's probability function      432
Approximation by incomplete B function      428
Approximation by incomplete r function      426
Approximation by Laplace's probability function      430
Approximation by Poisson's probability function      431
Approximation, principle of least squares      422
Approximation, principle of moments      422
Arithmetical progression      79
Bayes’ Theorem      86
Bernoulli numbers      23 62 130 140 181 182 227 229
Bernoulli numbers by Cmi      229 599
Bernoulli numbers by Stirling numbers      219 236
Bernoulli numbers, ex-pressed by b      250
Bernoulli numbers, symbolical formula of      233
Bernoulli numbers, table of      234 235
Bernoulli polynomials into Boole polynomials      323
Bernoulli polynomials into Euler polynomials      308
Bernoulli polynomials into Fourier series      242
Bernoulli polynomials, expansion into Bernoulli polynomials first kind      276
Bernoulli polynomials, expansion into factorials      235
Bernoulli polynomials, extrema      240
Bernoulli polynomials, first kind      60 62 181 230
Bernoulli polynomials, generating function of      279
Bernoulli polynomials, limits      272
Bernoulli polynomials, multiplication formula of      252
Bernoulli polynomials, roots      240
Bernoulli polynomials, second kind      64 72 134 147 224 265 277
Bernoulli polynomials, symmetry      268
Bernoulli series      253
Bernoulli series, second kind      280
Bernoulli’s solution of numerical equations      494
Beta functions      50 80
Binomial coefficients      62—70
Binomial coefficients, Cauchy’s formula of      73
Binomial coefficients, generalised      70
Binomial moments      424
Binomial moments by Stirling numbers      267 147
Binomial moments, computation of      460 613
Binomial moments, expressed by Bernoulli numbers      249 278
Binomial moments, generating function of      258 279 284
Boole polynomials      64 317 321
Boole polynomials, expanded into Bernoulli polynomials second kind      321
Boole series      323
Boole’s first summation formula      315 354
Boole’s first summation formula, second summation formula      323
Cauchy’s formula for binomials      24 41 48 68
Cauchy’s formula, integral      41
Cauchy’s formula, rule for summing binomials      73 133 135 136
Cauchy’s formula, rule of multiplication of series      25
Changing the origin of intervals      219
Changing the origin of intervals the length of intervals      220
Combinations, difference equation giving number of      578
Construction of tables      363
Correlation, coefficient of      455
cot z expansion of      259
Cotes numbers      224 388
Cotes numbers, determined by Stirling numbers      514
Decomposition into partial fractions      34—40 335
Derivatives expressed by $\theta$ operation      197
Derivatives expressed by differences      164—165
Difference equation linear variable coefficients first order      576
Difference equation, complete      583
Difference equation, homogeneous by Laplace’s method      579
Difference equation, reducible      584
Difference equation, resolution by Andre’s method      587
Difference equation, resolution by generating functions      586
Difference equations by method of arbitrary constants      569
Difference equations with complex roots      554
Difference equations with multiple roots      549—552
Difference equations with negative roots      552
Difference equations, characteristic equation with single roots      542
Difference equations, genesis of      543
Difference equations, linear mixed      632
Difference equations, linear, constant coefficients, complete      557—564
Difference equations, linear, constant coefficients, homogeneous      545
Difference equations, particular solution obtained directly      564
Difference equations, resolved by generating functions      572
Difference of a function with negative argument      5 69
Difference of a product      94—98
Differences (advancing)      2
Differences (advancing), central      15
Differences (advancing), divided      18
Differences (advancing), receding      14
Differences expressed by derivatives      189—192
Differences expressed by means      9
Differences expressed by the operation      200
Differences expressed by values of the function      8
Digamma function      58—60
Digamma function by Bernoulli series, of the second kind      283
Digamma function by powers series      327
Digamma function by reciprocal powers      328
Digamma function expressed by Bernoulli series      256
Digamma function, sum of      1
Displacement      6
Endpanel interpolation formula      379
Euler numbers      23 300
Euler numbers limits of      302
Euler polynomials      62 115 288
Euler polynomials, extrema of      293
Euler polynomials, generating function of      309
Euler polynomials, inverse difference of      297
Euler polynomials, inverse means of      297
Euler polynomials, limits of      302
Euler polynomials, multiplication theorem of      311
Euler polynomials, roots of      293
Euler polynomials, symmetry of      292
Euler’s constant      27 55 58 129 130 148 341
Euler’s formula for f functions      55 83
Euler’s formula for trigonometric functions      90
Euler’s polynomials expanded into Bernoulli polynomials      295
Expansion of a function by decomposition into, partial fractions      34 40
Expansion of a function of functions      31 33 204 205
Expansion of a function, complex intergrals      40—41
Expansion of a function, difference equations      41—44
Expansion of a function, symbolical methods      11
Expansion of functions into Bernoulli polynomials      307 248 250
Expansion of functions into Bernoulli series      253
Expansion of functions into binomials      74
Expansion of functions into Boole polynomials      322
Expansion of functions into Boole series      323
Expansion of functions into Euler polynomials      307
Expansion of functions into Euler series      313
Expansion of functions into Legendre polynomials      434
Expansion of functions into Newton series      74 358
Expansion of functions into orthogonal series      447
Expansion of functions into polynomials      355
Expansion of functions into powers      29—34
Expansion of functions into reciprocal factorials      192 212
Expansion of functions into reciprocal powers      212
Exponential functions      87—88
Faa Bruno’s expansion of a function of function      33 205
Factorial moments determined by generating functions      208
Factorials      45—53
Factorials, computation of      52
Factorials, definition by gamma functions      56
Factorials, difference of      51
Factorials, expansion into powers      142
Factorials, mean of      52
Fibonacci numbers      548
First panel interpolation formula      378
Fourier series      242 426 463
Fourier series, Newton series      289
Functions expressed by differences      10
Functions expressed by means      10
Functions expressed whose differences or means are zero      94
Functions expressed, product of two      94—98
G polynomials      426 473
Gamma function      53—56
Gamma function, computation of      55
Generating functions      20 29 109
Generating functions of binomial coefficients      71 73
Generating functions, determination by difference equations      27
Graduation by least squares and orthogonal polynomials      456
Gregory’s formula of numerical integration      525
Hermite polynomials      63 426 467
Hospital’s rule      133
Hyperbolic functions      38
Incomplete B function      83—37
Indefinite sum      101
Indefinite sum by difference equation      109
Indefinite sum by inversion      103
Indefinite sum by summation by parts      105
Infinite series, differences and means of      110
Interpolation by Bessel's formula      373
Interpolation formula, general case      420 421
Interpolation formula, precision of      417—420
Interpolation in a double entry table      532
Interpolation, case of three variables      541
Interpolation, Everett's formula      376
Interpolation, formula needing no differences      390
Interpolation, Gauss' formulae      363
Interpolation, Newton's formula      361
Interpolation, Stirling's formula      374
Inverse difference      101
Inverse difference, Everett      381
Inverse difference, Formula needing no differences      411
Inverse difference, Lagrange      390
Inverse mean      111—116
Inverse mean of a function      306
Inversion of sums and series      183 185
Iteration, method of, for solving numerical equations      492
Lacroix’s difference equation giving the sum of x      596
Lagrange’s formula      360
Lagrange’s formula, interpolation formula      386
Lagrange’s formula, polynomial      385
Legendre polynomials      389 434
Legendre polynomials, roots of the      435
Leibnitz’ formula of higher derivatives      96 143 167
log, Bernoulli series, second kind      284 287
log, expansion into Bernoulli polynomials, second kind      280
log, tables      399 402 407 409 410
Maclaurin series      201 216 246
Maclaurin — Euler summation formula      260 265
Mean of a function      7
Mean of a function of a product      98—99
Mean of a function, central      15
Mean, arithmetical      433
Mean, binomial moment      448
Mean, orthogonal moment      450
Mean-square deviation      427 433 452
Median      433
Midpanel interpolation formula      397
MODE      433
Moments, actorial      424
Moments, binomial      424
Moments, computation of binomial moments      615
Moments, expressed by semi-invariants      211
Moments, power      163 164 812
Newton — Rapbson solution of numerical equations      489
Newton's binomial formula      49
Newton's binomial formula, expansion      26 75 76 77 79 164 189 219 357
Newton's binomial formula, expansion for unequal intervals      20
Newton's binomial formula, formula for two variables      531
Numerical integration      512
Numerical integration, Euler — Maclaurin formula      525
Numerical resolution of difference equations      527
Orthogonal polynomials      436
Orthogonal polynomials, central value of the      445
Partial difference equations      604
Partial difference equations four independent variables      638
Partial difference equations, Boole's symbolical method      616
Partial difference equations, Fourier, Legendre, Ellis' method      619
Partial difference equations, Laplace's method of generating functions      607
Partial difference equations, linear, constant coefficients      60
Partial difference equations, three independent variables      633
Partial differences      530
Pascal’s arithmetical triangle      612
Power moments expressed by      6
Power moments expressed by operations      197
Power series, sum of      246
Powers expressed by factorials      181
Probability by Stirling numbers      166 177
Probability function, binomial moments of      424
Probability function, expanded into G polynomials      483 484
Probability function, table of      400 403 408
Probability, coincidences      595 610
Probability, determination of, by sums      140
Probability, Examples on, Bayes' theorem      86
Probability, parcours      630 633 638
Probability, problem of      575
Probability, problem of points      608
Probability, repeated trials      86 599 615
Probability, ruin      550 627
Product of prime numbers, decomposition of      179 181
Progression, arithmetical      118
Progression, geometrical      124
Rational fraction, sum of      335
Reciprocal factorial, derivation of      337
Reciprocal factorial, expanded into reciprocal powers      193—195
Reciprocal factorial, integration of      194 338
Reciprocal factorial, sum of      121
Reciprocal powers, alternate sum of      244
Reciprocal powers, difference of      194
Reciprocal powers, sum of      194 244 214 325 338
Reciprocal powers, sum of, by Stirling numbers      338
Regula Falsi      366 486
Remainder of the expansion into a series of polynomials      356
Remainder of the expansion, maximum of      362
Rule of false position      366 486
Schlomilch's expansion formula      31 204
Semi-invariants of Thiele      204 210
Semi-invariants of Thiele, expressed by moments      211
Simmons theorem      87
Simultaneous linear difference equation      601
Stirling numbers of the second lcind      32 134 168 179
Stirling numbers, formulae containing      182 185—189
Stirling polynomials      224—229
Sum without repetition transformed      153—158
Summation by parts      105 106 107
Summation by parts of alternate function      108 140 320 324
Symbolical calc.      7—14
tan z, expansion of      259
Tangent coefficients      130 298
Taylor’s series      13 165 189
Tchebichef polynomial      389
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