Boros G., Moll V. — Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals :: Электронная библиотека попечительского совета мехмата МГУ

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Boros G., Moll V. — Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals

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Название: Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals

Авторы: Boros G., Moll V.

Аннотация:

Boros (mathematics, Xavier U. of Louisiana before his death) and Moll (mathematics, Tulane U.) write on areas of mathematics that appear in the evaluation of integrals. Writing for junior or senior undergraduate students, they assume readers to have a good knowledge of one-variable calculus, and to have been exposed to rigorous proof in a course such as Discrete Mathematics. They emphasize the connection with number theory and the connection between the discrete—prime numbers, binomial coefficients—and the continuous—integrals, special functions.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2004

Количество страниц: 306

Добавлена в каталог: 11.06.2008

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Предметный указатель

Harmonic numbers of higher order      202
Harmonic numbers, definition      76
Harmonic numbers, relation to $\gamma$       173
Harmonic numbers, relation to Riemann hypothesis      77
Harmonic series, definition      78
Harmonic series, divergence      78
Hauss, M., series of $(sin^{-1}x)^n$       123
Havil, J., information on $\gamma$       173
Havil, J., the Riemann hypothesis      220
Hellman, M., solution of cubics and quarries      39
Hermite, Ch., reduction procedure      147
Hermite, Ch., transcendence of e      89
Hijab, O., differentiation with respect to a parameter      21
Hijab, O., infinite products      129
Hijab, O., manipulation of power series      63
Hijab, O., powers of irrational exponent      22
Holder, O., theorem on $\Gamma$       210
Homogeneous form      203 270
Hurwitz zeta function      see Advanced functions Hurwitz
Huylebrouch, D., irrationality proofs      232
Hypergeometric function      see Advanced functions Hypergeometric
Integral, combinations of exponentials and polynomials      103
Integral, combinations of logarithms and rational functions, linear denominators      239
Integral, combinations of logarithms and rational functions, quadratic denominators      239
Integral, combinations of polynomials and logarithms      81 238
Integral, combinations of trigonometric functions and polynomials      135
Integral, cosine      136
Integral, elliptic      88 110
Integral, exponential      98 104
Integral, Frullani      98 178
Integral, iterated of logarithms      82
Integral, Laplace      171
Integral, logarithmic      98
Integral, master formula      250
Integral, normal      162
Integral, polynomials      20
Integral, powers of loggamma      203
Integral, powers of logsine      245
Integral, products of logarithms      244
Integral, rational functions      25
Integral, rational functions, biquadratic denominator      44
Integral, rational functions, cubic denominator      42
Integral, rational functions, linear denominator      48
Integral, rational functions, normalization      27
Integral, rational functions, quadratic denominator      241
Integral, rational functions, quartic denominator      137
Integral, rational functions, Wallis* integral      see Wallis' formula
Integral, representations for psi-function      215
Integral, representations for Riemann zeta function      222
Integral, representations of beta and gamma      193
Integral, sine      136
Jacobi, C., polynomials      155
Jacobi, C., sums of two squares      166
Jacobi, C., theta function      205
Johnson, W., zeta series      214
Johnsonbaugh, R.F., existence of $\gamma$       175
Kalman, D., methods to sum $\zeta(2)$       225
Kazarinoff, N.D., definitions of e      86
Kerney, K., logarithmic integral      244
Klamkin, M.S., some harmonic series      78
Knulh, D., compulation of Euler's constant      184
Koecher, M., expression for $\zeta(5)$       235
Koepf.W.      14
Kontsevich, M., conjecture on periods      262
Korlram, R., normal integral and sums of two squares      166
Kummer, E., regular primes      101
Lagarias, J., harmonic numbers and the Riemann hypothesis      77
Lagrange's inversion formula      150
Lambert, J.H., e is irrational      89
Landen, J., special values of dilogarithm      245
Laplace, P., integral      171 251
Laplace, P., transform      192
Larson, R. et al      112
Laugwitz, D      187
Lech — Mahler — Skolem theorem      69
Legendre, A.M., definition of $\Gamma$       186
Legendre, A.M., duplication formula for $\Gamma$       195 197 198 205 255
Legendre, A.M., formula for the exponent of p in m!      6
Lehmer, D., interesting Taylor expansions      118
Leibnitz, G.      109
Lewin, L., logarithmic integral      63
Lindemann, F., $\pi$ is transcendental      117
Linis, V., logarithmic integral      242
Liouville, J., e is not a quadratic irrational      90
Liouville, J., proof of Legendre's duplication formula for $\Gamma$       197
Liouville, J., theorem on elementary primitives      162
Little, J., polynomials with roots on a vertical line      210
Logarithm, definition      19 73
Logarithm, growth and harmonic series      77
Logarithm, hypergeomelric representation      75
Logarithm, In x is not algebraic      80
Logarithm, In x is not rational      79
Logarithm, iterated integrals      82
Logarithm, power series      75
Logarithmic derivative of $\Gamma$       212
Logarithmic derivative, second      129 132 204
Logarithmic integral      see Advanced functions. Logarithmic integral
Logconcavity      157
Loggamma function      see Advanced functions Loggamma
Lossers O.P.      229
Loxton, J.H., special values of dilogarithm      245
Lucas, E., denominators of Bernoulli numbers      101
Maor, E., information about e      85
Marchisolto, E.A., integration in finite terms      162
Marsaglia, G. and J.C., proof of Stirling's formula      95
Master formula      250
Mathemalica commands, Apart      26
Mathemalica commands, Assumptions      116
Mathemalica commands, BernoulliB      100
Mathemalica commands, Binomial      17
Mathemalica commands, CoefficientList      9
Mathemalica commands, Complexity function      27
Mathemalica commands, Element      21
Mathemalica commands, Extension      149
Mathemalica commands, Factor      15
Mathemalica commands, Factorial      3
Mathemalica commands, Factorlnteger      4
Mathemalica commands, Floor      5
Mathemalica commands, FullSimplify      21
Mathemalica commands, If      11
Mathemalica commands, Pochhammer      17
Mathemalica commands, PolynomialQuotient      138
Mathemalica commands, PolynomialRemainder      138
Mathemalica commands, Root      40
Mathemalica commands, Series      9
Mathemalica commands, SlirlingSl      18
Mathemalica commands, Solve      40
Mathemalica commands, Sum      15
Mathemalica commands, Table      11
Matsuoka, Y., proof of $\zeta(2) = \pi^2/6$       227
McMillan, E.M., calculation of $\gamma$       184
Medina, H., polynomial approximation of $tan^{-1}x$       124
Mellin transform      55 196
Mendelson, N.S., limit form fort      88
Mordell, L., sign of Bernoulli numbers      100
Murty, R., update on Artin's conjecture      134
Myerson, G., Taylor expansions of rational functions      69
Nahim, P.      127
Napier, J., inequality      74
Nemes, I. el ah, examples of WZ-method      275
Newman, D., invariance of elliptic integrals      89
Newman, D., logconcavity and reciprocal of series      158
Newman, D., prime number theorem      220
Nielsen — Ramanujan constants      240
Niven, I., proof of irrationality of $\pi$       117
Number, algebraic      see Algebraic number
Number, bernoulli      see Bernoulli number
Number, Eulerian      see Eulerian number
Osier, T., relation between Vieta's and Wallis' formula for $\pi$       116
p-adic valuations      4

Parameters in integrals      20
Parameters, differentiation with respect to      21
Partial fractions      25
Pathak, P., proof of Stirling's formula      92
Pennisi, L., e is irrational      89
periods      262
Petkovsek, M., rational certificates      275
Plouffe, S., expression for $\pi$       122
Pochhammer symbol      see Ascending factorial symbol
Poly logarithms      see Advanced functions Polylogarithms
Polya, G. and Szego, G.      86
Polygamma function      see Advanced functions Polygamma
Polynomials      20
Polynomials,       139
Polynomials, Bernoulli      101
Polynomials, Chebyshev of the first kind      110
Polynomials, coefficients      20
Polynomials, division algorithm      68
Polynomials, Eulerian      64 134 269
Polynomials, Jacobi      155
Polynomials, primitive      20
Polynomials, roots of      36
Polynomials, symmetric      29
Poorlen, van der. A., Taylor expansions of rational functions      69
Posey, R., evaluation of master formula      251
Power series, $(1+x)^{\alpha}$       66
Power series,       62
Power series, 1/(1-x)      61
Power series, arctangent      109
Power series, cotangent      130
Power series, definition      61
Power series, dilogarithm      62
Power series, double square root      150
Power series, exponential      91
Power series, involving Bernoulli numbers      99
Power series, involving central binomial coefficients      66 118
Power series, logarithm      75
Power series, loggamma      201
Power series, logsine      131
Power series, poly logarithm      239
Power series, powers of arcsine      122
Power series, powers of logarithms      76
Power series, psi function      213
Power series, rational functions      70
Power series, Riemann zeta function      224
Power series, secant and cosecant      132
Power series, sine and cosine      110
Power series, tangent      63
Power series, triple square root      160
Prime numbers, definition      3
Prime numbers, Fermat      121
Prime numbers, prime number theorem      220
Prime numbers, regular      101
primitive      19
Primitive root      134
Psi function      see Advanced functions Psi
Qi,F, best bound on harmonic sequence      181
Ramanujan, S., expression for Euler's constant      179
Ramanujan, S., master theorem      151
Rao, S.K., integral representation of $\gamma$       177
Rao, S.K., Legendre's duplication formula      196
Rational certificate      275
Rational functions, a map on the space of rational functions      133
Rational functions, normalization      28—30
Rational functions, partial fraction decomposition      37
Rational functions, pole      49
Rational functions, Taylor series      67—72
Recurrences      1
Reversing order of summation      22
Ribenboim, P.      3 101
Riemann, B., hypothesis      77 219
Riemann, B., zeta function      130 201 219
Rivoal, T., irrationality of $\zeta(j)$       232
Rodewald, B      187
Rodriguez, D., logarithmic integral      241
Rodriguez, D., logsine integral      245
Rornik, D., proof of Stirling's formula      92 170
Roy, R., history of Gregory series      109
Ruffini, P., equations of degree at least five      43
Sandor, J., inequalities involving e      86
Scaling      51
Schlomilch transformation      251
Semifactorials      114
Serret, M.J.A., logarithmic integral      243
Serret, M.J.A., proof of Legendre's duplication formula      197
Shurman, J., solution of the quinlic      43
Sine integral      see Advanced functions. Sine integral
Slieltjes constants      224
Sloane, N.      45 243
Sondow, J., antisymmetric formula for $\gamma$       234
Sondow, J., irrationality of Eulers constant      184
Spivak, M., definition of $\pi$       106
Srivaslava, H.      203
Stanley, R., rational generating functions      72
Stanley, R., survey on unimodality      157
Stenger, A., logarithmic integral      244
Stirling formula, evaluation of constant      115 170
Stirling formula, proof by Blylh and Palhak      92
Stirling formula, proof by Feller      96
Stirling formula, proof by G. and J.C.Marsaglia      95
Stirling formula, proof by Romik      92 170
Stirling numbers of the first kind      18
Stirling's approximation      92—97
Stirling, J., biography      92
Stirling, J., formula      115
Stirling, J., numbers      18
Stromberg, K.      22
Swenney, D.W., computation of $\gamma$       184
Symbolic evaluation, blind      20
Taylor coefficients of rational functions      69
Taylor series      see Power series
Titchmarsh, E.C.      55
Totik, proof of Holder's theorem on $\Gamma$       211
Triangle number      243
Trigonometric functions, addition theorem      110
Trigonometric functions, cotangent      129
Trigonometric functions, infinite products      126
Trigonometric functions, inverse sine      107
Trigonometric functions, inverse tangent      107
Trigonometric functions, sine and cosine      109
Trigonometric functions, solution of cubics and quarlics      111
Trigonometric functions, Taylor expansions      110
Triple square root      160
Tweedle, I., biography of J.Stirling      92
Tyler, D., special zeta series      248
Umemura, H.      44
Underwood, R.S      84
Unimodality      156
Vacca, G., a series for $\gamma$       184
Valuations      see p-adic valuations
Vandermonde formula for ascending factorial      18
Vardi, I.      237
Venkalachaliengar, K., infinite product for sin x      127
Vieta, F., radical expression for $2/\pi$       116
von Mangoldt function, connection to Eulers constant      224
von Mangoldt function, connection to Riemann hypothesis      219
von Mangoldt function, iterate integrals of ln(1+ x)      84
Wallis' formula, Osier's relation between Vieta's and Wallis' formula for $\pi$       116
Wallis' formula, proof by recurrence      32
Wallis' formula, proof by WZ-melhod      273
Wallis' formula, proof from Legendre's duplication formula      198
Wallis' formula, proof from master formula      255
Weierstarss, K., $\wp$ -function      205
Weight      203 270
Weisstein, E.      18 84 111
Wilf, H.      15 157 271
Williams, K.S., formula for $\zeta(3)$       233
Williams, K.S., proof of $\zeta(2) = \pi^2/6$       231
Wrench, J., partial sums of harmonic series      78
WZ-method      156 166 271

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