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.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 2004

Количество страниц: 306

Добавлена в каталог: 11.06.2008

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Предметный указатель

, coefficients      208
, definition      139
, hypergeometric representation      155
, recurrence      148
, single sum      155
, triple sum      143
$\pi$ , definition      106
$\pi$ , irrationality      117
$\pi$ , Wallis' product representation      128
$\pi$ , websites      111
Abel, N., unsolvabiiity of the quintic      43
Adamchik, V., expressions for Catalan's constant      109
Adamchik, V., logarithmic integrals      237—238
Adamchik, V., negapolygamma function      215
Addison, A.W., series representation for $\gamma$       183
Advanced functions, Beta: B, definition      192
Advanced functions, Beta: B, functional equation      192
Advanced functions, Beta: B, integral representation      193
Advanced functions, Beta: B, trigonometric integrals      194
Advanced functions, Cosine integral: ci      136
Advanced functions, Dilogarithm: Dilog(x), definition      62
Advanced functions, Dilogarithm: special values      245
Advanced functions, Exponential integral: Explntegral(at x)      98 104
Advanced functions, Gamma: $\Gamma(x)$ , definition      186
Advanced functions, Gamma: $\Gamma(x)$ , Euler's definition      188
Advanced functions, Gamma: $\Gamma(x)$ , evaluation of sums      23
Advanced functions, Gamma: $\Gamma(x)$ , functional equation      187
Advanced functions, Gamma: $\Gamma(x)$ , Gauss's multiplicative formula      205
Advanced functions, Gamma: $\Gamma(x)$ , Holder's theorem      210
Advanced functions, Gamma: $\Gamma(x)$ , infinite product      204
Advanced functions, Gamma: $\Gamma(x)$ , integral representation      193
Advanced functions, Gamma: $\Gamma(x)$ , Legendre's duplication formula      195
Advanced functions, Gamma: $\Gamma(x)$ , reflection rule      189
Advanced functions, Gamma: $\Gamma(x)$ , relation to Euler's constant      190
Advanced functions, Gamma: $\Gamma(x)$ , relation to normal integral      166
Advanced functions, Hurwilz zeta: $\zeta (z, q)$       215 248
Advanced functions, Hypergeometric: arctangent      109
Advanced functions, Hypergeometric: logarithm      75
Advanced functions, Hypergeometric: polynomials       155
Advanced functions, Hypergeometric: relation to arcsine      122
Advanced functions, Hypergeometric: series evaluation      59
Advanced functions, Hypergeometric: symbolic evaluation      54
Advanced functions, Logarithmic integral, delinition      98
Advanced functions, Logarithmic integral, relation to prime numbers      220
Advanced functions, Loggamma function, definition      201
Advanced functions, Loggamma function, integrals      203
Advanced functions, Loggamma function, Taylor series      201
Advanced functions, Poly gamma, definition      214
Advanced functions, Poly gamma, logarithmic integrals      264
Advanced functions, Poly Logarithm, definition      239
Advanced functions, Poly Logarithm, logarithmic integrals      245
Advanced functions, Psi: $\psi$ , definition      212
Advanced functions, Psi: $\psi$ , integral representation      215—217
Advanced functions, Psi: $\psi$ , series expansion at 0      213
Advanced functions, Psi: $\psi$ , special values      212 213
Advanced functions, Sine integral: si      136
Algebraic functions, definition      79
Algebraic functions, double square root      80
Algebraic functions, lnx is not one      80
Algebraic numbers, definition      80
Almkvist, G., generating function for $(\zeta (4k + 3)$       236
Alzer, H., arithmetic-geometric mean inequality      88
Alzer, H., bounds on harmonic mean of $\Gamma^2$       191
Alzer, H., optimal growth for Bernoulli numbers      102
Alzer, H., sequences related to harmonic numbers      175
Amdeberham, T., iterative primitives of In x      82
Amdeberham, T., series for $\zeta(3)$       236
Andrews, G., Death of Proof?      271
Apery's number, Amdeberham's formula      236
Apery's number, Beukers' formula      234
Apery's number, definition      231
Apery's number, eslimates of Euler's constant      175
Apery's number, Ewell's formula      232
Apery's number, integral representation      232
Apery's number, integral with arcsine      122
Apery's number, irrationality      232
Apery's number, Sondow's antisymmetric formula      234
Apery's number, Yue and William's formula      233
Apostol, T., Dirichlet series      224
Apostol, T., Euler — MacLaurin summation      94
Apostol, T., proof of $\zeta(2) = \pi^2/6$       225
Apostol, T., proof of von Staudt — CIausen theorem      101
Arcsine function, definition      107
Arcsine function, power series      119
Arcsine function, power series of the cube      123
Arcsine function, power series of the square      122
Arctangent function, definition      19 107
Arctangent function, hypergeomelric representation      109
Arctangent function, polynomial approximation      124
Arctangent function, power series      109
Arctangent function, relation to Arcsine      107
Arithmetic geometric mean      88
Arora, A.K., logarithmic integral      241
Arora, A.K., logsine integral      245
Artin, E., conjecture on primitive roots      134
Ascending factorial symbol, definition      16
Ascending factorial symbol, dimidiation formula      17
Ascending factorial symbol, duplication formula      17
Ascending factorial symbol, extended binomial coefficients      65
Ascending factorial symbol, relation to $\Gamma$       187
Ascending factorial symbol, Vandermonde's formula      18
Assmus, E.F., $\pi$ in length and area of a circle      108
Atkinson. M.D., series for tan* and sec x      133
Ayoub. R., nonsolvabilily of polynomials      43
Barnes, C W., e as a limit      85
Barnes, C W., existence of Euler's constant      174
Bateman, manuscript project      55
Beatly, S., e is not a quadratic irrational      90
Berndt, B., Entry 21 of Chapter 26 in Ramanujan's Notebook      178
Berndt, B., history of Lagrange's inversion formula      150
Berndt, B., integral for Euler's constant      180
Berndt, B., optimal bounds on $\muj_p(m)$       145
Berndt, B., unpublished notes on $\Gamma$       187
Bernoulli numbers, definition      99
Bernoulli numbers, denominators      101
Bernoulli numbers, expansion of cosecant      132
Bernoulli numbers, expansion of cotangent      130
Bernoulli numbers, expansion of tangent      132
Bernoulli numbers, integral representation      223
Bernoulli numbers, optimal growth      102
Bernoulli numbers, relation to the $\zeta$ function      131
Bernoulli numbers, relation to the Euler numbers      132
Bernoulli numbers, sign      101
Bernoulli numbers, von Staudt — CIausen      101
Bernoulli polynomials      101—102
Bertrand's postulate      77
beta function      see Advanced functions. Beta
Beukers, F., formula for $\zeta(3)$       234
Beukers, F., integrals and coninued fractions of $\pi$       126
Beukers, F., triple integral for $\zeta(3)$       232
Beumer, M.G., recursion for logsine integrals      246
Bharghava, S., optimal bounds on $\mu_p(m)$       145
Binomial coefficients      10
Binomial recurrence      11
Binomial theorem      10
Binomial, central      14 66 94 256
Binomial, extended      65
Biquadratic integral, elementary evaluation      44
Biquadratic integral, explicit value      156
Biquadratic integral, relation to the polynomial       139
Biquadratic integral, Taylor expansion of double square root      150
Blind evaluation      20
Blyth, C., proof of Stirling's formula      92
Boas, R., partial sums of harmonic series      78
Bohr — Mollerup theorem      187
Boruein. J. and P., arithmetic-geometric mean      88
Boruein. J. and P., calculation of $\gamma$       185
Boruein. J. and P., elliptic integrals      110
Boruein. J. and P., evaluation of normal integral      164
Boruein. J. and P., integral for $\zeta(4)$       235

Boruein. J. and P., series for $\zeta(7)$       235
Bowman, D., integral for Euler's constant      180
Bradley, D., divergence of harmonic series      78
Bradley, D., formulas for Catalan's constant      109
Bradley, D., series for $\zeta(7)$       235
Brenner, J.L., sequences related to harmonic numbers      175
Brent, R.P., calculation of $\gamma$       184
Brenti, F., logconcavity criteria      157
Breusch, R., proof of irrationality of $\pi$       117
Bromwich, T.J., double square root series      151
Bromwich, T.J., zeta series      214
Bronstein, M., indefinite integration      147
Brown, J.W., relation between B and $\Gamma$       192
Calabi, E., proof of $\zeta(2) = \pi^2/6$       226
Calculus, Fundamental Theorem of Calculus      19
Calculus, primitive      19
Cardano, G., solution of the cubic equation      39
Cardano, G., solution of the quartic equation      42
Carlitz, L., logconcavity and reciprocal of series      158
Catalan's constant, Adamchik's expressions      109
Catalan's constant, Bradley's expressions      109
Catalan's constant, definition      109 221
Catalan, E., formula for $\gamma$       179
Cauchy, A., product of power series      66
Chebyshev, P.L., polynomials of the first kind      110
Chen, C.P., best bound on harmonic sequence      181
Chernhoff, P., special zeta series      248
Choe, B.R., proof of $\zeta(2) = \pi^2/6$       230
Choi, J.      203
Chong, K., proof of arithmetic-geometric mean inequality      88
Coleman, A.J., evaluation of the normal integral      169
Coolidge, J., information about c      85
Cotangent, definition      129
Cotangent, expansion at 0      130
Cox, D. et al.      160
Danese, A., a zeta series      248
Davis, P.J., Euler and history of the gamma function      187
Debnath, L., inequalities involving e      86
Derangment numbers, closed form      2
Derangment numbers, definition      2
Desbrow, D., proof of irrationality of $\pi^2$       117
Descartes, R., solution of the quartic equation      42
DeTemple. D., fast convergence to $\gamma$       181
Dilcher, K., website on Bernoulli numbers      99
Dilogarithm      see Advanced functions Dilogarithm
Dirichlet series      224 246
Discriminant curve      41
Discriminant of a cubic      39
Discriminant of a quadratic      31
Doetsch, G.      55
Dunham. W., biography of Euler      187
e, definition      84
e, irrationality      89
e, limit form      85
e, series form      85
EKHAD, automatic proofs      273
Elkies, N., story of Calabi's proof of $\zeta(2) = \pi^2/6$       226
Elkies, N., zeta series      221
Elliptic functions      110 204
Euler numbers, definition      132
Euler numbers, relation to Elkies series      222
Euler — McLaurin summation formula      93
Euler's constant, definition      173
Euler's constant, existence      173—174
Euler's constant, integral representations      176—179
Euler's constant, irrationality      184
Euler's constant, rale of convergence      181
Euler's constant, series representations      183
Euler, L., dilogarithm      239
Euler, L., infinite product for sine      126
Euler, L., integral definition of $\Gamma$       186
Euler, L., lemniscatic identity      194
Euler, L., logarithmic integrals      241
Euler, L., original definition of $\Gamma$       188
Euler, L., proof of $\zeta(2) = \pi^2/6$       225
Eulerian polynomials in integrals of exponentials      99
Eulerian polynomials, definition      64
Eulerian polynomials, limits of an iteration method      134
Everest. G. et al., power series of rational functions      69
Ewell, J., formula for $\pi^2$       123
Ewell, J., formula for $\zeta(3)$       232
Exponent of p in r: $\mu_p(r)$       4
Exponential function, definition      91
Exponential integral      see Advanced functions Exponential
Factorial definition      2
Feller, W., proof of Stirling's formula      96
Femiat, P., last theorem      101
Femiat, P., primes      121
Femiat, P., sums of squares      166
Fibonacci numbers, closed form      71
Fibonacci numbers, definition      1
Finch, S      85
Frullani integrals      98
Function, algebraic      79
Function, Beta      192
Function, cotangent      129
Function, dilogarithm      62
Function, error      162
Function, exponential      9
Function, floor      5
Function, Gamma      186
Function, generating      8
Function, hypergeometric      54
Function, incomplete gamma      9
Function, loggamma      201
Function, negapolygamma      215
Function, polygamma      214
Function, power      21
Function, psi      212
Function, rational      25
Function, tangent      63
Function, trigonometric      109
Function, zeta, Hurwitz      248
Function, zeta, Riemann      130 201 219
Fundamental Theorem of Arithmetic      3
Galois, E., unsolvability of the quintic      43
Gamma function      see Advanced functions Gamma
Gauss. K.F, arithmetic-geometric mean      88
Gauss. K.F, multiplicative formula for $\Gamma$       205
Gauss. K.F, radical values of sine      121
Gauss. K.F, rational values for $\psi$       213
Gautschi, W., inequalities for $\Gamma$       191
Generating function for $\zeta(4k + 3)$       236
Generating function, binomial coefficients       10
Generating function, definition      8
Generating function, factorials      9
Generating function, Fibonacci numbers      9
Generating function, generating polynomials      8
Geometric series      61
Goel, S., logarithmic integral      241
Goel, S., logsine integral      245
Goode, J., limit of harmonic sums      78
Gosper, W., evaluation of Nielsen — Ramanujan constants      240
Gosper, W., negapolygammas      215
Gouvea, F., p-adic analysis      4
Granville, A., generating function for $\zeta(4k + 3)$       236
Greene, R. — Krantz. S., details on infinite products      129
Greenstein, D.S., growth of In x      79
Gregory. J., series for $tan^{-1} x$       109
Grosswald, E., logarithmic integral      242
Hamming, R., In x is not algebraic      80
Hamming, R., In x is not rational      79
Hardy, G. and Wright, E., $\pi$ is transcendental      117
Hardy, G. and Wright, E., continued fractions      126
Hardy, G. and Wright, E., e is transcendental      89
Hardy, G. and Wright, E., p-adic analysis      4
Hardy, G. and Wright, E., prime numbers      3
Hardy, G. and Wright, E., proof of Bertrand's postulate      77
Hardy, G., integration of rational functions      44
Harmonic numbers are not integers      77

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