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Special integrals of Gradshteyn and Ryzhik : the proofs : v. 2 /

by Moll, Victor H [aut].
Material type: materialTypeLabelBookSeries: Monographs and research notes in mathematics: Publisher: Boca Raton : CRC Press, Taylor & Francis Group, 2016Description: xi, 263 p. ; 24 cm.ISBN: 9781482256536 (v. 2 : hardback : acidfree paper); 1482256533 (v. 2 : hardback : acidfree paper).Subject(s): Definite integrals | Integrals, Logarithmic | MATHEMATICS / General | MATHEMATICS / Applied | MATHEMATICS / Arithmetic
Contents:
Complete elliptic integrals Introduction Some examples An elementary transformation Some principal value integrals The hypergeometric connection Evaluation by series expansions A small correction to a formula in Gradshteyn and Ryzhik The Riemann zeta function Introduction A first integral representation Integrals involving partial sums of zeta(s) The alternate version The logarithmic scale The alternating logarithmic scale Integrals over the whole line Some automatic proofs Introduction The class of holonomic functions A first example: The indefinite form of Wallis' integral A differential equation for hypergeometric functions in two variables An integral involving Chebyshev polynomials An integral involving a hypergeometric function An integral involving Gegenbauer polynomials The product of two Bessel functions An example involving parabolic cylinder functions An elementary trigonometric integral The error function Introduction Elementary integrals Elementary scaling A series representation for the error function An integral of Laplace Some elementary changes of variables Some more challenging elementary integrals Differentiation with respect to a parameter A family of Laplace transforms A family involving the complementary error function A final collection of examples Hypergeometric functions Introduction Integrals over [0, 1] A linear scaling Powers of linear factors Some quadratic factors A single factor of higher degree Integrals over a half-line An exponential scale A more challenging example One last example: A combination of algebraic factors and exponentials Hyperbolic functions Introduction Some elementary examples An example that is evaluated in terms of the Hurwitz zeta function A direct series expansion An example involving Catalan's constant Quotients of hyperbolic functions An evaluation by residues An evaluation via differential equations Squares in denominators Two integrals giving beta function values The last two entries of Section 3.525 Bessel-K functions Introduction A first integral representation of modified Bessel functions A second integral representation of modified Bessel functions A family with typos The Mellin transform method A family of integrals and a recurrence A hyperexponential example Combination of logarithms and rational functions Introduction Combinations of logarithms and linear rational functions Combinations of logarithms and rational functions with denominators that are squares of linear terms Combinations of logarithms and rational functions with quadratic denominators An example via recurrences An elementary example Some parametric examples Integrals yielding partial sums of the zeta function A singular integral Polylogarithm functions Introduction Some examples from the table by Gradshteyn and Ryzhik Evaluation by series Introduction A hypergeometric example An integral involving the binomial theorem A product of logarithms Some integrals involving the exponential function Some combinations of powers and algebraic functions Some examples related to geometric series The exponential integral Introduction Some simple changes of variables Entries obtained by differentiation Entries with quadratic denominators Some higher degree denominators Entries involving absolute values Some integrals involving the logarithm function The exponential scale More logarithmic integrals Introduction Some examples involving rational functions An entry involving the Poisson kernel for the disk Some rational integrands with a pole at x = 1 Some singular integrals Combinations of logarithms and algebraic functions An example producing a trigonometric answer Confluent hypergeometric and Whittaker functions Introduction A sample of formulas Evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets Introduction The method of brackets Examples of index 0 Examples of index 1 Examples of index 2 The goal is to minimize the index The evaluation of a Mellin transform The introduction of a parameter The list of integrals The list References
Summary: "This provides a compilation of papers published in Revista Scientia, a journal published by the Department of Mathematics from the University of Tecnica Frederico Santa Maria in Chilie. It details interesting approaches and techniques that help readers study other areas in mathematics. In addition to the original papers by the author, the book includes commentary to further clarify and provide instruction on the proofs."-- Provided by publisher.Summary: A Guide to the Evaluation of IntegralsSpecial Integrals of Gradshetyn and Ryzhik: The Proofs provides self-contained proofs of a variety of entries in the frequently used table of integrals by I.S. Gradshteyn and I.M. Ryzhik. The book gives the most elementary arguments possible and uses Mathematica® to verify the formulas. Readers discover the beauty, patterns, and unexpected connections behind the formulas. Volume I collects 15 papers from Revista Scientia covering logarithmic integrals, the gamma function, trigonometric integrals, the beta function, the digamma function, the incomplete beta.
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515.43 M7262.v2 (Browse shelf) Available

Includes bibliographical references (p. 255-257) and index.

Complete elliptic integrals Introduction Some examples An elementary transformation Some principal value integrals The hypergeometric connection Evaluation by series expansions A small correction to a formula in Gradshteyn and Ryzhik The Riemann zeta function Introduction A first integral representation Integrals involving partial sums of zeta(s) The alternate version The logarithmic scale The alternating logarithmic scale Integrals over the whole line Some automatic proofs Introduction The class of holonomic functions A first example: The indefinite form of Wallis' integral A differential equation for hypergeometric functions in two variables An integral involving Chebyshev polynomials An integral involving a hypergeometric function An integral involving Gegenbauer polynomials The product of two Bessel functions An example involving parabolic cylinder functions An elementary trigonometric integral The error function Introduction Elementary integrals Elementary scaling A series representation for the error function An integral of Laplace Some elementary changes of variables Some more challenging elementary integrals Differentiation with respect to a parameter A family of Laplace transforms A family involving the complementary error function A final collection of examples Hypergeometric functions Introduction Integrals over [0, 1] A linear scaling Powers of linear factors Some quadratic factors A single factor of higher degree Integrals over a half-line An exponential scale A more challenging example One last example: A combination of algebraic factors and exponentials Hyperbolic functions Introduction Some elementary examples An example that is evaluated in terms of the Hurwitz zeta function A direct series expansion An example involving Catalan's constant Quotients of hyperbolic functions An evaluation by residues An evaluation via differential equations Squares in denominators Two integrals giving beta function values The last two entries of Section 3.525 Bessel-K functions Introduction A first integral representation of modified Bessel functions A second integral representation of modified Bessel functions A family with typos The Mellin transform method A family of integrals and a recurrence A hyperexponential example Combination of logarithms and rational functions Introduction Combinations of logarithms and linear rational functions Combinations of logarithms and rational functions with denominators that are squares of linear terms Combinations of logarithms and rational functions with quadratic denominators An example via recurrences An elementary example Some parametric examples Integrals yielding partial sums of the zeta function A singular integral Polylogarithm functions Introduction Some examples from the table by Gradshteyn and Ryzhik Evaluation by series Introduction A hypergeometric example An integral involving the binomial theorem A product of logarithms Some integrals involving the exponential function Some combinations of powers and algebraic functions Some examples related to geometric series The exponential integral Introduction Some simple changes of variables Entries obtained by differentiation Entries with quadratic denominators Some higher degree denominators Entries involving absolute values Some integrals involving the logarithm function The exponential scale More logarithmic integrals Introduction Some examples involving rational functions An entry involving the Poisson kernel for the disk Some rational integrands with a pole at x = 1 Some singular integrals Combinations of logarithms and algebraic functions An example producing a trigonometric answer Confluent hypergeometric and Whittaker functions Introduction A sample of formulas Evaluation of entries in Gradshteyn and Ryzhik employing the method of brackets Introduction The method of brackets Examples of index 0 Examples of index 1 Examples of index 2 The goal is to minimize the index The evaluation of a Mellin transform The introduction of a parameter The list of integrals The list References

"This provides a compilation of papers published in Revista Scientia, a journal published by the Department of Mathematics from the University of Tecnica Frederico Santa Maria in Chilie. It details interesting approaches and techniques that help readers study other areas in mathematics. In addition to the original papers by the author, the book includes commentary to further clarify and provide instruction on the proofs."-- Provided by publisher.

A Guide to the Evaluation of IntegralsSpecial Integrals of Gradshetyn and Ryzhik: The Proofs provides self-contained proofs of a variety of entries in the frequently used table of integrals by I.S. Gradshteyn and I.M. Ryzhik. The book gives the most elementary arguments possible and uses Mathematica® to verify the formulas. Readers discover the beauty, patterns, and unexpected connections behind the formulas. Volume I collects 15 papers from Revista Scientia covering logarithmic integrals, the gamma function, trigonometric integrals, the beta function, the digamma function, the incomplete beta.