04558nam 22004093 450 991087898960332120240802080341.03-031-62123-9(MiAaPQ)EBC31575156(Au-PeEL)EBL31575156(CKB)33566381300041(EXLCZ)993356638130004120240802d2024 uy 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierInnovative Integrals and Their Applications II1st ed.Cham :Springer,2024.©2024.1 online resource (318 pages)STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics and Health Series3-031-62122-0 Intro -- Preface -- References -- Contents -- 1 An Overview of the Methods -- 1 Special Functions -- 1.1 The Gamma Function and the Factorial -- 1.2 The Error Function and the Inverse Error Function -- 1.3 Bessel Functions and Modified Bessel Functions -- 1.4 The Rectangle Function and the Heaviside Step Function -- 1.5 The Signum Function -- 1.6 The Owen T-Function -- 1.7 The Exponential Integral and the Logarithmic Integral -- 1.8 The Cosine Integral and the Sine Integral -- 1.9 Hypergeometric Functions -- 1.10 Elliptic Integrals and Jacobi Elliptic Functions -- 1.11 The Polygamma Function and the Harmonic Number -- 1.12 The Polylogarithm -- 1.13 The Riemann Zeta Function and the Hurwitz Zeta Function -- 1.14 The Lerch Transcendent -- 1.15 The Incomplete Beta Function -- 1.16 Struve Functions and Modified Struve Functions -- 1.17 The Lambert W-Function -- 1.18 Fresnel Integrals -- 1.19 The Laguerre Polynomial -- 1.20 The Meijer G-Function -- 1.21 Dawson's Integral -- 2 Fundamental Constants -- 2.1 The Euler-Mascheroni Constant -- 2.2 Catalan's Constant -- 2.3 The Stieltjes Constants -- 3 The Generalized Method of Exhaustion -- 3.1 An Infinite Product -- 3.2 An Alternative Form -- 4 Multivariate Substitution -- 4.1 Some Examples -- 5 Permutation Symmetry -- 6 Interchanging the Order of Integration -- 7 Is It New? -- 8 A Word on Innovation -- 9 Exercises -- References -- 2 Warm-Up: Interchanging the Order of Integration -- 1 Preliminary -- 1.1 Theoretical Considerations -- 1.2 Examples -- 2 Triple Integrals -- 3 More Double Integrals -- 4 Integrals with Finite Limits of Integration -- 5 Exercises -- 3 Permutation Symmetry -- 1 Preliminary -- 2 Results Involving the Exponential Integral Function -- 3 Results Involving the Polylogarithm -- 4 Results Involving Bessel Functions -- 5 Miscellaneous Results.6 Integrals with Finite Limits of Integration -- 7 Exercises -- References -- 4 Identities Involving Laplace Transforms and Fourier Transforms -- 1 Introduction -- 1.1 The Laplace Transform -- 1.2 The Fourier Transform -- 2 The Use of Laplace Transform Pairs -- 3 Laplace Transform Pairs -- 4 Fourier Transform Pairs -- 4.1 Previous Results, Reinterpreted -- 4.2 New Results -- 4.3 Related Fourier Transform and Laplace Transform Pairs -- 5 The Use of Fourier Transform Pairs -- 6 Exercises -- References -- 5 A Potpourri of Methods and Results -- 1 Preliminary Results -- 2 Identities Involving Bessel Functions and Struve Functions -- 3 Identities Involving the Riemann Zeta Function and the Hurwitz Zeta Function -- 4 Identities Involving the Owen T-Function -- 5 Results Involving the Polygamma Function and the Harmonic Number -- 6 Identities Involving the Polylogarithm -- 7 Postliminary Results -- 8 Exercises -- 6 Applications in the Sciences, Technology, and Engineering -- 1 Non-Gaussian Distributions -- 1.1 An Improved Proof -- 1.2 Finite Limits of Integration -- 2 Weighted Gaussian Distributions -- 2.1 An Improved Proof -- 2.2 Finite Limits of Integration -- 3 Weighted Non-Gaussian Distributions -- 4 Benchmarks for Cubature Formulas -- 4.1 An Old Result -- 4.2 New Results -- 4.3 Finite Limits of Integration -- 5 The Reciprocal of a Complex Number -- 5.1 The Sommerfeld Integral -- 5.2 A Family of Expansions -- 5.3 Other Applications -- 6 Exercises -- References -- Bibliography -- Index.STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics and Health SeriesRuffa Anthony A1264622Toni Bourama1739712MiAaPQMiAaPQMiAaPQBOOK9910878989603321Innovative Integrals and Their Applications II4205162UNINA