Innovative Integrals and Their Applications II (STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics & Health)

In its second installment, Innovative Integrals and Their Applications II explores multidimensional integral identities, unveiling powerful techniques for attacking otherwise intractable integrals, thus demanding ingenuity and novel approaches. This volume focuses on novel approaches for evaluating definite integrals, with the aid of tools such as Mathematica as a means of obtaining useful results. Building upon the previous methodologies, this volume introduces additional concepts such as interchanging the order of integration, permutation symmetry, and the use of pairs of Laplace transforms and Fourier transforms, offering readers a comprehensive array of integral identities. The content further elucidates the techniques of permutation symmetry and extends the multivariate substitution approach to integrals with finite limits of integration. These insights culminate in a collection of integral identities involving gamma functions, incomplete beta functions, Bessel functions, polylogarithms, and the Meijer G-function. Additionally, readers will encounter applications of error functions, inverse error functions, hypergeometric functions, the Lambert W-function, elliptic integrals, Jacobi elliptic functions, and the Riemann zeta function, among many others, with a focus on their relevance in various scientific disciplines and cutting-edge technologies. Each chapter in this volume concludes with many interesting exercises for the reader to practice. A key tenet is that such approaches work best when applied to integrals having certain characteristics as a starting point. Most integrals, if used as a starting point, lead to no result at all, or lead to a known result. However, there is a special class of integrals (i.e., innovative integrals), which, if used as a starting point for such approaches, lead to new and useful results, and can also enable the reader to generate other new results that do not appear in the book. The intended readership includes science, technology, engineering, and mathematics (STEM) undergraduates and graduates, as well as STEM researchers and the community of engineers, scientists, and physicists; most of these potential readers have experienced the importance and/or the applications of integration from finding areas, volumes, lengths, and velocities to more advanced applications. The pedagogical approach of the exposition empowers students to comprehend and efficiently wield multidimensional integrals from their foundations, fostering a deeper understanding of advanced mathematical concepts. 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