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This book develops integral identities, mostly involving multidimensional functions and infinite limits of integration, whose evaluations are intractable by common means. It exposes a methodology based on the multivariate power substitution and its variants, assisted by the software tool Mathematica. The approaches introduced comprise the generalized method of exhaustion, the multivariate power substitution and its variants, and the use of permutation symmetry to evaluate definite integrals, which are very important both in their own right, and as necessary intermediate steps towards more involved computation.
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, will lead to no result at all, or will 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, will lead to new and useful results, and can also enable the reader to generate many other new results that are not in the book.
The reader will find a myriad of novel approaches for evaluating integrals, with a focus on tools such as Mathematica as a means of obtaining useful results, and also checking whether they are already known. Results presented involve the gamma function, the hypergeometric functions, the complementary error function, the exponential integral function, the Riemann zeta function, and others that will be introduced as they arise. The book concludes with selected engineering applications, e.g., involving wave propagation, antenna theory, non-Gaussian and weighted Gaussian distributions, and other areas.
The intended audience comprises junior and senior sciences majors planning to continue in the pure and applied sciences at the graduate level, graduate students in mathematics and the sciences, and junior and established researchers in mathematical physics, engineering, and mathematics. Indeed, the pedagogical inclination of the exposition will have students work out, understand, and efficiently use multidimensional integrals from first principles.
The Generalized Method of Exhaustion
Preliminary Concepts
The Residue Theorem
The Laplace Transform
Approximating the Area Under a Curve with Triangles
A Simple Illustration
Analytic Derivation
The Fundamental Theorem of Calculus
Infinite Products for the Logarithm and the Sine
Application to Other Functions
Improper Integrals
Summary and Further Reading
The Multivariate Power Substitution and Its Variants
The Multivariate Power Substitution
Illustrative Examples
Some Preliminary Results
Permutation Symmetry
The Laplace Transform
Multivariate Power Substitution Variants
More Variants
Triple Integrals
Further Reading
Additional Multivariate Substitution Variants
Polar Coordinates and Spherical Coordinates
The Generalized Method of Exhaustion Revisited
The Complementary Error Function
A New Variant
The Laplace Transform Revisited
Triple and Quadruple Integrals
The Owen T-Function
Further Reading
Miscellaneous Integral Identities
Identities Involving the Lerch Transcendent
Integrals Involving the Logarithm
Various Other Integral Identities
Summary and Further Reading
The Exponential Integral Function, the Sine Integral and Cosine Integrals
Integral Identities Involving E(x)
Identities Involving Ea(x)
The Laplace Transform
Permutation Symmetry
Further Reading
The Riemann Zeta Function and the Hurwitz Zeta Function
The Riemann Zeta Function
Integrals Involving the Polygamma Function
The Hurwitz Zeta Function
Integrals Involving Trigonometric Functions
The Laplace Transform
Identities Involving the Polylogarithm
Identities Involving the Lerch Transcendent
Summary and Further Reading
Engineering Applications
Plane Waves and Spherical Waves
Plane Waves
The Angular Spectrum of Plane Waves
Spherical Waves
The Sommerfeld Identity
An Alternative Identity
The Sommerfeld Integral
Bessel Functions and Frequency Modulation
Non-Gaussian and Weighted Gaussian Distributions
Further Reading