Inside Interesting Integrals

Preface.- 1. Introduction.- 1.1 The Riemann Integral.- 1.2 An Example of Riemann Integration.- 1.3 The Lebesgue Integral.- 1.4 ‘Interesting’ and ‘Inside’.- 1.5 An Example of a Trick.- 1.6 Singularities.- 1.7 Dalzell’s Integral.- 1.8 Where Integrals Come From.- 1.9 Last Words.- 1.10 Challenge Problems.- 2. ‘Easy’ Integrals.- 2.1 Six ‘Easy’ Warm-ups.- 2.2 A New Trick.- 2.3 Two Old Tricks, Plus a New One.- 2.4 Another Old Trick: Euler’s Log-Sine Integral.- 2.5 Challenge Problems.- 3. Feynman’s Favorite Trick.- 3.1 Leibniz’s Formula.- 3.2 Dirichlet’s Amazing Integral.- 3.3 Frullani’s Integral.- 3.4 The Flip-Side of Feynman’s Trick.- 3.5 Combining Two Tricks.- 3.6 Uhler’s Integral and Symbolic Integration.- 3.7 The Probability Integral Revisited.- 3.8 Dini’s Integral.- 3.9 Feynman’s Favorite Trick Solves a Physics Equation .- 3.10 Challenge Problems.- 4. Gamma and Beta Function Integrals.- 4.1 Euler’s Gamma Function.- 4.2 Wallis’ Integral and the Beta Function.- 4.3 Double Integration Reversal.- 4.4 The Gamma Function Meets Physics.- 4.5 Challenge Problems.- 5. Using Power Series to Evaluate Integrals.- 5.1 Catalan’s Constant.- 5.2 Power Series for the Log Function.- 5.3 Zeta Function Integrals.- 5.4 Euler’s Constant and Related Integrals.- 5.5 Challenge Problems.- 6. Seven Not-So-Easy Integrals.- 6.1 Bernoulli’s Integral .- 6.2 Ahmed’s Integral.- 6.3 Coxeter’s Integral.- 6.4 The Hardy-Schuster Optical Integral.- 6.5 The Watson/van Peype Triple Integrals.- 6.6 Elliptic Integrals in a Physical Problem.- 6.7 Challenge Problems.- 7. Using √(-1) to Evaluate Integrals.- 7.1 Euler’s Formula.- 7.2 The Fresnel Integrals.- 7.3 (3) and More Log-Sine Integrals .- 7.4 (2), At Last!.- 7.5 The Probability Integral Again.- 7.6 Beyond Dirichlet’s Integral.- 7.7 Dirichlet Meets the Gamma Function.- 7.8 Fourier Transforms and Energy Integrals.- 7.9 ‘Weird’ Integrals from Radio Engineering.- 7.10 Causality and Hilbert Transform Integrals.- 7.11 Challenge Problems.- 8. Contour Integration.- 8.1 Prelude.- 8.2 Line Integrals.- 8.3 Functions of a Complex Variable.- 8.4 The Cauchy-Riemann Equations and Analytic Functions.- 8.5 Green’s Integral Theorem.- 8.6 Cauchy’s First Integral Theorem.- 8.7 Cauchy’s Second Integral Theorem.- 8.8 Singularities and the Residue Theorem.- 8.9 Integrals with Multi-valued Integrands.- 8.10 Challenge Problems.- 9. Epilogue.- 9.1 Riemann, Prime Numbers, and the Zeta Function.- 9.2 Deriving the Functional Equation for (s).- 9.3 Challenge Questions.- Solutions to the Challenge Problems.