Inside Interesting Integrals

Paul Nahin was born in California, and did all of his schooling there (Brea-Olinda High 1958, Stanford BS 1962, Caltech MS 1963 and - as a Howard Hughes Staff Doctoral Fellow - UC/Irvine PhD 1972, with all degrees in electrical engineering). He has taught at Harvey Mudd College, the Naval Postgraduate School and the Universities of New Hampshire (where he is now emeritus professor of electrical engineering) and Virginia. Prof. Nahin has published a couple of dozen short science fiction stories in ANALOG, OMNI and TWILIGHT ZONE magazines, and has written 14 books on mathematics and physics. He has given invited talks on mathematics at Bowdoin College, the Claremont Graduate School, the University of Tennessee and Caltech, has appeared on National Public Radio's "Science Friday" show (discussing time travel) as well as on New Hampshire Public Radio's "The Front Porch" show (discussing imaginary numbers) and advised Boston's WGBH Public Television's "Nova" program on the script for their time travel episode. He gave the invited Sampson Lectures for 2011 in Mathematics at Bates College (Lewiston, Maine).

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.