Description
Book SynopsisThis book tells the story of the probability integral, the approaches to analyzing it throughout history, and the many areas of science where it arises. The so-called probability integral, the integral over the real line of a Gaussian function, occurs ubiquitously in mathematics, physics, engineering and probability theory. Stubbornly resistant to the undergraduate toolkit for handling integrals, calculating its value and investigating its properties occupied such mathematical luminaries as De Moivre, Laplace, Poisson, and Liouville. This book introduces the probability integral, puts it into a historical context, and describes the different approaches throughout history to evaluate and analyze it. The author also takes entertaining diversions into areas of math, science, and engineering where the probability integral arises: as well as being indispensable to probability theory and statistics, it also shows up naturally in thermodynamics and signal processing. Designed to be accessible to anyone at the undergraduate level and above, this book will appeal to anyone interested in integration techniques, as well as historians of math, science, and statistics.
Table of ContentsPreface
Chapter 1: De Moivre and the
Discovery of the Probability Integral
Evaluating the Probability Integral
— Part 1
Chapter 2: Laplace’s First
Derivation
Chapter 3: How Euler Could Have Done
It Before Laplace (but did he?)
Chapter 4: Laplace’s Second
Derivation
Chapter 5: Generalizing the
Probability Integral
Chapter 6: Poisson’s Derivation
Interlude
Chapter 7: Rice’s Radar Integral
Chapter 8: Liouville’s Proof That
∫e−x2dx Has No Finite Form
Chapter 9: How the Error Function
Appeared in the Electrical Response of the Trans-Atlantic Telegraph Cable
Evaluating the Probability Integral
— Part 2
Chapter 10: Doing the Probability
Integral with Differentiation
Chapter 11: The Probability Integral
as a Volume
Chapter 12: How Cauchy Could Have
Done It (but didn’t)
Chapter 13: Fourier Has the Last Word
