The use of special functions, and in particular Airy functions, is rather common in physics. The reason may be found in the need, and even in the necessity, to express a physical phenomenon in terms of an effective and comprehensive analytical form for the whole scientific community. However, for the past twenty years, many physical problems have been resolved by computers. This trend is now becoming the norm as the importance of computers continues to grow. As a last resort, the special functions employed in physics will have to be calculated numerically, even if the analytic formulation of physics is of primary importance.
Airy functions have periodically been the subject of many review articles, but no noteworthy compilation on this subject has been published since the 1950s. In this work, we provide an exhaustive compilation of the current knowledge on the analytical properties of Airy functions, developing with care the calculus implying the Airy functions.
The book is divided into 2 parts: the first is devoted to the mathematical properties of Airy functions, whilst the second presents some applications of Airy functions to various fields of physics. The examples provided succinctly illustrate the use of Airy functions in classical and quantum physics.
Sample Chapter(s)
Chapter 1: A Historical Introduction: Sir George Biddell Airy (211 KB)
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Contents:
- A Historical Introduction: Sir George Biddell Airy
- Definitions and Properties
- Primitives and Integrals of Airy Functions
- Transformations of Airy Functions
- The Uniform Approximation
- Generalisation of Airy Functions
- Applications to Classical Physics
- Applications to Quantum Physics
Readership: Physicists or chemical physicists who use closed and analytical formulae in their field, graduate students in physics and chemical physics.
“This small (nearly 200 pages) monograph on the Airy functions is an important addition to the literature on special functions. It is competently written, with mathematical equations nicely embedded in the text and with many beautiful figures, which help in following the discussed subject. I strongly recommend this book, not only to people who like analytical solutions and the beauty and elegance of special functions, but also to the large group of mathematicians, researchers and graduate students involved in solving applied problems in physics, chemistry and technology.”
Professor Alexander Apelblat
Ben Gurion University of the Negev, Israel
“These authors have collected a great deal of useful material that will be unfamiliar to most scientists. I am especially enthusiastic about the large collection of integrals; I was already aware of the integral (2.152) relating an Airy function to its square, and the Fourier generalization (3.94-5) of this, but most of the others were new to me — in particular the Airy transform which is remarkable.”
Professor Michael Berry
University of Bristol, UK
“The text by Vallee and Soares provides a relatively compact discussion of these functions and their applications to both classical and quantum problems. The book takes a tutorial approach and the mathematical presentations are logical … The text will make a good reference source.”
Barrie S H Royce
Professor of Mechanical and Aerospace Engineering
Princeton University
“In conclusion, I am very enthusiastic that this book exists. It fills a long-standing gap in a very special but essential segment of applied mathematics, or mathematics for applications. I am aware of no other text that would rival it for the coverage of this subject. It will be a highly useful reference work for a broad community, from mathematicians to engineers, as it collates thorough, up-to-date and almost encyclopaedic data inside a reasonable size (below 200 pages) and price.”
André Voros
CEA-Service de Physique Théorique de Saclay
“In spite of the book's flaws, I did enjoy reading it for the many beautiful formulas it contains. Certain integrals involving Airy functions are surprisingly easy because of the simplicity of the differential equation, and there is an excellent list of these on pages 43–45 that was great fun to go through.”
MAA Online Book Review
“This book may serve as a comprehensive collection of formulas as well as a textbook in a course on mathematical techniuqes in physics.”
Mathematical Reviews
