Higher Transcendental Functions

Higher Transcendental Functions is a three-volume mathematics reference work published by McGraw-Hill in 1953-1955. The volumes compile extensive information about special functions and their properties, including hypergeometric functions, Legendre functions, and various orthogonal polynomials. The text presents systematic derivations, integral representations, and series expansions for these functions. Each chapter contains detailed mathematical proofs along with tables of important relations and transformations. Extensive citations and historical notes document the development of special function theory through the mid-20th century. The bibliography provides a comprehensive survey of the mathematical literature in this field up to the time of publication. This work standardized much of the modern theory and notation for special functions, becoming a foundational reference that influenced subsequent research in mathematical physics and applied mathematics. The rigorous treatment and organizational structure established a template for how advanced mathematical concepts could be systematically presented.

Readers consistently note this is a comprehensive reference work for special functions in mathematics, with detailed derivations and extensive formulas. Many mathematicians and physicists rely on it for looking up properties of special functions. Liked: - Complete coverage of special functions - Clear presentation of formulas and relationships - Reliable reference for research papers - High quality printing and typesetting - Includes less common functions not found elsewhere Disliked: - Dense mathematical notation can be challenging - Some sections assume advanced knowledge - Original edition is expensive and hard to find - A few readers noted minor errors in some formulas - Paper quality in newer reprints isn't as good Ratings: Goodreads: 4.5/5 (12 ratings) Amazon: 4.7/5 (6 ratings) One researcher wrote: "These volumes saved me countless hours deriving special function properties." Another noted: "The binding on my copy is falling apart from constant use - it's that invaluable."

Special Functions of Mathematical Physics by N. N. Lebedev This text covers orthogonal polynomials, Bessel functions, spherical harmonics, and hypergeometric functions with rigorous mathematical treatment and applications in physics.

Table of Integrals, Series, and Products by Ryzhik This comprehensive reference compiles formulas for the integration of products and special functions encountered in mathematics and physics.

Asymptotic Approximations of Integrals by R. Wong The book presents methods for evaluating complex integrals and asymptotic expansions of special functions used in applied mathematics.

Special Functions: An Introduction to the Classical Functions of Mathematical Physics by F.W.J. Olver The text develops special functions from fundamental principles with emphasis on uniform asymptotic expansions and computational methods.

Special Functions and Their Applications by N.N. Lebedev and Richard A. Silverman This work presents special functions through differential equations and integral representations with connections to physical applications.

🔷 The book, commonly known as "The Bateman Manuscript Project," was created under the direction of mathematician Harry Bateman and published after his death in 1953. His notes formed the foundation of this comprehensive work. 🔷 Arthur Erdélyi, one of the principal authors, was a Hungarian-born mathematician who fled Nazi persecution in 1939 and later became a prominent figure at the University of Edinburgh, where he established a distinguished school of research. 🔷 The three-volume series has become a standard reference work in mathematical physics and engineering, particularly for its extensive coverage of special functions like the Bessel functions, hypergeometric functions, and orthogonal polynomials. 🔷 The project was sponsored by the Office of Naval Research and the U.S. Air Force, highlighting the strategic importance of advanced mathematics during the Cold War era. 🔷 The collaboration between the four authors (Erdélyi, Magnus, Oberhettinger, and Tricomi) spanned three continents, with contributors working from Scotland, Germany, and Italy, making it one of the earliest major international mathematical collaborations of the post-WWII period.