This is a very interesting and very brief (39 pages) introduction to the gamma function, based on log-convexity (a log-convex function is one whose logarithm is convex). The book covers all the standard gamma facts, including the integral and limit representations, an especially nice derivation of an effective Stirling’s formula, and some of the multiplication and reflection formulas. The book appeared originally in German in 1931 and in a corrected English translation from Holt, Rinehart & Winston in 1964. The present volume is a Dover 2015 unaltered reprint of the 1964 edition.
The gamma function is log convex, and in fact (the Bohr–Mollerup theorem) it is the only log-convex function satisfying its recursion \(\Gamma(x+1) = x \Gamma(x)\) and the initial condition \(\Gamma(1) =1\) . In most accounts this theorem is presented as a curiosity, but Artin’s exposition makes log-convexity the central concept from which the gamma function’s properties are derived.
The book is full of clever (but simple) proofs and is worth reading just for those. I was especially impressed with how far you can go with log-convexity. The sum of convex functions is convex, and so the product of log-convex functions is log-convex, but surprisingly the sum of log-convex functions is also log-convex. The book makes heavy use of these properties to build up the functions of interest from simple log-convex functions.
An unusual feature of the book is that it only considers the gamma function for a real argument. Euler did this too, so it is historically accurate, but most expositions today approach it as an application of complex analysis. I like the approach in the present book, because it requires very little machinery (just definite integrals and infinite series). As the author points out on p. vi, most of the results extend immediately to a complex argument by analytic continuation. The only things you might miss are some of the contour-integral representations and Stirling’s formula for complex arguments.
There is another monograph on this subject, Nielsen’s 1906 Die Gammafunktion, but it is very difficult to use because the notations have changed so much since it was written. Most books on complex analysis and on special functions have good coverage of the gamma function; one thorough example is Whittaker & Watson’s Modern Analysis. There are also good summaries in most handbooks of mathematical functions, for example the NIST Handbook of Mathematical Functions.
Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.
2. The Euler Integrals and the Gauss Formula
3. Large Values of \(x\) and the Multiplication Formula
4. The Connection with \(\sin x\)
5. Applications to Definite Integrals
6. Determining \(\Gamma(x)\) by Functional Equations