The Gamma Function

by Emil Artin, Michael Butler (Translator)

This brief monograph on the gamma function was designed by the author to fill what he perceived as a gap in the literature of mathematics, which often treated the gamma function in a manner he described as both sketchy and overly complicated. Author Emil Artin, one of the twentieth century's leading mathematicians, wrote in his Preface to this book, "I feel that this monograph will help to show that the gamma function can be thought of as one of the elementary functions, and that all of its basic properties can be established using elementary methods of the calculus."
Generations of teachers and students have benefitted from Artin's masterly arguments and precise results. Suitable for advanced undergraduates and graduate students of mathematics, his treatment examines functions, the Euler integrals and the Gauss formula, large values of x and the multiplication formula, the connection with sin x, applications to definite integrals, and other subjects.

Genres: Mathematics

48 pages, Paperback

First published January 1, 2002

Reviews

[Matt Pillsbury · Rating 5 out of 5]

I've been mildly obsessed with the gamma function since high school, when I first learned that the factorial could have a value for an argument that isn't a non-negative integer. The familiar definition in terms of permutations of elements could be extended far beyond that, using a dizzying array of integral representations and intimidating infinite products. The knowledge I had was picked up piecemeal, and amounted more to a collection of tricks--wonderful, useful tricks, mind you--than a coherent, unified understanding.

Emil Artin's brief book changed that. He starts out with a set of properties that characterize the gamma function and proves that any function that has those properties must be the gamma function, and from there shows how most of the familiar identities follow from those requirements. He includes the integral you might have learned in calculus, several infinite product representations, and the Stirling approximation, and along the way to proving the association between the gamma function and the sine, provides a rigorous proof of the Euler infinite product for the sine (which is "derived" in a very hand-wavey way most of the time).

The book is very clearly written (with the exception of one too-slick step in an early proof), and most of it relies on nothing more advanced than a college-level calculus course. I don't normally rate or review the math books I read on Goodreads, but it's been quite a while since I've been so delighted with one of them. If you have any interest in the subject matter, I strongly recommend this one.

[Guilherme · Rating 5 out of 5]

Compact review of the Gamma function, providing a good understanding of the motivation for this function.

[Michael Barnes · Rating 4 out of 5]

It's a very precise and entertaining book that mostly uses nothing more than advanced calculus.