The Gamma Function

Emil Artin, Michael Butler

• Publisher: Dover Publications
• Year: 2015
• ISBN:
  9780486803005
• Topic: Mathematics
• Subtopic: Functional Analysis
• Index: Mathematics

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.

[ 1 ]

Convex Functions

Let f(x) be a real-valued function defined on an open interval a < x < b of the real line. For each pair x₁, x₂ of distinct numbers in the interval we form the difference quotient

and for each triple of distinct numbers x₁, x₂, x₃ the quotient

The value of the function Ψ(x₁, x₂, x₃) does not change when the arguments x₁, x₂, x₃ are permuted.

f(x) is called convex (on the interval (a, b)) if, for every number x₃ of our interval, φ(x₁, x₃) is a monotonically increasing function of x₁. This means, of course, that for any pair of numbers x₁ > x₂ distinct from x₃ the inequality

holds; in other words, that

. Since the value of Ψ is not changed by permuting the arguments, the convexity of f(x) is equivalent to the inequality

for all triples of distinct numbers in our interval.

Suppose g(x) is another function that is defined and convex on the same interval. By adding (1.3) to the corresponding inequality for g(x), we can easily see that the sum f(x) + g(x) is also convex. More generally, suppose f₁(x), f₂(x), f₃(x) ··· is a sequence of functions that are all defined and convex on the same interval. Furthermore, suppose that the limit lim_n→∞f_n(x) = f(x) exists and is finite for all x in the interval. By forming the inequality (1.3) for f_n(x) with arbitrary but fixed numbers x₁, x₂, x₃, and then taking the limit as n → ∞, we see that f(x) is likewise convex. This proves the following theorem:

Theorem 1.1

The sum of convex functions is again convex. The limit function of a convergent sequence of convex functions is convex. A convergent infinite series whose terms are all convex has a convex sum.

The last statement of this theorem follows from the fact that each partial sum of the series is a convex function and the sum of the series is merely the limit of these partial sums.

We are now going to investigate some important properties of a function f(x) defined and convex on the open interval (a, b). For a fixed x₀ in the interva...