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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.
Product Details
| ISBN-13: | 9780486803005 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 01/01/2015 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 48 |
| File size: | 20 MB |
| Note: | This product may take a few minutes to download. |
About the Author
Read an Excerpt
The Gamma Function
By Emil Artin, Michael Butler
Dover Publications, Inc.
Copyright © 1964 Holt, Rinehart and Winston, Inc.All rights reserved.
ISBN: 978-0-486-80300-5
CHAPTER 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
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.1)
and for each triple of distinct numbers x1, x2, x3 the quotient
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2)
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 φ(x1, x3)≥ (x2, x3) holds; in other words, that Ψ(x1, x2, x3) ≥ 0. Since the value of Ψ is not changed by permuting the arguments, the convexity of f(x) is equivalent to the inequality
Ψ(x1, x2, x3) ≥ 0 (1.3)
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 limn->∞fn(x) = f(x) exists and is finite for all x in the interval. By forming the inequality (1.3) for fn(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 interval let x1 range over all numbers >x0 and x2 range over all numbers < x0. We have
φ(x1, x0) ≥ φ(x2, x0). (1.4)
If x2 is kept fixed and x1 decreases approaching x0, the left side of Eq. (1.4) will decrease but always remain greater than the right side. This implies that the "right-handed" derivative of f(x) exists; that is to say, the limit
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
for which we shall use the intuitive notation f'(x0 + 0). Furthermore, the inequality (1.4) also shows that
f'(x0 + 0) ≥ φ(x2, x0).
If we let x2 increase, approaching x0, we see that the "left-handed" derivative f'(x0 – 0) also exists, and that
f'(x0 + 0) ≥ f'(x0 - 0). (1.5)
Given two numbers x0< x1 in our interval, we can choose x2, x3 such that x0< x2< x3< x1. Then
φ(x2, x0) ≤ φ(x3, x0) = φ(x0, x3) ≤ φ(x1, x3) = φ(x3, x1).
If we let x2 approach x0 and x3 approach x1, we obtain
f'(x0 + 0) ≤ f'(x1 - 0) for x0< x1. (1.6)
This proves that the one-sided derivatives of a convex function always exist and that they satisfy the inequalities (1.5) and (1.6). We shall refer to the properties (1.5) and (1.6) by saying that the one- sided derivatives are monotonically increasing.
In order to show the converse, we must generalize the ordinary mean-value theorem to cover the case of functions for which only the one-sided derivatives exist. The analogue to Rolle's theorem is the following:
Theorem 1.2
Let f(x) be a function, defined and continuous on a ≤ x ≤ b, whose one-sided derivatives exist in the open interval a< x< b. Suppose f(a) = f(b). Then there exists a value [xi] with a< [xi] < b such that one of the values f'([xi] + 0) and f'([xi] – 0) is ≥ 0 and the other ≤ 0.
Proof
(1) If f(x) takes on its maximum [xi] in the interior of our interval, then
[f([xi] + h) - f([xi])]/h
is ≤ 0 for positive h, ≥ 0 for negative h. Taking limits, we get f'([xi] + 0) ≤ 0, f'([xi] + 0) ≥ 0.
(2) If the minimum [xi] is taken on in the interior, we obtain similarly f'([xi] + 0) ≥ 0, f'([xi] - 0) ≤ 0.
(3) If both maximum and minimum are at a or b, then f(x) is constant, f'(x) = 0, and [xi] can be taken anywhere in the interior. This completes the proof.
The substitute for the mean-value theorem is the following:
Theorem 1.3
Let f(x) be defined and continuous on a ≤ x ≤ b and have one-sided derivatives in the interior. Then there exists a value [xi] in the interior such that (f(b) – f(a))/(b – a) lies between f'([xi] – 0) and f'([xi] + 0).
Proof
The function
F(x) = f(x) – [[f(b) - f(a)]/[b – a]] (x - a)
is continuous, has one-sided derivatives
F'(x ± 0) = f'(x ± 0) – [[f(b) - f(a)]/[b – a]],
and F(a) = f(a), F(b) = f(a). According to our extension of Rolle's theorem, there is a [xi]in the interior such that one of the values
f'([xi] + 0) – [[f(b) - f(a)]/[b – a]] or f'([xi] - 0) – [[f(b) - f(a)]/[b – a]]
is ≤ 0, the other ≥ 0. This completes the proof.
We are now in a position to prove the desired converse. Let f(x) be a function defined on the open interval a< x< b. Suppose f(x) has one-sided derivatives that are monotonically increasing. We contend that f(x) is convex.
Let x1, x2, x3 be distinct numbers in our interval. Since the value of Ψ does not change under permutation of the subscripts, we may assume that x2< x3< x1. According to the mean-value theorem, we Can find [xi], η with x2< η < x3< [xi] < x1 such that φ(x1, x3) lies between f'([xi] – 0) and f'([xi] + 0), and φ(x2, x3) between f'(η – 0) and f'(η + 0). Therefore (1.5) implies that
φ(x1, x3) ≤ f'([xi] - 0) and φ(x2, x3) ≥ f'(η + 0).
From Eq. (1.2) we obtain
Ψ(x1, x2, x3) ≤ [f'([xi] - 0) - f'(η + 0)]/[x1 - x2].
Finally we conclude from (1.6) that
Ψ(x1, x2, x3) ≥ 0,
and this is the contention.
Theorem 1.4
f(x) is an convex function if, and only if, f(x) has monotonically increasing one-sided derivatives.
Corollary
Let f(x) be a twice differentiable function. Then f(x) is convex if, and only if, f"(x) ≤ for all x of our interval.
Proof
f'(x) is monotonically increasing if, and only if, f"(x) ≥ 0.
We now return to Eq. (1.2) and select for x3 the midpoint (x1 + x2)/2 of x1 and x2. Assuming for a moment that x2< x1, we have
x3 - x2 = x1 - x3 = 1/2(x1 - x2).
The numerator of Ψ(x1, x2, x3) becomes
(x1 - x2)(1/2f(x1) + 1/2f(x2) - f(x3)),
and the denominator is positive. For a convex function we obtain the inequality
f([x1 + x2]/2) ≥ 1/2(f(x1)+f(x2)), (1.7)
which is symmetric in x1 and x2 and therefore also holds for x1< x2. For x1 = x2 it is trival.
We shall call a function defined on an interval weakly convex if it satisfies the inequality (1.7) for all x1, x2 of the interval. It is obvious that the sum of two weakly convex functions, both defined on the same interval, is again weakly convex. It is also obvious that the limit function of a sequence of weakly convex functions, all defined on the same interval, is weakly convex.
Let f(x) be weakly convex. The inequality (1.7) can be generalized to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.8)
Proof
(1) We first show that if (1.8) holds for a certain integer n, then it also holds for 2n. Indeed, suppose x1, x2, ···, x2n are numbers in our interval. Replacing x1 and x2 in Eq. (1.7) by
[x1 + ··· + xn]/n and [xn+1 + ··· + x2n]/n,
respectively, we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Applying the inequality (1.8) to both terms on the right-hand side, we get the desired formula
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
(2) Next we show that if (1.8) holds for n + 1, then it also holds for n. With n numbers (x1, x2, ···, xn) the number
xn+1 = 1/n (x1 + ··· + xn)
also belongs to our interval. If (1.8) holds for n + 1, then
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Transposing the term 1(n + 1) f(xn+1) to the left side, we obtain (1.8) for the n given numbers.
(3) We now combine steps (1) and (2) to attain the desired result. If (1.8) holds for any integer n, then step (2) implies that it also holds for all smaller integers. Because of step (1) the contention is true for arbitrarily large integers. Therefore it must be true for all n. This completes the proof.
We wish to prove the following theorem:
Theorem 1.5
A function is convex if, and only if, it is continuous and weakly convex.
Proof
(1) A convex function is continuous since it has one-sided derivatives. It is also weakly convex, as has already been shown.
(2) Suppose that f(x) is weakly convex, that there are x2< x1 numbers in our interval, and that 0 ≤ p ≤ n are two arbitrary integers. Apply (1.8) to the case where p of the n numbers have the value x1 and the remaining n – p numbers have the value x2. We obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.9)
Assume now that f(x) is continuous and let t be any real number such that 0 ≤ t ≤ 1. Select a sequence of rational numbers between 0 and 1 that converges to t. Every term of this sequence is of the form p/n for suitable integers p and n; therefore Eq. (1.9) can be applied. Since f(x) is continuous, we can go to the limit. We obtain
f(tx1 + (1 - t)x2) ≤ tf(x1) + (1 - t)f(x2). (1.10)
For any distinct numbers (x1, x2, x3) in our interval we must show that ψ(x1, x2, x3). Since ψ is symmetric, we may assume that x2< x3< x1. The denominator of Eq. (1.2) is positive.
We set t = (x3 – x2)/ (x1 – x2); then
0 < t< 1, 1 - t = x1 - x3/x1 - x2
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Hence Eq. (1.10) implies that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
which shows that the numerator of Ψ is ≥ 0. This completes the proof.
Numerous inequalities useful in analysis can be obtained from Eq. (1.8) by a suitable choice for f(x). As an example, consider f(x) = – log x for x > 0. We have f"(x) = 1/x2 and our function is convex. Therefore Eq. (1.8) implies that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
hence
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
and consequently
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
We now introduce an important concept closely related to that of convexity. A function f(x) defined and positive on a certain interval is called log convex (weakly log convex) if the function log f(x) is convex (weakly convex). The condition that f(x) be positive is obviously necessary, for otherwise the function log f(x) could not be formed. As an immediate consequence of our previous results, we have the following:
Theorem 1.6
A product of log-convex (weakly log-convex) functions is again log convex (weakly log convex). A convergent sequence of log-convex weakly log-convex) functions has a log-convex (weakly log-convex) limit function, provided the limit is positive.
Instead of the condition that the limit function be positive, we could require that the sequence of the logarithms of the individual terms be convergent.
(Continues...)
Excerpted from The Gamma Function by Emil Artin, Michael Butler. Copyright © 1964 Holt, Rinehart and Winston, Inc.. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Contents
Editor's Preface,Preface,
1. Functions,
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 Γ(x) by Functional Equations,
Index,