伽马函数《The gamma function》

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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.
The german original Einfuhrung in die Theorie der Gammafunktion appearcd in the Hamburger Mathematische Einzelschriften 1. Heft/931, published by verlag B. G. Teubner, Leipzig Editor's Preface a generation has passed since the late Emil Artin, s little classic on the gamma function appeared in the Hamburger Mathematische Einaelschriften Since that time, it has been read with joy and fascination by many thousand of mathematicians and students of mathematics. In the United States(and presumably elsewhere as well), it has for many years been hard to find, and dog-eared copies and crude photocopies have been passed from hand to hand Professor Artin s monograph has given many a student his first look at genuine analysis-the delicacy of its arguments, the precision of its results. Artin had a deep feeling for these aspects of analysis, and he treated them with a master's hand. His undergraduate lectures in the calculus, for example were flled with elegant constructions and theorems which, alas, Artin never had time to put into printed form. We may be all the more grateful for this beautiful essay, and for its appearance in a new English edition, Various changes made by Artin himself have been incorporated in the present edition. In particular a small error following formula(59)(this edition)was corrected on the basis of gge tion by professor Be Finally, thanks are due to the translator, Mr. Michael Butler, and to the firm of B. G. Teubner for English-language rights. EDWIN HEWITT Seattle, Washington English Translation May, 1964 Copyright 1964 br Holt, Rinehart and Winston, Inc. Library of Congress Catalog Card Number: 64-22994 20526-0114 Printed in the United States of America All Rights Reserved Preface Contents I have written this monogram h with the hope of filling in a certain gap which has often been felt to exist in the mathematical literature. Despite th Editor's Preface importance of the gamma function in many different parts of mathematics, calculus books often treat this function in a very sketchy and complicated P reface fashion. 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 1. function As far as prerequisites are concerned, the reader need only be well acquainted 2. The euler Integrals and the Gauss Formula with calculus, including improper integrals. Some of the more important 3. Large Values of x and the Multiplication Formula concepts needed will even be introduced and discussed again in the first chapter 4. The Connection with sin x 25 With this background the reader should have no trouble understanding every 5. Applications to Defnite Integrals 28 thing but the later parts of the last two chapters, which do assume some knowl- 6. Determining T() by Functional Equations 33 edge of Fourier series. But then, these parts of the monograph can be passed over on a first reading without any difficulty whatsoever The following parts of the theory will not be discussed Index 39 ()Extension to complex variables. For those familiar with the theory of complex variables, it will suffice to point out that for the most part the expressions used are analytic, and hence they retain their validity in the complex case because of the principle of analytic continuation. The only parts of the theory that really need to be changed are those dealing with approximations. This certainly should not be much of an obstacle (2)Holder's theorem showing that the gamma function does not satisfy any algebraic differential equation. (3)Kummer's series and the integral representation of log r(r) (4)The formula for the logarithmic derivative of r(x). All the necessary expressions for this can easily be worked out by the reader I have chosen the integral as my original definition of the gamma function because this approach saves us the trouble of proving the convergence of Gauss'product, Any other analytic expression having the characteristic proper- he gamma function could just as well have been used. The whole theory will then be deduced using the concept of log convexity. This method comes from Bohr and Mollerup EMIL ARTIN 1 H. Bohr and ]. Mollerup, laer natem Analyse ( Kopenhagen 1922) vol. III,149-164 Convex Functions Let f(r)be a real-valued function defined on an open interval a<x<b of the real line For each pair xn, xg of distinct numbers in the interval we form the difference quotient q(x1,x2) f(x1)~f(x2) =y{x2,x1) and for each triple of distinct numbers xi, #2, xg the quotient 平(x1,,与)叫x,x。)一(x2,x) (3-x2)f(x1) x3)f(x2)+(x2-x1)f( (x1-x2)(x2-x3)(x The value of the function Y(x1, x2, xg) does not change when the arguments x3 are permuted f(e) is called convex(on the interval (a, b))if, for every number xa of our interval, (#1, x3)is a monotonically increasing function of *,. This means, of course, that for any pair of numbers, > x 2 distinct from x, the inequality F(=2,*)2 o(=2, x3)holds the ds, that Y( 0. Since the is not changed by permuting the arguments, the convexity of f() Is equivalent to the inequality x)≥0 (L.3) for all triples of distinct numbers in our interval Suppose g(a) 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(=)+g(r) is also convex. Mlore generall f2(x), f3(r) is a sequence of functions that are all defined and convex on the same interval. Turthermore, suppose that the limit limn xfn(x)=f(x)exists and is finite for all x in the interval. By forming the inequality(1.3)for fn() b en taking the lin n-0o, we see that f(x)is likewise convex. This proves the following theorem THE GAMMA FUNCTION CONVEX FUNCTIONS Theorem 1.1 In order to show the converse, we must generalize the ordinary mean-value The sum of convex functions is again convex, The limit function theorem to cover the case of functions for which only the one-sided derivatives of a convergent sequence of convex functions is convex, A convergent exist. The analogue to rolle,s theorem is the following infinite series whose terms are all convex has a convex sum. heorem 1.2 The last statement of this theorem follows from the fact that each partial et f(r)be a function, defined and continuous on a< x< b, whose sum of the series is a convex function and the sum of the series is merely the one-sided derivatives exist in the open interval a<x <b. Suppose limit of these partial sums. f(a=f(6). Then there exists a value s with a<5b such that one of We are now going to investigate some important properties of a function the values '(s+O)and(s-o)is >>0 and the other o f()defined and convex on the open interval(a, b). For a fixed xo in the interval let x, range over all numbers >xo and 2 range over all numbers *o. We Proof have (1)If f(r)takes on its maximum s in the interior of our interval, then q(r1,xo)≥q(x2,% (1.4) f(6+h)一f() h If x2 is kept fixed and xn decreases approaching xo, the left side of eq(1. 4)will " right-handed"derivative of f(x) exists; that is to say, the limi plies that the decrease but always remain greater than the right side. This s≤ 0 for positive h,≥0 for negative h. Taking limits, we get f(+0)≤0, ∫(5-0)≥0 (2) If the minimum 5 is taken on in the interior, we obtain similarly lim p(x1, xo)=lim ∫(x1)-f(xo) ∫(+0)≥0,f(-0)≤0 C。+ (3)If both maximum and minimum are at a or b, then f(x)is constant, f()=0, and s can be taken any where in the interior. This completes the proof for which we shall use the intuitive notation f (=o+O). Furthermore, the The substitute for the mean-value theorem is the following inequality(1. 4)also shows that f(xo10)≥叭(x2,o) Theorem 1.3 If we let x2 increase, approaching *o, we see that the"left-handed"derivative Let f(x be defined and continuous on a s xs b and have one- sided derivatives in the interior. Then there exists a value f in the interior f'lra-0)also exists, and that such that((6)f(a)) (b-a)lies between f'(s-O)andf(s+o) f(x0+0)≥f(x0-O) (1.5) Given two numbers xo <xn in our interval we can choose x2, *3 such that The function to xa<N3 <.5n. Then F(x)=f(r) f(b-f(a) 6-a g(x2,x≤g(x3,x)=g(N,≤9(x1,x=x,x) is continuous, has one-sided derivatives If we let x2 approach o and ta approach , we obtain F(x±0)=f(x±o-1)-/a f(ro +0)sf' (nn-0) for to<ru 1.6) and F(a)=f(a), F(b)=f(a). According to our extension of Rolle's theorem, there is a s in the interior such that one of the values + This proves that the one- sided derivatives of a convex function always exist nd that they satisfy the inequalities([ 5 )and(1. 6). we shall refer to the pro- f(5+-0)一 f(6)-f(a perties(1. 5 )and(1.6)by saying that the one-sided derivatives are monotonically 6-a OT /(s-O)-f(6)-f(a creasy is≥0, the ot her≤0. This completes the proof THE GAMMA FUNCTION CONVEX FUNCTIONS We are now in a position to prove the desired converse. Letf(a) be a func which is symmetric in #1 and x2 and therefore also holds for x,<x2.For tion defined on the open interval a <t<b. Suppose f(x)has one-sided deriva tives that are monotonically increasing. We contend that f(x)is convex. We shall call a function defined on an interval weakly convex if it satisfies t.t. Let xi, i2, a be distinct numbers in our interval. Since the value of y the inequality(1. 7)for all #1, te of the interval It is obvious that the sum of does not change under permutation of the subscripts, we may assume that two weakly convex functions, both defined on the same interval, is again weakly 2<3< 1. According to the mean-value theorem, we can find 5, n with convex. It is also obvious that the limit function of a sequence of weakly convex 2<n<x3 <<5 <x, such that p(1, x,)lies between(5-0)andf'(s+O), functions, all defned on the same interval, is weakly convex. and (I2, r,) between/'(n-0)andf'(n+O). Therefore (1.5)implies that Let f(r)be weakly convex. The inequality(1.7)can be generalized to q(1,x3)≥∫(-0)and(x2,x2)≤∫(+0 /{十当“+)≤1(x)+1)+…+)(18 From Eq. (. 2)we obtain 9(x1:x2,x3)≥ f(5-0)-f(n+0) (1)We first show that if ( 1.8) holds for a certain integer n, then it also holds for 2n. Indeed, suppose x1, x2,",xen are numbers in our interval Finally we conclude from(.6) that Replacing #1 and x2 in Eq (1. 7)by H(x1,2,x3)≥0, 十…十x a and this is the contention respectively, we have Theorem 1.4 f(c)is an convex function if, and only if, f(=) has monotonically increas- (+2+)≤((+n+)+/(+”+) ing one-sided derivatives Corollary Applying the inequality(1.8)to both terms on the right-hand side, we get the desired formula Let f(r)be a twice differentiable function. Then f(a)is convex if and only if,f"(=)20 for all x of our interval f(+…+)≤(fx)+fx2)+…+(x2) 2n Proof (2)Next we show that if(1. 8)holds for n + 1, then it also holds for n f(r)is monotonically increasing if, and only if, f"(x):>0. with n numbers(=,, x2,-, tn) the number i and x2. Assuming for a moment that 2 < 1, we have oint (x+x2 of W'e now return to Eq. (1. 2)and select for #3 the mid also belongs to our interval. If (1. 8) holds for n+1, then The numerator of Y(=1, x2, x3)becomes fun)=f nn+ +…十x+xn (x1-x2)(是号(x1)÷是f(x2)-f(x3), and the denominator is positive. For a convex function we obtain the inequality n (f(x1)+…+f(xn)+f(xn+1) (2)≤量(/x)+1x) Transposing the term 1/(n +1)f(n+,)to the left side, we obtain(.8)for the n given numbers. THE GAMMA FUNCTION CONVEX FUNCTIONS (3)We now combine steps(1)and (2)to attain the desired result. If Numerous inequalities useful in analysis can be obtained from Eq(1.8) 1 8)holds for any integer n, then step(2)implies that it also holds for all smaller by a suitable choice forf(=). As an example, consider f(x)=-log x for x>0. integers. Because of step(1)the contention is true for arbitrarily large integers We have f(x)=1/x' and our function is convex. Therefore Eq (1. 8)implies Therefore it must be true for all n. This completes the proof. hat We wish to prove the following theorem 1g(+"+)≤-n(egx+1gx+…+gx Theorem 1.5 hence A function is convex if, and only if, it is continuous and weakly con vex and consequently (I A convex function is continuous since it has one-sided derivatives v十“十馬≤十…+ It is also weakly convex, as has already been shown (2) Suppose that f(e)is weakly convex, that there are x2<x, numbers We now introduce an important concept closely related to that of convexity in our interval, and that 0 s psn are two arbitrary integers. Apply ( 1.8 A function f(x) defined and positive on a certain interval is called log convex to the case where p of the n numbers have the value x, and the remaining (weakly log convex) if the function log f(x) is convex(weakly convex). The cot n-p numbers have the value 2, We obtain dition that f(s) be positive is obviously necessary, for otherwise the function px1十 f(2+(1-2)x)≤2f()+{- log f(a) could not be formed. As an immediate consequence of our previous (】.9) resuits, we have the following: Assume now that f()is continuous and let t be any real number such that 0<ts1. Select a sequence of rational numbers between 0 and i that converges Theorem 1.6 to t. Every term of this sequence is of the form pin for suitable integers p and n; A product of log-convex (weakly log-convex) functions is again therefore Eq(1.9)can be applied. Since f(r)is continuous, we can go to the log convex (weakly log convex). A convergent sequence of log-convex limit We obtain weakly log-convex)functions has a log-convex(weakly log-convex)limit f(tx1+(-)x2)≤(x1)+(1-0)f(x2) (1.10) function, provided the limit is positive. For any distinct numbers(5, 3) in our interval we must show that Instead of the condition that the limit function be positive, we could pAr1, x2,3)>0. Since y is symmetric, we may assume that x2 <x3< 1 require that the sequence of the logarithms of the individual terms be con- The denominator of Eg.(1. 2) is positive vergent Wie set t=(rg-x2)/*1-x2);then 0<t<l.i-t Theorem 1.7 X1-x2 and Suppose f( )is a twice differentiable function. If the inequalities tx1+(l-0x2= (x3-x)x1+(x1-xx2 f(x)>0,f(x)∫"(x)-(f(x)2≥0 hold, then f(=) is log convex Hence Eq (1.10)implies that This theorem follows directly from the fact that the second derivative of f()sx-1fx1)+x1一考 og f()has the value f(x)∫"(x)-(f(x)2 which shows that the numerator of p is>0. This completes the proof THE GAMMA FUNCTION CONVEX FUNCTIONS The properties of log-convex functions mentioned thus far are all more or This completes the proof of the theorem ss Immediate consea uences of the definition. The following remarkable eorem, ho wever, is a much deeper result he reader who enjoys working with identities can check the validity of the following, for an alternate proof Theorem 1.8 a142(a1+a)(e1+c2)-(b1+b2)2) Suppose f(a and g()are functions, defined on a common interval If both are weakly log convex, then their sum f(x)+ g(r)is also weakly =a(a1+a)(a11-b+a1(a1+a2)(ag2-b+(a1h2-ab1) log convex. If both are log convex, then f()+g(x)is log convex Proof b≥0, ≥0, It suffices to prove the first statement. The second then follows immediately with the addition of continuity. the right side is>0 and the conclusion follows. Both f(=)and g()are positive. If xi, 2 are numbers in out interval, then Other important facts can be obtained by combining our previous results. e f(t, r)is a function of the two variables x and t, which is defined and 22)</)3)ad(()≤a continuous for t in the interval a< t< b and x in some arbitrary interval Furthermore, for any fixed value of t, suppose that f(t, *)is a log-convex,twice We have to show that differentiable function of x. For every integer n we can build the function Fn(x)=从f(a,x)十f(a十b,x)+f(a+2h,x)+…+f(a+(n-1)h,x), 2)+8(为十)2 ≤(f(x1)+8(x1)(f(x2)+g(x2) where h=(b-a)/n, Being the sum of log-convex functions, Fn(*)is also In other words the proof of our theorem amounts to showing that if a1, b,, u g convex As n approaches infinity, the functions Fn(=)converge to the integral a2,如, C2 are positive real numbers with a~b3≥0anda2g2-的≥0 ∫(,x)dt (a1+a)(1+c)-(b1+b2≥0 hence this integral is also log convex Consider the quadratic form a, r2+ 2b,-ry +Gry2 where a1>0. We have Suppose that f(t, x)only satisfies our conditions in the interior of the f interval, or that the upper bound of the interval is infinite. If the improper an(a1x222+c1y2)=(a1x+b1y)2+(a21-6)y2 integral Ⅱa11-姓≥0, the quadratic for rm never takes on a negative value, what f(t, x) ever t, y may be On the other hand, if a,c-61<0, the quadratic form takes on the negative value an 1 --bi fory (b:如a1) ur conditions imply that neither exists, then it is log convex. This follows directly from the fact that an improper integral is the limit of proper integrals over subintervals. Hence, as the limit av2+2b ry +G,y2 nor a2x2+ 2b2ry i c2y2 function of log-convex functions it is also log convex. In this book we will only have to test integrals of the form takes on negative values. Therefore (a1+a2)x2+2(+b2)xy+(a1+c2)y2 9(2)ta-l di will not take c on negative values. Consequentl for log convexity, where (t) is a positive continuous function in the interior of the integration interval. If we take the logarithm of the integrand and then (a1+a2)(1+c2)-(b1+b2)≥0 differentiate twice with respect to r, we get 0. THE GAMMA FUNCT】oN Theorem 1.9 If (t)is a positive continuous function defined on the interior of he integration interval, then The Euler integrals and the g() Gauss product Formula is a log-convex function of x for every interval on which the proper or improper integral exists The follo wing theorem is quite obvious The theory of the gamma function was developed in connection with the Theorem 1. 10 problem of generalizing the factorial function of the natural numbers, that is the problem of finding an expression that has the value n! for positive integers n, If f(*)is log convex on a certain interval, and if c is any real number and that can be extended to arbitrary real numbers at the same time. In looking #+0, then both the functions f(x+ c)and f(ex)are log convex on the corres for such an expression, we come upon the following well-known improper ponding intervals Integ e-t in dt= n This suggests replacing the integer n on the left side by an arbitrary real number (provided the integral still converges)and defining x! for arbitrary t as the value of this integral. Rather than doing precisely that, we will follow the cu of introducing a function that has the value(n-1)! for positive integers n Namely T(x) 4=-l dt (21) We still must determine the values of x for which this integral converges. The integrand is smaller than t*- when t is positive; therefore For x>0 e-t*-l dt is bounded from above by lf-x. If we hold x fixed and let e decrease, the value of the integral increase monotonically. This means that exists for all positive x

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