THE GAMMA FUNCTION THU NGỌC DƯƠNG

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1 THE GAMMA FUNCTION THU NGỌC DƯƠNG The Gamma unction was discovered during the search or a actorial analog deined on real numbers. This paper will explore the properties o the actorial unction and use them to introduce the Gamma unction.. The Factorial The actorial unction, n!, was initially deined over the positive integers as, n! = (n)(n )(n ) (3)()(). This expresses the number o ways to permute an n element set. Since a set with element can be permuted only one way,! =. With this base case, the actorial may be deined recursively as: n! = n(n )!. An interesting property o the actorial unction is its rate o growth. The actorial grows in a weak log convex ashion. But to understand this behaviour, we must irst understand what it means or a unction to be convex, log convex, and weakly convex. These concepts will be explored in the ollowing section... Weak Log Convexity.... Convexity. Consider a real valued unction : (a, b) R. For x, y (a, b), deine the symmetric unction φ: φ(x, y) = φ(y, x) = (x) (y). x y Then is convex i φ(x, y) is monotonically increasing with respect to each variable. This means φ(x, y ) φ(x, y ) when y y and symmetrically, φ(x, y) φ(x, y) when x x. I in addition, is dierentiable, then is convex when (x) (y) or all a < x < y < b.

2 THU NGỌC DƯƠNG The convexity o may also be analyzed by considering the unction Φ: Φ(x, y, z) = = φ(x, z) φ(y, z) x y (x)(y z) + (y)(z x) + (z)(x y) (x z)(y z)(x y) or x, y, z (a, b). Then is convex i Φ is nonnegative. I is also twicedierentiable, then is convex when (x) or all a < x < b.... Log Convexity. Log convexity is similarly deined. A unction g is log convex i log g is convex. Suppose g is also twice dierentiable. Then, (log g). Dierentiating log g gives us, (log g) = g g (log g) = g g g. g It ollows that g g g. Note that log convex unctions are also convex. This is because log is deined or nonnegative reals, so g. Thereore g g /g when g =...3. Weak Convexity. However, convex unctions are continuous in R and the actorial is not. So the actorial is not a convex unction, though it behaves like one. Let us deine a weaker orm o convexity which encompasses noncontinuous unctions as well. There exists certain unctions deined on (a, b) R or whom Φ(x, y, z) may not be nonnegative or all x, y, z (a, b), but or whom Φ(x, (x+z)/, z) is nonnegative or all x, y, z (a, b). Such unctions are called weakly convex. Suppose : (a, b) R is a weakly convex unctions. Consider a < x < y < z < b such that y = (x + z)/ and let, This implies, Φ(x, y, z) = m = (z y) = (y x) = z x. (x)(y z) + (y)(z x) + (z)(x y) (x z)(y z)(x y) [ (x) + (y) (z)]m =. (x z)(y z)(x y)

3 THE GAMMA FUNCTION 3 Because (x z)(y z)(x y) < and m >, it ollows that Φ(x, y, z) when (x) + (y) (z). Thereore, is weakly convex i and only i, ( ) x + z [(x) + (z)]. In general, given a weak convex unction deined on (a, b) and x,..., x n (a, b), ( ) x + + x n [ (x ) + + (x n )]. n n We know this statement holds or n =. Suppose the statement is true or an arbitrarily given n N. Then it is true or n. ( ) [ ( ) ( )] x + + x n x + + x n x n+ + + x n + n n n [ (x ) + + (xn )]. n And it is true or (n ). Let x,..., x n (a, b) and deine + x n x n = x +. n which is also in (a, b). This gives us, ( ) ( ) x + + x n (n )x n + x n = (x n ) = n n (x n ) [(x ) + + (x n ) + (x n )] ( ) n x + + x n [ (x ) + + (x n )]. n n Since the relation holds or (n ) when it holds or n, and since it holds or arbitrarily large n, the relation holds or all n N by induction...4. Weak Log Convexity. Weak log convexity is similarly deined. A unction g is weakly log convex i log g is weakly convex. Suppose g is a weakly log convex unction on (a, b). Then or x, y, z (a, b), ( ) x + z log g [log g(x) + log g(z)] x + z log g log g(x)g(z) x + z g g(x)g(z).

4 4 THU NGỌC DƯƠNG.. The Factorial is Weakly Log Convex. Having introduced the concept o weak log convexity, we will now show that the actorial, (n) = n! or n N, is a weakly log convex unction. This means m + n (m)(n), or all m, n N. Proo. Let m = u and n = v. Then the above statement becomes, (u + v)! (u)!(v)!. Without loss o generality, assume u v. (u + v)! = (u + v)!(u + v) (u + v (v u ))(u)! (v) (v (v u ))(u + v)!(u)! = (v)!(u)!. Then the question is, does a more general version o the actorial unction exist? And i it exists, what is it? Such a unction would possess the ollowing properties,. () =,. (x + ) = (x + ) (x), 3. (x) is weakly log convex or x N, and would be deined or all nonnegative real numbers.. The Existence o the Gamma Function In the early 8th century, the proliic mathematician Leonhard Euler discovered a unction deined or (, ) which mimicked the actorial on the positive integers. It possesses the ollowing properties, which are similar to those o the actorial:. () =,. (x + ) = x (x), 3. (x) is log convex or x (, ). This unction is known as the gamma unction and is deined as, Γ(x) = e t t x dt. Figure. is a graph o the gamma unction over both positive and negative values.

5 THE GAMMA FUNCTION 5 e+6 Γ(x). e 4 e Figure. The Γ unction and It is easy to veriy the irst two properties o the gamma unction. Γ() = e t dt = e t =. Γ(x + ) = e t t x dt = t x e t + = xγ(x). xt x e t dt However, to understand why Γ is log convex, irst note that sums o weakly log convex unctions are also weakly log convex. Suppose and g are weakly log convex unctions. Then, u + v (u)(v). u + v g g(u)g(v). [ ( ) ( )] u+v u+v E. Artin showed that this implies + g [(u) + g(u)] [(v) + g(v)] by considering the polynomial, h(x, y) = a(ax + bxy + cy ) = (ax + by) + (ac b )y, or a.

6 6 THU NGỌC DƯƠNG When b ac, h is nonnegative. ( ) ( ) u+v u+v Let a = (u), b =, c = (v) and a g = g(u), b g = g, c a c and b g = g(z). Then the convexity o and g imply b g a g c g, so we have a x + b xy + c y a g x + b g xy + c g y (a + a g )x + (b + b g )xy + (c + c g )y. Thereore, (b + b g ) (a + a g )(c + c g ). It ollows that + g is weakly log convex. Likewise, integrals o weakly log convex unctions are weakly log convex. And since log convex unctions are also weakly log convex, integrals o log convex unctions are log convex. (x) = e t t x is a log convex unction because it is twice dierentiable and satisies e t t (x ) (ln t) e t t (x ) (ln t). Thereore, Γ is a log convex unction because it is an integral o a log convex unction. Though the gamma unction is not equivalent to the actorial unction since (x + )! = (x + )x! and Γ(x + ) = xγ(x), both unctions are similarly deined recursively and both unctions grow weak log convexly. On the positive integers, the gamma unction may be reduced to Γ(x) = (x )!. Now that we have ound a real analog o (x )!, we may ask, do other such unctions exist? 3. Uniqueness o the Gamma Function We have shown that there exists a gamma unction which behaves like the actorial on the positive integers. We will now show that any unction, which is deined or (, ) and behaves like (x )! on the positive integers, must be the gamma unction. Let be such a unction. Then,. () =.

7 THE GAMMA FUNCTION 7. (x + ) = x (x). 3. is log convex or x (, ). Because (x + ) = x (x), it is suicient to determine the behaviour o on the interval (, ]. So consider < x and let n N such that n. From the log convexity o, we have the ollowing, log ( +n) log (n) log (x+n) log (n) log (+n) log (n) ( +n) n (x+n) n (+n) n. Since behaves like the actorial on the positive integers and () =, we know (n) = (n )!. This implies that, log(n )! log(n )! log (x+n) log(n )! log n! log(n )! x log (x+n) log(n )! log(n ) x log n (n ) x (n )! (x + n) n x (n )!. Because (x + ) = x (x), it ollows that (x + n) = (x + n ) (x) (x). (n ) x (n )! (x + n ) (x) (x) n x (n )! (n ) x (n )! (x+n ) (x) (x) n x n! x+n (x+n)(x+n ) (x) n. (n ) may be replaced with n on the let hand side o the above inequality since the relation holds or all integers n. n x n! (x+n) (x) (x) n x n! x+n (x+n) (x) n n (x) n x n! x+n (x+n) (x) (x). Taking the limit o both sides o the inequality as n tends to ininity gives, Thereore, (x) lim n n x n! (x). (x+n) (x) n x n! (x) = lim n (x + n) (x). Because this limit is unique, is unique. Thus, there exists only one unction which possesses the actorial like properties discussed. Since the gamma unction possesses these properties, the gamma unction must be the only such generalization o the actorial to real numbers. Figure illustrates how well Γ approximates actorial. 4. Acknowledgements I would like to thank Punyashloka Biswal teaching me how to use L A TEX, and or creating Figure and Figure. I would also to thank Justin Curry or his L A TEX suggestions.

8 8 THU NGỌC DƯƠNG e+8 e+6 x! Γ(x + ) e+4 e+ e+ e+8 e Figure. A comparison o the actorial and gamma unctions Reerences [] Artin, Emil. (964). The Gamma Function. New York: Holt, Rinehart, and Winston. Translated by: Michael Butler.