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GAMMA AND BETA FUNCTIONS
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The equation (2) is a recurrence relationship that leads to the factorial concept. First observe that if x 1, then (1) can be evaluated, and in particular, 1 1: From (2) x 1 x x x x 1 x 1 x x 1 x 2 x k x k If x n, where n is a positive integer, then n 1 n n 1 n 2 . . . 1 n! 3
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If x is a real number, then x! x 1 is de ned by x 1 . The value of this identi cation is in intuitive guidance. If the recurrence relation (2) is characterized as a di erential equation, then the de nition of x can be extended to negative real numbers by a process called analytic continuation. The key idea is that 1 even though x is de ned in (1) is not convergent for x < 0, the relation x x 1 allows the x meaning to be extended to the interval 1 < x < 0, and from there to 2 < x < 1, and so on. A general development of this concept is beyond the scope of this presentation; however, some information is presented in Problem 15.7. The factorial notion guides us to information about x 1 in more than one way. In the eighteenth century, Sterling introduced the formula (for positive integer values n) p n 1 n 2 n e lim 1 4 n!1 n! p n 1 n This is called Sterling s formula and it indicates that n! asymptotically approaches 2 n e for large values of n. This information has proved useful, since n! is di cult to calculate for large values of n. There is another consequence of Sterling s formula. It suggests the possibility that for su ciently large values of x, p 5a x! x 1 % 2 xx 1 e x (An argument supporting this is made in Problem 15.20.) It is known that x 1 satis es the inequality p x 1 x p 1 2 x e < x 1 < 2 xx 1 e x e12 x 1
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Since the factor e ! 0 for large values of x, the suggested value (5a) of x 1 is consistent with (5b). An exact representation of x 1 is suggested by the following manipulation of n!. (It depends on n k ! k n !.) n! lim 12 . . . n n 1 n 2 . . . n k k! kn k 1 k 2 . . . k n lim lim : k!1 k!1 n 1 . . . n k k!1 n 1 n 2 . . . n k kn
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k! kn : (This must be read as an Since n is xed the second limit is one, therefore, n! lim k!1 n 1 . . . n k in nite product.)
CHAP. 15]
GAMMA AND BETA FUNCTIONS
This factorial representation for positive integers suggests the possibility that x 1 x! lim k! kx x 1 . . . x k x 6 1; 2; k 6
Gauss veri ed this identi cation back in the nineteenth century. This in nite product is symbolized by x; k , i.e., x; k function and through this symbolism, x 1 lim x; k
k! kx . It is called Gauss s x 1 x k 7
The expression for follows.
1 (which has some advantage in developing the derivative of x ) results as x Put (6a) in the form kx k!1 1 x 1 x=2 . . . 1 x=k lim 1 1 1 x 6 ; ; . . . ; 2 3 k