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k 1 ln k 2 3 k Then
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is Euler s constant. This constant has been calculated to many places, a few of which are
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1 Y 1 1 x 1 x=2 1 x=k xe
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x 1 x=k e x=k x x=2 k!1 e x e e  1 1 Y  1 1 Y
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x x ln k  ex ex=2 ex=k x e
x e e = 1 k!1 1 x 1 x=2 k 1 x=k k 1 k!1 x
kx k! k x lim
1 2 3 k xx lim x; k k!1 1 x 2 x k
Another result of special interest emanates from a comparison of x 1 x with the famous formula & ' Y 1 x 1 1 1 lim f1 x=k 2 g 10 sin x k!1 1 x2 1 x=2 2 1 x=k 2  1 (See Di erential and Integral Calculus, by R. Courant (translated by E. J. McShane), Blackie & Son Limited.) 1 1 x is obtained from y y 1 by letting y x, i.e., y 1 x 1 x or 1 x x x x
GAMMA AND BETA FUNCTIONS
[CHAP. 15
Now use (8) to produce x 1 x ( x Thus x 1 x Observe that (11) yields the result 1 2  ; sin x p  0<x<1 11a
1
x 1 Y  1
)! 1 x=k e
1 x=k
1 Y  1
! 1 x=k
1 x=k
1 Y 1 1 x=k 2 lim x k!1  1
11b
Another exact representation of x 1 is & ' p x 1 x 1 1 139 x 1 2 x e 1 12x 288x2 51840x3
12
The method of obtaining this result is closely related to Sterling s asymptotic series for the gamma function. (See Problem 15.20 and Problem 15.74.) The duplication formula p 22x 1 x x 1  2x 13a 2 also is part of the literature. Its proof is given in Problem 15.24. The duplication formula is a special case m 2 of the following product formula:       1 m 1 1 2 m 1 x x x X m2 mx 2 2 mx m m m It can be shown that the gamma function has continuous derivatives of all orders. obtained by di erentiating (with respect to the parameter) under the integral sign.
13b They are
It helps to recall that x 1 Therefore, y 0 ln t. y It follows that
tx 1 e yt dt and that if y tx 1 , then ln y ln tx 1 x 1 ln t.
x
tx 1 e t ln t dt:
14a
This result can be obtained (after making assumptions about the interchange of di erentiation with limits) by taking the logarithm of both sides of (9) and then di erentiating. In particular, 0 1
It also may be shown that
is Euler s constant.) 14b
      0 x 1 1 1 1 1 1
x 1 x 2 x 1 n x n 1
15
(See Problem 15.73 for further information.)
THE BETA FUNCTION The beta function is a two-parameter composition of gamma functions that has been useful enough in application to gain its own name. Its de nition is
CHAP. 15]
GAMMA AND BETA FUNCTIONS
B x; y
tx 1 1 t y 1 dt
16
If x ! 1 and y ! 1, this is a proper integral. If x > 0; y > 0 and either or both x < 1 or y < 1, the integral is improper but convergent. It is shown in Problem 15.11 that the beta function can be expressed through gamma functions in the following way B x; y x y x y 17
Many integrals can be expressed through beta and gamma functions. Two of special interest are =2 1 1 x y 18 sin2x 1  cos2y 1  d B x; y 2 2 x y 0 1 p 1 x  0<p<1 19 dx p p 1 sin p 0 1 x See Problem 15.17. Also see Page 377 where a classical reference is given. Finally, see 16, Problem 16.38 where an elegant complex variable resolution of the integral is presented.
DIRICHLET INTEGRALS
x p yq zr If V denotes the closed region in the rst octant bounded by the surface 1 and a b c the coordinate planes, then if all the constants are positive,      
a b c
p q r   x 1 y 1 z
1 dx dy dz 20
pqr 1 V p q r
Integrals of this type are called Dirichlet integrals and are often useful in evaluating multiple integrals (see Problem 15.21).
Solved Problems
THE GAMMA FUNCTION 15.1. Prove: (a) x 1 x x ; x > 0;
a v 1
xv e x dx lim xv e x dx M!1 0 & ' M lim xv e x jM e x vxv 1 dx 0 M!1 0 & ' M v M v xv 1 e x dx v v lim M e
M!1 0
b n 1 n!; n 1; 2; 3; . . . .
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