Concrete mathematics : a foundation for computer science

468 ASYMPTOTICS
In Chapter 5, equation (5.83), we generalized factorials to arbitrary real OL
by using a definition
1_ = lim n + O1 Nina
a! n-+m ( n 1
suggested by Euler. Suppose a is a large number; then
lncx! = Ji+mJalnn+lnn!-fln(a+k)),
k=l
and Euler’s summation formula can be used with f(x) = ln(x+ a) to estimate
this sum:
k=l
ln(k+ a) = F,(a,n) - F,(a,O) + Rz,,,(a,n) ,
ln(x + a)
F,(a,x) = (x+a)ln(x+cx-xf 2
n Bz,n(IxI) dx
RI,,,(a,n) = - -
J0
.2m (x + a)2m ’
B2k
k=, 2k(2k - 1 )(x + a)2kp1 ’
(Here we have used (9.67) with a = 0 and b = n, then added ln(n + a) -
lna to both sides.) If we subtract this approximation for xE=, ln(k + a)
from Stirling’s approximation for Inn!, then add alnn and take the limit as
n + 00, we get
lna! = alna-a+lnf+o
B2k
+fm-l)a2kP1
k=l
m
J o
B2m(b>) dx
2m (x+ OoZrn ’
because alnn+nlnn-n+i.lnn-(n+a) ln(nt-a)+n-i ln(n+a) + -a and
the other terms not shown ‘here tend to zero. Thus Stirling’s approximation
behaves for generalized factorials (and for the Gamma function r( a + 1) = a!)
exactly as for ordinary factorials.
Summation 4: A bell-shaped summand.
Let’s turn now to a sum that has quite a different flavor:
. + e-9/n + e-4/n + e l/n + , + e-l/n + e -4/n + e-9/n +. . . .
(9.92)