Concrete mathematics : a foundation for computer science

9.6 FINAL SUMMATIONS 471
when they haven’t been tabulated. Elementary calculus suffices to evaluate C
if we are clever enough to look at the double integral
+CC
+CC +m +a2
C2 = epxz dx e-y’ dy = e-(X'+Yz) dx dy.
J-M
J-0c)
J -03 J-00
Converting to polar coordinates gives
c2 =
2n co
J J
0 0
12" 00
EZ-
J J0
1
=-
2x
e -“T dr d0
2 0 d0 epu du
J
2 0
d0 = rr.
(u = l-2)
So C = ,/%. The fact that x2 + y2 = r2 is the equation of a circle whose
circumference is 27rr somehow explains why rr gets into the act.
Another way to evaluate C is to replace x by fi and dx by itt’/2 dt:
J +oO
C = epx2 dx = 2
J”
epxz dx =
-DC, 0
t-‘/+t dt
This integral equals r(i), since I == jr taP’ePt dt according to (5.84).
Therefore we have demonstrated that r(i) = ,,/?r.
Our final formula, then, is
0, = x eCkzin = &iii+ O(neM) , for all fixed M. (9.93)
k
The constant in the 0 depends on M; that’s why we say that M is “fixed!’
When n = 2, for example, the infinite sum 02 is equal to 2.506628288;
this is already an excellent approximation to fi = 2.506628275, even though
n is quite small. The value of @loo agrees with 1Ofi to 427 decimal places!
Exercise 59 uses ad.vanced methods to derive a rapidly convergent series
for 0,; it turns out ,that
@,/&ii = 1 -I- 2eCnnz + 0( eC4nnL ) . (9.94)
Summation 5: The clincher.
Now we will do one last sum, which will turn out to tell us the value
of Stirling’s constant cr. This last sum also illustrates many of the other
techniques of this last chapter (and of this whole book), so it will be a fitting
way for us to conclude our explorations of Concrete Mathematics.