CHAP. 9]
GAMMA, BETA AND OTHER SPECIAL FUNCTIONS
219
STIRLING'S FORMULA
9.23. Show that for large n, n\ = \/2irnn"e-n approximately. We have
W The function n In x — x has a relative maximum for x = n, as is easily shown by elementary calculus. This leads us to the substitution x = n + y. Then (1) becomes (2)
Up to now the analysis is rigorous. The following procedures in which we proceed formally can be made rigorous by suitable limiting procedures, but the proofs become involved and we shall omit them. In (2) use the result
(S) with x — y/n. Then on letting y = \/n v, we find
(4) When n is large a close approximation is (*>
It is of interest that from (4) we can also obtain the miscellaneous result 4 on page 211 l[see Problem 9.38].
DIRICHLET INTEGRALS 9.24. Evaluate
where V is the region in the first octant bounded by the sphere x2 + yz + za = 1 and the coordinate planes. Let *2 = u, y* = v, zz = w. Then
where ^ is the region in the uvw space bounded by the plane u + v + w = 1 and the uv, vw and uw planes as in Fig. 9-2. Thus
Fig 9.2
(*)