Calculus: Analytic Geometry and Calculus, with Vectors

(^646) Series
In this formula we substitute the expression for log n! given in (7) and the expres-
sion for the last sum given in (1). With the aid of the fact that
(10) lim (z + n + Y')[log (z + n) - log n] = z,
we find that
(11) log z! = log V + (z +') log (z + 1) - (z + 1)

• M

I

Bi^1

j=2 (9 - 1)] (z +1)i-i-J1

Co(m - 1)!Bm(x)

dx.

(Z+x)m

In case z > 0, we can add log (z + 1) to both sides of (11) and then replacez

by z - 1 to obtain the alternative formula

(12) log z! = log \/2ir + (z + I) log z - z

+

m

Bi 1 Co (m - 1)!Bm(x)

5= (1 - 1)9 zl-1-./0 (Z+ x)m

which reduces to (7) when z = n. The derivation of Stirling formulas is not

yet complete. To finish the task, we must study the theory of analytic functions

of a complex variable. It will then be possible to observe that the members of

(12) are analytic over the set consisting of complex numbers which are neither

0 nor negative. The principle of analytic extension then implies that the mem-

bers of (12) are equal for each z which is neither 0 nor negative. We conclude

with some remarks about (12). Let

Bi 1 (m - 1)!Bm(x)

(13) E(z)_ -12 &- 1)j zi1 o (z + x)m dx.

We can then put (12) in the forms

(14) log z! = log 2zu + log zz - z + E(z)

and

(15) z! = /2zxr zze =eacz>.

The formulas (12) and (14) are Stirling formulas for log z!, and (15) is the Stirling

formula for z!. In many applications, E(z) is so near 0 that it can be neglected.

Information about E(z) is obtained from (13). When n = 1, the first sum in the

right member of (13) contains no terms. Hence putting m = 1 in (13) gives

(16) E(z) (Co Bo(x)dx.

o z+x

Putting m = 3 and 5 and 7 in (13) gives

(17)

B

E(Z) =12z-Io, ))adx

(z x

(18) E() _^11 6B5(x)

m

z -

12z 360z'-Io (z+x)5dx

(19) E I^11 ( 120B7(x)

()z - 12z 360za + 1260z5 - o (z + x), dx,