ME. E. W. BAENES ON INTEGEAL FUNCTIONS. 
465 
If now we substitute in the general formula 
yo. 
log F (z) = - i log 2 + | log <t> {n) dn - t f ? f (n) dn 
1 8=1 
yjr (t) dt 
i - s (—)*(— + 
s=l 
t s 
we shall obtain log F (z) = — \ log z - p - log 2tt — Z ( 0 ; e _~’' ' ' 
^ \ ^1> • • • 
+ 5 
S=1 
+ f 
5=1 
2 1—2e 
1-e, 
'+I &■. U-a p — ei &, 
tp — — t p 4- H „ 1 ~ t p — —t p . . . 
b. 
dt 
1 + 
b (- o s 
And when we sum the Fourier series which result from the last integral, we find 
log F ( 2 ) = — i log 2 - -J- log 2 tt - Z ^0 ; ‘ * 
CO ( — V 1 CO ( V 1 / 
+ - F (p 6 ‘) + 2 —— Z(p, S, 
s=l SZT s=l S2" \ 0 1; On, . • . 
, . e r e 2- • • 
+ 
7T 
1 5,7r 
2P- 
2 p 
Sill 
7 r 
P -L Cl 
r Sill 7T -± 
TT^ . P + 1 ~ 2 U 
1 - * ^ 2 
l-2t, 
ZP 
sin 7 r 
1-2^ 
b.,7T 
P 
1-6 
2 P. 
Sill 7T 
1 — 6„ 
§ 59. This expansion is valid for all values of any z which lie between — it and 7r. 
It is arithmetically asymptotic in the same way as the expansion from which it is 
derived. 
We see from the results just obtained that the asymptotic approximation for 
log ri 
n = 1 
1 + - 
au 
, where a n — u p + b 1 n p “ ej + . . . exceeds that for log II 
n= 1 
1 + 
nP 
by 
a quantity whose first term is — Z (0 ; y 2 ’ ’ ’when e l > 1, and by a quantity 
\ b l! bn, ■ • ./ 
whose first term is 
b { 7T 
P 
1-6 
Z P 
, when e x < 1. 
Sill 7T 
When e x = 1, the difference of the two asymptotic approximations commences with 
z 9 
sin 7 t9 
the indeterminate form 
bpr 
P U 
b x f^ (m ) dt 
V J 7 
6=0 
, which arises from the integral 
1 4 
(-0 3 
s=—k 
VOL. OXC1X.—A. 
o O 
