MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
455 
f &' ( -- OTIS'S 1 
or log F (z) = — log 2irz h + 2m <{ 1 -j- ^ ---— > • 
& W & 1 ^ ,=-* (2s + 1)*J 
Suppose now that \z\ is large, and that 
m 2 = ze~ ie where 6 = arg z. 
[This assumes that \z\ is a large integer, a restriction which, as will he seen later, 
can easily be removed.] 
Then, when \z\ is large, we have asymptotically 
log F (z) = — log 2772* + z h e 2 
>-■ (_y 
2 + 2 ' 
»=-k S + 2 
The sum of the Fourier’s series inside the square bracket is, when — tt < 6 < 77, 
equal to ire* . 
Therefore, when \z\ is large, we have asymptotically 
log F ( 2 ) = — log 27 TZ h + 772". 
§ 51. In the preceding investigation we have assumed that the Maclaurin sum 
formula expresses asymptotically the values, when m is large, of the functions 
m— 1 
log (m !) and 2 n 2s (s positive or negative). 
n = 1 
Accurately we have of course 
If 70 d 
log-m — 1 t — (m — I) log m — lo g — -—— , 
& \ 2 / & I V 1 J q e^y — 1 
m— l 
2 n* 3 = 
«=1 
m 
2j+1 
irt“ 
2s + 1 2 t Jo e 2 *y — 1 
dy 
log (/u -f- /.//) — log (m — iy) 
(m + L y)~ ~ ( m — li/) 2s , when s is positive ; 
and 
S _L _ 1 1 _ 1 r dy 
„ = m n 2s (2-s — 1) m 2 * -1 2m 2j " 1 J 0 e 2 ^ — 
positive: 
Hence, in the limit when k is infinite, 
(m + iy) 2,9 — (m — vy)~ 
■2s 
when s is 
log F ( 2 ) = — log ( 2772 ") + 2 m ] 1 + 2 
i, (-y^ 
s= - k .z 3 (2s + 1) 
(-) s-1 _ ( m + wT 
S Z* 
+ tL^^vi{- 21o «G + ‘'/)+A 
- 7 l j - 2 lo S - ,?/) +-77-} 
This formula is accurate and holds whatever positive integral value m may have. 
