INTEGRAL FUNCTIONS DEFINED BY TAYLOR’S SERIES. 
259 
The second part of the theorem is true, since when s = u + iv and \v\ is very large, 
| T (—s ) | behaves like exp I — — y \ v | j . 
L 2 J 
The first part follows from this fact in combination with the asymptotic expansion 
of T (— s) for complex values of s. 
§ 10. If Lj be a contour parallel to the imaginary axis and cutting the real axis 
between s = 0 and s = — 1, the contour, if necessary, having a loop to ensure that 
— 6 is to its left as in the figure, then 
For by the lemma we may bend the contour round until it becomes the contour 
L 0 of the figure. 
The residue of the subject of integration at s = n is 
/ _ '-y» V 1 
T. f— lC cTl -n - A = ' ’ 
e—o (n+ey K ’ (n+eyr{n+i) 
Hence by Cauchy’s theorem we have the proposition stated. 
§ 11. Let L 2 be a contour parallel to the imaginary axis (except for a loop round 
— d) which cuts the real axis in s = —X between s — — k and s — — (k+ 1), then 
G„(-z;0) = -- 1 -[ x ' r < ds 
2 ttl J l., (s + 6 y 
This follows from Cauchy’s theorem combined with the second part of the lemma. 
§12. The integral along the straight parts of the contour may be denoted by I*. 
2 l 2 
