312 ML O. N. WATSON: A THEORY OF ASYMPTOTIC SEKIES. 



When x is real and positive it is known that* 





"11 * * real and positive, e-li(e-) has an asymptotic expansion of which 



characteristics are 



A; = 1, p - i> 



1=1, <r=l. 



Suppose that X is complex, but not a real negative quantity. Then we may prove 

 that 



"H*- 



Jo X + V 



so 



that 



where 



If R (x) > 0, on the path of integration | x + v \ > | x \ . 

 If R(a-)<0, then |z+||I(x)|. 

 Thus, if R (*)>(), 



I R B I < | x \ " n " 1 f "e"' a!v | a; | " B " 1 . n! . 



Jo 



Whereas, if 



i7r<iargxi<^7r + a (a < ^TT), 



we have 



1 1 (.r) | > | x \ cos a, 



nn that 



|R|Sseca.|x|~' "'. n!. 



Thus, if iargxjsir+, the function e x li(e~ I ) possesses an asymptotic expansion 

 of which characteristics and constants are 



k= 1, p = 1, A= 1, 



1=1, <r = 1, B = 1 or sec a, 



the value unity being taken for B if | arg x \ S ^ir. 



* WHITTAKER, ' Modern Analysis,' 87. 



