2G8 
MR. E. W. BARNES ON THE ASYMPTOTIC EXPANSION OF 
admits the expansion 
“ (-)T( ff) 
n=o F (/3—n) n ! 
log p+L (0+(f) — TT ) 
log r + l (6— 7t) . 
Evidently the only trouble which arises is with the specification of the imaginary 
parts of the logarithms. Now when r is large and r> p> 1, the imaginary part oi 
log [“log (-V®)] = log Pog r /p- L ^ is -tan" 1 {<f>/(\ogr/p)}, the principal value of 
the inverse tangent which lies between 0 and ir/2 being taken. 
The imaginary part of log (log r+iO-n) is similarly -tan -1 {(ir-0)/log r], the 
inverse tangent again lying between 0 and 7t/2. 
.i i • • , p , Ti log p + L (0 + <j>-n) 
Also the imaginary part of logjJL-- j Q h. r+ 1 (6>-tt) ~ 
which is represented within the shaded area by the series 
when the value is taken 
S - 
n = 0 
log p+i (0 + (f) — 7T ) 
Ion r + L ( 0 —tt) . 
is also negative and lies between 0 and tt/ 2 . 
When r is large and p is just greater than unity, all three imaginary parts are very 
small. The equality (2) is therefore established. 
8 22. We see then that within that part of BD which is within the shaded area 
lo ° ( _ X 
5-1 
— n 0O - (— aW 3-1 4 x i_ YHiPl 
-Ll° b ( • )J \* 0 T(/3-n)n\ 
fogy 
log(-x)_ 
the principal values of log rj and log (—as) being taken. And on the remainder of BD 
betv/een e and oo the expansion continues to hold as one which is divergent but 
summable. [An exceptional case occurs when x is real and negative, when a slight 
modification of the contour must be taken to avoid the point ().] 
Therefore, by the fundamental theorem of § 5, we have, when 3&(0)> 0, for the 
asymptotic expansion of the second integral, 
oo p 
t (-£)-* 
71 = 0 J < 
e~ y if _1 * 
(-)” [log yf 
nZ 0 r ((3-n) r (n+l) [log (-a)] 
—« d n 
71+ 1 — /3 
= (-*)■• pog 
(_)" r (n)(g) _ 
r (n +1) r (/3- n) [log (—»)]“ 
Finally, therefore, if 2&(0)>O, we have G p (x;0) represented by the sum of the 
two asymptotic expansions 
— % (-)" r (^±^)- + (-x)~ e [log (-a:)]^ 1 i 
/ Q\ n-n ' ' L & V /J Tin — 71 A loC 1 
X^ n =0 
r (/8) x : 
T(P-n) [log (-.A)]" 
This result is valid for all values of arg x, with the assigned prescription for log (— x). 
