DOUBLE GAMMA FUNCTION. 
280 
or, utilising § 19 Corollary, provided 
+ (n + 1) Wo I Wo] + [z + (n + 1) wj Wo] 
— [5; + (?i + 1 ) • (<^i + I <^3] ~ ^ • 
j«i = 0 III., = 0 _ 
But (“Gamma Function,” Part IV.) we know that when Izj is very large, and 
2- not real and negative, 
log Ti (2; + « I w) = (- i) jlog f + log i log 
CO / [ 0 ) J W CO 
+ terms which vanish when |2:| becomes infinite. 
In every case the principal value of the logarithm is to be taken, i.e., that value 
whose amplitude lies between —n and tt. 
Now log — + log w = log 2: 
in all cases except when 2 lies in the region formed by lines from the origin to the 
points —CO and —1 (shaded in the figure). 
/ \ I -1 T 
Wiyofiwi, Wo) “T 2-— 
Wo )i = X 
(Cl) 
When 2 does lie Avithin this region, we readily see that 
log + log w = log : + 2 
TTL 
CO 
if I (w), the imaginary part of w, is positive, and 
log- + log w = log 2 — 
ITTi 
if I (w), the imaginary part of w, is negative. 
Thus 
log fo(2 -f a \ w) = ( i ) ■' log 2 + 2/t77C ~ + 2 log 
(O 
CO 
CO 
+ terms Avhich vanish when 121 is infinite. 
where k = 0, unless 2 lie within the region between the axes to — 1 and 
VOL. cxcvi.—A. 2 r 
w. 
