THE GAMMA, OR FACTORIAL, FUNCTION. 



645 



PROPOSITION X. 



IT// 



The ex I ended Binomial Coefficient («)„ = 777 777- is the contour 



•■' ^ ' n(» — a)IIa 



integral — \{i+z)"z-"-'^dz round the unit circle^ ivhen «>— i 



and a has any value. 



r iiz 



Consider .i'*( I +5:)/^-" over the closed path OttOtt'O (unit 



circle). The whole integral being zero, the integral over the 

 circle tt'ott , 



J'^dx , 

 — ] {xe"'y{i—xy — {xe-"'y{i —xY\ 



which 



= 2i sin (ITT .r"'~'(i —xy^lx = 



Jo 



2i sni air . 



lI(«-l)a/?_ 



n(a+/3y 



o > o and /5 > — I, 



and, by Proposition IV, Ii(-«) . 11« =a7r/sin ott, 



r 



11/5 



:/. . .«-(! +.). = ... •mi^+am^) = '^' • ^^^^-'- 



therefore when /5 >- I and a > o, (/3)_ . = -7-U/^.^"-'(i+2)'«round 



the unit circle ; call the right side Av z-'-'{i-\-zy ;i or (a— i, /j). 

 We proceed to prove that the restriction o > o may be removed. 



