638 



Till': (;amma, or factorial, function'. 



Now consider the more general integral /. 



'=ih'e 



(id 



where x is not a positive integer. We shall prove (i) that it is 

 finite if .v> — i, but infinite (i.e. meaningless) when a; < — i ; and 

 that when x > —i, (2) its logarithm is a 'simple' function, and (3) 

 it obeys the factorial law. 



Since Gauss' function has been proved to be the unique 

 'simple' function satisfying the law and agreeing with i\, it will 

 follow that the integral (/^) = n(A) whenever a;> — i. 



(i) We know that log r ^ "^-^ ^; when n is positive. 



^ -^ > 7/(1- v-'") ^ 



do 



n%i - d""y 



dd ! if X is positive, 



o 1 ^ ' 



n' 



therefore if x is positive, /^ < q-, n being chosen greater 



than X. Therefore /,. is finite when x is positive. 

 Now, integrating by parts, 



.-=[<.o.i)-]; 



o •'o 



+ \ dd. x[ log 



Now if A- > o, wflog-^j = o at both limits; but if a- < o, 



H lc)g nj — > + 00 when 6 — >i. 



And when a- > o flog ^j dd has been proved finite, 



.'. /,._, is also finite, .*. I^. is finite if a> — i. 



Again, if .r 



o 



, 1^ is finite, but /^ — a7,_, — > ao 



/^_, — > (+ 00 ), i.e. /^ is + 00 if A-<^ 



— I 



— 2 



and as a- increases negatively (log i)^ becomes 00 of a higher order, 

 therefore I-^ is meaningless if a- < — i, but finite if a; > — i. 



We have also proved that when a' > — i, I^^ obeys the factorial 

 law. 



