Till-: (;.\MAI.\. OK FA( 'I'MRIAI,, I'^l ' NcTION . 63; 



Now in the limit 



n(Nn + nx) = (Nx)'- . n(.V//) = {Nn)'--^ . 7(27r) . e"^'" (Nny"-^^ 



and in general, n(A^ + r) = A^^ . UN = A^-^ . J{27r) . e'" . iV-''+* 



llnx 



Ux.mx-^]...n X-" ^^ 



which = V(27r)-« . j^ „.->.;.+i,.v,. + n/. • -1j^'A^= { 7(27r)} "- Q-^"^' 



C.™«.n..-,f.=o..n(-i).n(-f)..,n(-'I^) = y((^) 



PROPOSITION VI. 



The Factorial .Function as ^Ruler's Second IntegraV or the ''Gamma 



Function.^ 



We have approached the Factorial Function from Gauss' 

 point of view, which has the advantage of a definition holding 

 for all values of the variable. Euler studied it as a definite 

 integral in a somewhat different form from that which we shall 

 choose. 



We know that 



6-^ - 1 +-V log-^ + f-; (iog-;)v . . . + ^' (log ij+... . 



for all positive values of 8 and for all values of x. 

 .-. |V'^0^sfj£(log^)'./e, 



but «-\/0 = — -- (for all values of .v less than i) = Xv'. 



Hence i log -p. jd6 = i\ 



