644 



THE GAMMA, OR FACTORIAL. FUNCTION. 



Now (/''+' ~''"d ') increases or diminishes, as / increases, according 



/ i\ I > , I . 



as {s-\-i — -jj^i, .-. L = X + I — gives a maxnnum value, 



therefore the integral is numerically less than ( - )"a" + 'e~ ''Z,^- . IT/;, 



Also it will be found that a''^'e~^L^ increases as a increases : 

 therefore a = i/(.r + i) gives the minimum value. 



Hence this first part of the integral lies between 



{-)"{x+ I)-"-' . nn 



I 



r — 1 



/•CO 



For the second part iU . /^e^'(log /)", let / = X~^ where /> 

 is positive. 



/"I / -r \ II 



; and, as before, 



We get /3" + " . c/\ log^ . i /«+'+'//3e-' 



/.r + i +//3^-.' has a maximum when / = .v + i + i//3 = L. 



Moreover ft"'^'L^e~^-^ as /3 increases, has a rate of increase 

 varying with (?; + i//3— i//3* log I), which is positive if ;/ is big 

 enough ; and, choosing ft to be small, ft"^'U-e~'^ is small : therefore 

 the second part of the integral bears a negligibly small ratio to II/z, 

 when n is big. 



•'n 



Now, the remainder after x-'"' ' in the expansion of T\{a-\-x) is 



T\^'^\a -\- X — z) . -r^, which by the above results 



(_)A- ^a+x-z+i) 



-JV-jJ^.V 



h' 





<-'l(^ 



a-\-x — z-\-i\~^~' ^dz 



< . , I "^'^ —. where ^ = ^ - - - 1 



therefore remainder 



a-\-x-\-i 



This gives the required result for the expansion of lix^ when 

 a = o, with the additional result that, when a-\-i > o, Tl{a-\-x) can 

 be explained in powers of x when, and only when, \x\ < i. 



In all cases the range of {a + x) must not include — i, —2, . . . , 

 where discontinuity occurs. 



