181)3.] On Operators in Physical Mathematics. 521 



and this is also expressed by the definite integral 



fn^ * ( 49 > 



J o I 



But when n is less than 1 we have the oscillatory curve of tV 

 inverse factorial, given by (44). We cannot use the definite integral 

 to express \n when n is less than 1. The one-sided reckoning of 

 the gamma function expressed in \n = T(n + l) is so exceedingly 

 inconvenient in generalized differentiation that the factorial function 

 had better be used constantly. For completeness and reference, we 

 may add the general formula. Take the logarithm of (42) and 

 arrange the terms suitably, and we obtain 



.., (50) 



j O 



where SM > = 1+^+i + ...., (51) 



and C = Si log r = 0-5772; (52) 



it being understood in (42) and (52) that r is made infinite. 



From (50) we may obtain a series for the inverse factorial in rising 

 powers of n. Thus, 



+ (C 4 -6C 2 S 2 +8CS 3 +3S 2 2 -6S 4 )+ ..... (53) 



As before remarked, only the value of \n from n = to J needs to 

 be calculated. Any number of special formulae for \n may be obtained 

 from algebraical expansions involving this function. 



A Suggested Cosine Split. 



19. The above split of the function (sin^7r)/n7r into 1/jw and 

 l/l"~ w > suggests other similar splits. In passing, one may be briefly 

 noticed, the cosine split. Thus, take 



and let/( n) be the same with the sign of n changed. Then, when. 

 r =. oo, we shall have 



/(n)/(-n)=coBw*. (55) 



