ON THE GAMMA, OR FACTORIAL, 



FUNCTION. 



By Prof. W. N. Roseveare, M.A. 



Summary of Paper. 



11 ! 

 The Binomial Coefficient which can be written t- — ■, . . , when 



{n — i) I I I 



11 is a positive integer, cannot be reduced to this simple form when 



n is more general, unless we can devise a meaning for n ! for other 



values of n. The continuous function so defined is known variously 



as the Gamma Function or Gauss' Function. I prefer to call it the 



Factorial Function and to represent it in Gauss' notation by Un. 



The general binomial coefficient I denote by («),, extending the 



symbol (;/)„ to represent the generalized =77 . ^ . where ;/, a 



■' ^ ^ ^ '=' n(« — flj.na' ' 



have any values. 



In this paper I have attempted, first, to give a series of 



connected proofs of the main theorems. 



Proposition I. — On the existence of the function. 



(Excursus on ' a fair curve ' and * simple ' 

 functions.) 



,, II. — That Gauss' function is the unique ' simple ' 

 function which generalizes / !. 



,, III. — A complete analytical expression for e'Ux/x^'^K 



,, IV. — Connection between Ux and n( — :x;). 



„ v.-n- n.v.n(.r-^)n(.r-?y . . n(.v-^yn(;u>). 



,, VI. — The Factorial Function as Euler's Second 

 Integral J AogM dx. 



„ VII.— f r( I - ByiW= ^ "f "I' , when finite (two 

 I ^ ^ U{x-\-y+i) ^ 



proofs). 



,, VIII. — Expansion of log Ux and n.r in power series. 



;. ,, IX. — Fundamental proof that IT.v can be expanded 



in powers of .v when | .v ] < i. 



