Till'. GAMMA. OK !• ACTdKl Al., I'UNCTION. C>;^() 



We now prove that log 7^. is a ' simple ' function. 



and in this case / ^ Av( log ^ ) , 



/'^^z^l^logy .loglog^j, f'^Avmog-'^ .(loglog^j ^ 



thus, writing / for log (i/9), we see that 



f"x .fx-{f'xy={Avl')Av{l'{\og l)^] - {AvFlogl}', 



which varies as S/j^// 1 (log Ij)^ + (log l^y — 2 log /, . log I^ \ (where 

 the suffixes indicate any two values of /). 



Since this expression is necessarily positive, d^/dv" (log 7^.) is 

 positive, therefore log7_j is 'simple' ; and, since Ilx is unique by 

 Proposition II, 7^ = ILr when it has a meaning 



I d = e~' reduces 7^ to e~'z'dz 



PROPOSITION VII. 



f n.v iiv 



To prove ihat\ 6%1-eydd = „,, ,, . when (a + i) and (>' + i) arc 

 positive, and for other values of x and y is infinite. 



This integral is known as Euler's First, or Beta, Integral : 

 though it is usual in both the Gamma and Beta integrals to write 

 (a — i) and {y—i) for our x, y. 



Calling the integral [x, y), we notice that (a, v) = {y, x). 

 Integrating (a-, y) by parts, 



\ d^i-ey^'dd = ^^^\j)'-\i+dy+'^ + ^^U'^'{i-ii)\w . (i) 



•X) '1 ■■ ' •'o 



If j'+i>o and A-+i>c, the middle term vanishes; and 

 we have 



(a-,j+i) = ^^(a+i, V) 



But writing (i + 9)^'^' = (i -9)-'(i -«), 



we get (a-, v+ i) = {x, y) - (a-+ i, v)- 



