634 THE CAM MA, OR FACTORIAL. Fl^NCTION. 



Expand the last fraction in powers of 6 and carry out the 

 integration (which is legitimate when x > o), and then we get 



^n.r ,,.,^ . ^ V^ I (a- 1 , I I 



log ^^^ - (ditto m ^^^+^)=2_j{^+Yr'\j+^ + T+2\ 



o 



2 1 ai+{2a — l) 



{.v+iy+' ' (/ + i)(/4-2)" 



o 



Hence, taking a = J, 



e'nx ,,. ^ . V^ I I- 1 



log ^^i - (ditto in .r+i) = Z^V^i] 



[)' 2i{i-\-i) 



2 



[This choice of a makes log ^'T[.r/A-^+" converge most rapidly 

 to a definite value when .v is big.] 



Writing down corresponding equations for values of -r 

 increasing by unity, and adding them up, we get 



..v + .Vt 



log _^-+i - log(^^.^;^).r+A+^ ~2_j /_j ' {x^ny 2i{i^\) 



;i = I 1 = 2 



and the second term on the left converges as A^ increases to 



(.v+iV)+log (.V-^nAO _(.r + Ar+ 1) log (.r+A^ 

 wh ich -^ X + A^ + .V log A^ + log nA^ - (.v + A^ + \) (log A^ + xj'N) 



and this is independent of .r, when A' — ^ cc . 

 Let L be the limit of e'^'nA^/A^^^^*. 



ii^en u — 1.1 ^A.^-i — Li (2A^)'^'^'+^ 



and so 



,_r/Mll i^^y'^' _ J. 2.^...2N /(2\ 



^ " n(2A0 ■ A'^^+' "1.3... 2N-1 ■ V \A7 



^^ . . Xir Xtrf X^\[ X- 



Now, since sin — = — (i — — l(i — — 



2 2 y ~ J \ 4 



we have Wallis's formula for tt, 



TT ^ ^^ 2- . 4- . . . (2A^)- 



2 1.3=^.5^ ... (2A^-l)='. (2A^+ l) 



Hence J(») = Li ,^3^;;,^, . ^ (,VVt) = '" 



