230 
A representation of the exponential function, etc. 
Proceeding from logarithms to exponentials, 
e z = IT (1 4- z n ) ku 
1 
= (!+*) (1 +z*f{ 1 +^ 3 ) __ 3(1 
and if we make use of the formula 
1 = (1 -y) (1 + y) (1 + 2 / 2 )...(l + 2 / 2 ")... 
this can be transformed into 
e z 
(1 + z' 
U —z. 
1 
2 
n 
(-i T 
1 + z p \ 2P } 
.1 ~z p ) 
( 3 ) 
(4) 
where P = p 1 p 2 ... p^, a product of fi different odd primes, and we 
are to take all such values of P. Thus, writing out the first few 
factors in the order of magnitude of P, 
g __ /I 4- /I 4- z 3 \ /I 4- z 5 \ “To /I 4 - 2 7 \ ~ iV /I + z n \~^ 
6 u-w U - Z 5 ) [T^Z V Vl 3 ^ 1 / 
w /I 4- Z 13 y 2 v /I 4 2 15 \3<j /I 4- - 34 /I + ^9\ - Jff 
x u-^v vi-^ i5 y u-^ 17 / vl-^ 9 J 
/I +£ 21 \4 J 2 /I +^ 23 \-?V 
x (i - z 21 ) (r^J 
The formulas (3) and (4) give good illustrations of function- 
theory. They are only valid for \ z\ <1, and the behaviour of the 
products on the right hand when \z\^l requires special 
examination. If 
n 
0n(«) = n(l + X n ) k ", 
1 
we have (f) n (# -1 ) = x~ x cf> n ( x ) 
U 
where X - Xnk n . 
i 
Now X is always an integer, when it is not zero ; it is found by 
calculation that X = 0 for n = 7, 12, so that 
(f) 7 (x x ) = (f) 7 (x), (fiw (j& 0 = ^12 if)- 
There are no other values of n below 101 for which X = 0 : the 
values of X, for successive values of n, fluctuate in a curious manner 
which might repay examination, and there may possibly be an 
unlimited number of values of n for which X = 0. Apparently, 
<f) n (1) converges to the value e. 
