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FUNCTION. 
[787 
6. Each of the functions exp u, sin u, cos u, tan u, &c., as a one-valued function 
of u, is in this respect analogous to a rational function of u; and there are further 
analogies of exp u, sin u, cos u, to a rational and integral function; and of tan u, sec u, &c., 
to a rational non-integral function. 
A rational and integral function has a certain number of roots, or zeros, each of 
a given multiplicity, and is completely determined (except as to a constant factor) 
when the several roots and the multiplicity of each of them is given; i.e., if a, b, c,... 
u\p 
be the roots, p, q, r,... their multiplicities, then the form is A 1 ) (1 
u\ q 
b) 
a rational (non-integral) function has a certain number of infinities, or poles, each of 
them of a given multiplicity, viz. the infinities are the roots or zeros of the rational 
and integral function which is its denominator. 
The function exp u has no finite roots, but an infinity of roots each = — oo; this 
appears from the equation exp u = {1 + -J , where n is indefinitely large and positive. 
The function sin u has the roots sir where s is any positive or negative integer, zero 
included; or, what is the same thing, its roots are 0 and + sir, s now denoting any 
positive integer from 1 to oo; each of these is a simple root, and we in fact have 
sin u = uli (1 — U „V Similarly the roots of cos u are (s + |) 7r, s denoting any positive 
S' 7J~ 
or negative integer, zero included, or, what is the same thing, they are + (s + 7r, 
s now denoting any positive integer from 0 to oo; each root is simple, and we have 
'2/2 \ 
cos u = II (1 + t—„). Obviously tan u, as the quotient sin u 4- cos u, has both roots 
(s + |) 2 7r 
and infinities, its roots being the roots of sin u, its infinities the roots of cos u; sec u 
as the reciprocal of cos u has infinities only, these being the roots of cos u, &c. 
In the foregoing expression sin u = ull f 1 — , ), the product must be understood 
V S"7T / 
to mean the limit of 11^ (1 
S 2 7T : 
for an indefinitely large positive integer value of n, 
viz. the product is first to be formed for the values s=l, 2, 3,... up to a determinate 
number n, and then n is to be taken indefinitely large. If, separating the positive 
and the negative values of s, we consider the product uTYj 1 ^1 +^^II 1 OT ^1 — , (where 
in the first product s has all the positive integer values from 1 to n, and in the 
second product s has all the positive integer values from 1 to m), then by making 
to and n each of them indefinitely large, the function does not approximate to sin«, 
unless m : n be a ratio of equality*. And similarly as regards cos«, the product 
to and n indefinitely large, does not approximate to 
n 
+ 
n 
(s + ^) 7Tj \(s + ^) 7T 
n [ 1 
° \ 
cos u, unless m : n be a ratio of equality. 
* The value of the function in question ul\p ^1 + —'j II/' 1 ^1 - f when m, n are each indefinitely 
u 
large, but — not =1, is =( - ) ,r sintt. 
6 71 \mj