787] 
FUNCTION. 
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viz. the variables are regarded as numbers; and we attend to the case of a function 
of one variable v =fu. But it has been remarked (see Equation) that it is not 
allowable to confine the attention to real numbers; a number u must in general be 
taken to be a complex number u = x + iy, x and y being real numbers, each suscept 
ible of continuous variation between the limits — oo, + oo, and i denoting V —1. In 
regard to any particular function, fu, although it may for some purposes be sufficient 
to know the value of the function for any real value whatever of u, yet to attend 
only to the real values of u is an essentially incomplete view of the question; to 
properly know the function, it is necessary to consider u under the aforesaid imaginary 
or complex form u — x + iy. 
To a given value x + iy of a there corresponds in general for v a value or values 
of the like form v = x' + iy', and we obtain a geometrical notion of the meaning of 
the functional relation v = fu by regarding x, y as rectangular coordinates of a point P 
in a plane II, and x', y as rectangular coordinates of a point P' in a plane (for 
greater convenience a different plane) IT; P, P' are thus the geometrical represent 
ations, or representative points, of the variables u = x + iy and u’ = x 4- iy' respectively ; 
and, according to a locution above referred to, the point P' might be regarded as a 
function of the point P; a given value of u = x-\-iy is thus represented by a point 
P in the plane II, and corresponding hereto we have a point or points P' in the 
plane IT, representing (if more than one, each of them) a value of the variable 
v —x + iy'. And, if we attend only to the values of u as corresponding to a series 
of positions of the representative point P, we have the notion of the “ path ” of a 
complex variable u = x + iy. 
Known Functions. 
1. The most simple kind of function is the rational and integral function. We 
have the series of powers u 3 , u 3 ,... each calculable not only for a real but also for a 
complex value of u, (x + iy)" = x 1 — iy 2 + 2ixy, (x + iy) 3 = x 3 — 3xy- -f- i (3x 3 y — y 3 ), &c., and 
thence, if a, b, ... be real or complex numbers, the general form a + bu+cu 2 + ... + ku m , 
of a rational and integral function of the order m. And taking two such functions, 
say of the orders m and n respectively, the quotient of one of these by the other 
represents the general form of a rational function of u. 
The function which next presents itself is the algebraical function, and in particular 
the algebraical function expressible by radicals. To take the most simple case, suppose 
i 
(m being a positive integer) that v m = u\ v is here the irrational function =u m . 
Obviously, if u is real and positive, there is always a real and positive value of v, 
calculable to any extent of approximation from the equation v m = u, which serves as 
y 
the definition of u m ; but it is known (see Equation) that, as well in this case as 
in the general case where u is a complex number, there are in fact m values of the 
y 
function u m ; and that for their determination we require the theory of the so-called 
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