An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

130

Functions of a complex variable.

Bk. II. ch. IT..

By these methods we shall be ultimately enabled to carry out the
development of implicit algebraic functions.
1. The power with a positive integer exponent: w == -m
is a one-valued function and continuous for the entire plane, because
u = rm cos mp, v rm sin mcp are continuous functions of r and g
that do not alter as the amplitude (p is increased by multiples of 2z.
Such a function having everywhere in the neighbourhood of any
definite point s a finite value that changes continuously with s, we
describe as behaving regularly in the neighbourhood of that point,
or as having that point as a regular point.
To investigate how this function behaves for infinite values of z,
let us puts =, then w = -I and to the infinite corresponds the point
0 = 0. At this point w becomes infinite, that is, its modulus increases
determinately beyond all limits in whatever way the point is approached.
The point infinity is therefore a singular point for this function. But
inasmuch as the function w - 1 is such that its product by z"' = (1)f
at the point ' = 0 has the finite value 1, this singular point is called
non-essential; the general definition being:,")
If a function f (z) become infinite at a finite point z- a or at
infinity s = o, this singular point is called non-essential, provided an
integer m can be assigned, such that the product:
((z - a)", f ()), or: (D) f(s)
is equal to a finite quantity G, in whatever manner z is maIde converge
to a, or to oo, as the case may be; or more strictly: a domain must be
assignable about the point, within which the above product shall differ
from G by less than an arbitrarily small number A. Moreover, when
the value of f(z) G for = a, or = — Gn for z = o =, we
I - a)m
say that for such a value of z the function becomes infinite in
the order m, or, that in the point its infinitude is equal to m.
It follows hence: Every rational integer function of the nti degree
in z is one-valued and continuous for the entire plane and has no singularity except one, which is non-essential, in the point infinity. For,
the function f(z) a== ao + al + a2z2 -+.'. a ln, in which an 0, is
for every finite s a sum of functions that are all continuous, and
only for z== c= becomes infinite, as fl) = ao- a 4 Z2+. 'n
does for '= 0; but for z'==0, 'nf(-) = ' f(z) ao' - a'-+a 'n-1 * a,,
is equal to an. Thus in s =- oc the infinitude of f(z) is equal to n.
*) Weierstrass: Zur Theorie der eindeutigen analytischen Functionen. Abhandl. d. Akad. d. Wissensch., Berlin 1876. Reprinted in his: Abhandlungen aus
der Functionenlehre. 1886.