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.

132

Functions of a complex variable.

Bk. II. ch. II.

when - > 0, at z - 0 the value 0, at z -= the value cc;
when - < 0, at S = 0 the value oc, at z - oo the value 0.
These two points are called branching points or ramifications
of the function; it has moreover, in the first case the point z =oo,
and in the second the point =- 0, as an infinity point.
The next inquiry is, how can we group together the values
belonging to a single branch of the function, so as to have each
branch by itself in general a continuous function.
Draw from the origin out to infinity any curve that does not
cross itself; the simplest that can be chosen is one of the axes of
coordinates, ex. gr. the positive part of the axis of abscissae.
To each point of this right line belong q values of the function
p
w =- (=cos (cos -- 21;-f) + i sin a (cp + 2Ikr))
( = 0,1,... q -1).
Selecting one of these values for a point at an arbitrarily small
finite distance from the origin, for instance the value belonging to
- 0, let us attribute to all other points of the curve those values
that proceed front the assumed value by continuous change of r and 9q,
for which therefore k is likewise zero. In this manner the values
p
w = rq are chosen for the positive axis of abscisse. Now in order
to construct the values of one branch of the function for other points
of the plane, suppose concentric circles drawn round the origin with
all possible values of r and attribute to their points those values of
w that result on continuous change of (p, the circles being described
in one and the same direction, ex. gr. from the positive axis of abscisse
to the positive axis of ordinates.
Along a circle with radius r corresponds in this way to the point:
1p
p 0= 0; z = r, w ri,
P
cp =J-; z =ri, w = Ar (cos 2 + isill )
Gc=;- - r r, T V }= -T (cos q 3- + i sin- ) 2,
q 2 c2
P
p= 27; z = ri, t = rT (cos- 2 +- i sin P
wp=2r; z}2 T rq (COS P 2+-isin-P2,).
q Q,
Thus w is a continuous function all along each circle, only its
final values for (p = 2 — do not coincide with its initial values for