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.

136

Functions of a complex variable.

Bk. IF. ch. II.

Another property distinguishing the essential singular point from
the non-essential, is, that with the latter an integer m can be assigned
for which Lim (a - a)lmf(z) = G. With the former this is not possible.
For, putting here: e- 1 =l+ - 1 + L- +t- + *, we have:
mpez _= n + I m-1 I+ I m- 2 + 1+ 1 +..
and however great we may choose m, we cannot give it any finite
value such that for - 0 the right side shall remain finite.
A second property is illustrated by the exponential function: it
is a periodic function; the period is 2ir.
ez+2i = ex { cos (y -- 2 z) +- isin (y+ 2 z) } = ez.
If we divide the plane into infinite strips by right lines parallel to
the axis of abscissa at distances 2z, the function reproduces itself
symmetrically in each of these strips.
5. The logarithm u - +iv of the number x - iy = "rp in
regard to the base e is by the definition (~ 74) an infinitely manyvalued function. But as long as the simplest value
u + iv = Z(x + iy) = 1(r) + i9p
is considered, it is a one-valued function. Only the points r = 0 and
r = oe are branching points; at these the real constituent of the
function increases beyond any limit, and the imaginary is completely
indeterminate. Conceiving therefore a branching section laid from
the zero point to the infinity point, as in the example of the irrational
function, one branch of the infinitely many-valued function is continuous in this perforated plane.
It is important for a subsequent application to interpret further
in the following manner
y -Zt\ ' the significance of the
branching section for the
different values of the
logarithm. If we make the
d~/ Hi~ \ ~ variable z describe a finite
I _/ / X _______ __-___r closed curve, beginning at
a point A and returning to
it, which curve neither
Fig. 6. crosses itself nor includes
the origin, and so meets
the positive part of the axis of abscissae either in an even number of points
or not at all, then as r and 9p vary continuously, the value of (z)
on the return to the point A is just the same as at first; for, z has