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.

~ 197-199. The point infinity being a branching point. 373
The integrations refer to the point infinity, i. e. they are to be taken
along an arbitrarily remote curve enclosing the point zero, and in the
direction that keep, the finite surface likewise on the left.
When the point z = oo is at the same time a singular point, we
obtain by the same substitution from Formula V. ~ 197 the expansion:
if(')= irm { z mF - 1 - d
+ *1 {j~ff(z)gz + ' f(z)dz 3- f(z) d
-(X,) ( ) ( (0. z
When fifinity is a non-essential singular point, an integer can be
f(UQ): _t__
is finite. The function then becomes infinite of the order n: m in
the branching point, and the second part of the above expansion
1 n
contains only the powers from tm to Um.
The value of ff(z)dz integrated round the point infinity is zero
when the coefficient of the term vanishes.
The same integral taken up to the point infinity is finite when
tf()1gI
vanishes for ' = 0; therefore also { f(z) z } = 0. In the (m - l)-branching
point = oo therefore the function must vanish in a higher order
than the first, i.e. this cannot be a singular point, and the first
m-part of the expansion must begin with the term 1:
part of the expansion must begin with the term 1:u m.
199. The investigations of the constitution of ambiguous functions
in the neighbourhood of a branching point are necessary in order
that we may attain a definitive insight into the theory of algebraic
functions.
As a culmination to these investigations we may establish a
theorem by which algebraic functions are completely characterised as a
special class among ambiguous functions.
For, in a similar manner as among unique analytic functions
rational algebraic functions admitted of being defined as those which