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.

168

The implicit algebraic function.

Bk. II. ch. lV.

Lastly we obtain upon the radius OF:
2iin 2i7t 5ii 2i
w10 ( r=2e r, 3~ 1/'3 re 3, OI/rGe e3 - 1/3r e3,
and in the same manner it is established, that /33 is a branching point
for w1 and w2.
Accordingly the system of branching sections is established. A
circuit of all the branching points leads each value back into the
initial value; consequently the point infinity also is only a non-essential
singularity.
99. We are not going to enter into the methods of simplifying
the system of branching sections - such investigations are important for
the theory of algebraic integrals and their periods - but we must raise
the question: By what general method can the different values of w,
which continuously follow each other as S varies continuously, be
calculated? For, the previous investigations have only demonstrated
that this problem is determinate, and in our simple examples, rising
no higher than the 2nd and the 3rd degree in w, only some methods
of treating the values at the branching points have found application.
The general problem of calculating the algebraic function therefore
still remains to be solved (see Book IV), and will find its accomplishment
by means of Taylor's series for complex functions.
We shall have occasion at the same time also for showing how
to determine the higher derivates of an algebraic function.