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.

~ 191.

Inversion of the unique analytic function.

359

not zero, only one of these values of Az can become zero when Aw
converges to zero. The others converge to the values that satisfy:
f '() + f (a) + ) +.... 0,
and the roots of this equation each increased by a are the remaining
values of z that belong to the point w- = -.
This one value of A z depends continuously on A w, inasmuch as the
Az
quotient A- tends to a determinate magnitude, for we have:
A] () + ) + / ("() -+..,
therefore when AL converges to zero, the limiting value is:
dz 1
This holds for every point at which f'(o) is not zero. The function
S can be continued in a determinate unique manner.
It is otherwise when -we come to a point a for which f'(a) = 0;
we have then:
Aw A
* A^ 1 f 3 +/.
Aw=] -- ' (a) + -k- ((a) +'"
or more generally
_ -- fm' (,a) "+ A,_ /'+() + ' *,
and to the point w- 3 belongs a value 2 = a to be counted doubly
(or m-ly) as there are at the neighbouring point P + Aw two values
S= a + Alz and =- a -+ A,2 (or m values) which converge to
zero with A w. Therefore when w arrives at a point for which
f'(z) = 0, a branching point is reached and the function can no longer
be continued uniquely.
In fact the point w= (, in whose neighbourhood we have:
1f"'(' TA +,+ 1I
A v^ -- - t (a) + ^ 5 f+i() +
is a branching point for the required function =z-= (w) in which ma
of its leaves are cyclically connected; or expressed otherwise: The
point being enclosed by an arbitrarily small circle let Aw go through
the values corresponding to the points of this circle, then by a single
circuit, Az will change from whichever of the m values it may have
begun with into some other; when A w repeats the circuit, A passes
from this second value into a third; after m - 1 circuits, A2 assumes
an mt value and only when m circuits are completed does it resume
the original value. For, from the equation: