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.

~ 193. 194.

The branching point of the order m - 1.

365

continuously into another root differing from the original in amplitude
by 2z: m. The first return of g to its initial value occurs when
2 has completed the entire m circuits. Interpreting the values of
g in a plane, g first completes the circle round the point C = 0
when 2 has completed its circuit on all the mn leaves of the winding
surface. To the points z of a single leaf within a circle of radius r,
1
correspond only the points C within a circular sector of radius r"' and
1
central angle 2 i: n. Having fixed which value of the radical (z - a)2m
shall be chosen initially, we have established a definite relation between
the consecutive circular sectors and the various leaves.
The values which the function f(s) assumes in the various leaves
can be coordinated uniquely and continuously to the points of the m
circular sectors, so that we can say, the function
f(A) = f(fm + -) = s ()
is a unique and continuous function for the interior of the circle
round the point =- 0. But it is also an analytic function. For, the
function f(S) must have a determinate finite derivate at every point,
except the branching point; therefore we have:
f^)~~ <'~* -- -1
ft(Z) 9,(:) d:,(g) W O ( -
i.e. the quantity:
m-1
g = ( -nf'(z). (a - a)
is everywhere determinate and finite. In the point a = o itself, f(z)
and therefore also p () must remain finite and continuous; hence this
cannot be a singular point for the otherwise unique analytic function
(c); consequently (~ 189) '( ) also has a determinate value. When
this value is finite and different from zero, the equation:
f (( -- ) 12
- -^
shows that the derived function f'() becomes infinite of the order
m —1 in the branching point. When p'(I) vanishes, f'(2) may be
finite or even zero, as shall be more strictly determined hereafter.
To the integral ff())dS formed for the complete circular circuit
in all the m leaves, beginning in the first leaf with one of the in
possible values of f(S) and changing this continuously with,
corresponds thus the integral
m p (P) l dt