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.

164

The implicit algebraic function.

Bk. I1. ch. IV.

When on the other hand the circuit of all finite branching points
in ac deterninate direction brings on a cycle: w, into w2, w2 into
W3,... Wp_ into tvp, lastly Wp into w,, the point infinity is a
branching point for the p leaves. The circuit of the branching point
in the prescribed direction, which becomes reversed in transforming by
Inversion, brings on the same cycle.
98. We shall work out the general theory in the following:
Examples.
1) w2 - ( - P1)(z - ) = 0
Let the quantities b, i1, f2 have any arbitrary complex values; and
let p1 be different from 0j2. By the substitution, it appears that
there are no finite non-essential singular points. The critical points
can easily be determined by means of the explicit form:
MV { =(D-( 1) (\- - 02) -
For, at points at which the two values of a square root are equal,
the function under the root must vanish, therefore the critical points
are: z == /, z = /.
In going round such a point the values are interchanged. For
if we consider a neighbouring point z = — +- reiS, where r is arbitrarily
small, the amplitudes of the corresponding pair of values
w = { breiT (i, - P2 + reiP)},
differ by z. If we choose one value and make p go through all values
from zero to 2r, e22i+i' occurs instead of eiT. In the last factor the
amplitude will not increase by 2r, since it always differs arbitrarily
little from the amplitude of the constant number 3 - /2. Accordingly
the root undergoes the change by the factor ei, i. e. the two values
interchange. The same is true for the point P2. Therefore they are
both branching points. The two leaves are connected along sections
which start from them.
If we call leaf 1 that which has at the origin O the value:
w~ = ((bj f) (2B, B, B2 (cos " F +- 2 + sin "+ )
Io 0 O', 2) 2 2
b = B (cos a + i sin a), / == B1 (cos V1 + i sin '1),
P2 - B2 (cos '2 + i sin ~2),
and leaf 2 that for which:
(T B a +) (V _ 2_ + _+zsin + VI + _2 + 2+
Cos -- [ i sin
w20 = — (BB, B2)1 (cos 2 2 + isiu 2
we can decide what value of w belongs to each point in each leaf,
as soon as we have settled about the branching section. If this be
drawn from ji parallel to the positive axis of abscissae, we have to
determine what values belong to leaf 1 along the right line 0/,.
'~^'" "'Y VV~LVUIX -— VYn