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.

166

The implicit algebraic function.

3lk. II. ch. IV.

and from this we substitute zu2 = z in 2); thus they will be determined
by the equation z3 (z3 - 4) = 0.
At the point z == 0 three roots of this equation vanish. To decide
whether this is a branching point of the three leaves of not, we take
an arbitrarily small value of r and put z =- reeP; then
w3 - 3reiPw + r3e3i'p - 0.
Since the algebraic function is continuous, the modulus of each value of
w must decrease arbitrarily at the same time as r does; but as the
ratio - is the root of an equation:
/ \ ' 3
-17- (v-)- eiP + eOi(P = 0,
and thus can never become indeterminate, it may tend to a value
either finite or zero or oo.
Now we see by this equation that the limit of the ratio wcannot be finite, for, its middle term increases beyond any. finite
amount when r -- 0.
If we assume that the limit of - is zero, the first term vanishes
in comparison with the second and third, so that we have
Lim (- 3 w i)_ -_ e3ip, therefore: Lim wv = r2Cs21 -,w 0.
this shows that close to the origin one root of the equation differs
arbitrarily little from i-re2i' =c -i2. This root is unique in the
neighbourhood of the point zero.
Again, - can only tend to an infinitely great value if
r i
Lim (-a - - Lim - C'P + e3i(p — 0,
or, as results from dividing this by -, if
Lim (t) eiP _ 0.
\ r / -r
It follows hence that either:
i(P
Lim w- = /. ri2 e2 == w2O
or: i_,
Lim zuw /= 3. r'e t- 30.
Two roots of the equation differ arbitrarily slightly from these values,
and for these two the origin is a branching point. Let us choose its
branching section along the negative axis of ordinates.
The other branching points are determined by the equation
- 4 = 0, which gives three values:
2inz
P= j1/4, /, = -T (cos 2 + isin.) )- -,
33 = 1X4 (COS + 'i sin 4j) 4e e