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.

~ 92. 93.

Continuity.

157

therefore Zo must vanish, while the other finite roots each increased
by wl, express the other n - 1 values of the algebraic function
at %. Since w1 is not a multiple root at that point, Z, certainly
does not vanish for Az =- 0. Now let the modulus of Az be chosen
so small, that, for all values within the circle described with it as radius
around zo, mod ZO shall not exceed a determinate arbitrarily small
number A; the modulus of Z1 will then not fall below a certain
amount B. This requirement can be satisfied; a superior limit of A 2
is determined by the assigned value A, for, in the polynomial Z(
there is no term free from Az, whereas Z1 contains such a term.
Now if we consider the form
f(o + A A, t, 1 +A w) -ZA w jZ + 1 + g +2 A +.
Z, Z, ZI
Z, iAw(l + P)
and put mod Aw = 6, then, if B denote the smallest amount of Z1,
dP=m od[- - + Aw * * Aw VI -
Z,, Aj ZW Zt nJi _
< - + r mod Z. + * - mod Z]
therefore if C denote the greatest of the moduli of Z2... Z within
the limit assumed for Az,
modp < 0A + C( + + d-) < - K + -
1 where
We can choose 6 so as to make - smaller than where E is
Bi-6 2'
arbitrarily small. In like manner we can determine the value of A and
thereby the superior limit for Az, so that however small 6 is, we shall
have - I < —; for this we must choose A < E -. The corresponding
limit for Az we call h. Now it has to be shown, that within the
circle having the radius 6 there is one and only one root of the equation:
f(Z + A., wv, + A w) = ZAw (l + F) = 0;
for any value of Az, whose modulus < h. This we show by taking:
log {f( + +,, + Aw)}= log Z, + log Awz + log (1 + P).
When we conduct Aw along the circle round w% with radius 6, log Z1
remains a constant, log Aw increases by 2iz, since the zero point
is included, but as mod P remains smaller than the arbitrarily small
number E for all points on the circle, log (1 + P) does not change
its imaginary part by the circuit; in the circle consequently there is
one root Aw, whose modulus is smaller than 6, as was to be proved.
Accordingly it follows, that on a determinate path, leading from
zo to Z without crossing the boundary of a singular point, each value
of the algebraic function w varies uniquely and continuously.