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.

152

The vanishing values of a series of powers.

Bk. 1I. (ch. IlI.

left. Since the argument z - z, vanishes only once within the circle,
namely for = — -s, the value of the logarithm when s returns to
the point A will differ by 2 iz from its initial value, ~ 82, 5; the
same will happen with l(Z - Z2),.. l(Z - s); on the other hand
l1g(z) on its return to the point A will resume the value it had at
first, because ~ (s) does not vanish at any point within the circle, but
has for every point a determinate finite continuously changing value,
so that no branching point of its logarithm is included. Accordingly
we see that: If there be v vanishing points of the finction tf() within
the circle of convergence, the value of If(z) changes by 2inv when the
argument z describes the entire circumference; and conversely: If the
logarithm of the function change by 2 iz v twhen its argument z describes
the circle of convergence, v is the number of vanishing points of f(z) within
this circle.
90. Applying this Theorem to the algebraic function, we can
take the circle) for whose points the values of z are to be formed, with
a radius so large that the amount of the term ansn shall far exceed
all the rest, and accordingly the amount of:
aO + a z + aCZ2 +.*. an _ n1
an n
n
be smaller than an arbitrarily small quantity 6'; to attain this, we have
only to take mod s greater than unity and then to determine that
Ao q- A-..i+. A_ l 1
mod Cs shall be also > +1 — A1,
each A denoting the modulus of the corresponding a; then let us put:
( o+a + a+2z +..a - ---
ans
writing E for the complex quantity, whose modulus is smaller than 6.
Now 1(1 +- ), formed from a determinate point upon the bounding
circle, differs everywhere inappreciably from 1(1) = - 2kiz; thus if
we begin with any value of the logarithm, ex. gr. the simplest, since
when E changes its value, the corresponding logarithm must vary
continuously, it will always differ only inappreciably from the simplest
value of 1(1) namely zero; therefore when z has returned to the
original point, the value of 1(1 + e) will not have increased by a
multiple of 2iz.
1
But while z describes the circle, 1 (an,)) =nl(ann s) changes
by 2iz. n, since the point = 0 is included within this circle.
Accordingly lf(z) undergoes the change 2izn, i. e. in the arbitrarily
great circle of convergence there are always n values, for which the
rational function
aO + a.1 + a2- ' + * ansn