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.

Third Chapter.
The vanishing values of a series of powers, specially those of the
integer rational algebraic function.
87. The Theorem last proved in the previous Section for series
of powers assists us in enquiring: how many points are there within
a circle of convergence, for which the function f(z) vanishes?
The function vanishes in a point oz, when, taking this point as
centre, it gives rise to an expansion in which the first term f(so) is
zero. When other succeeding terms also vanish, so that the expansion
begins with the term L: f(z), the point is called a vanishing or
zero point (nullity) of the order (nullitude) n; it must be counted as
f(zo 0+h)
n vanishing points; the quotient 7+ then remains finite at the
point h = 0. We have first to prove:
The function f(s) cannot be zero at infinitely many points within
a finite circle of convergence, unless it be zero identically, i. e. everywhere in the circle, so that all the coefficients of the series vanish.
In fact, if there be infinitely many vanishing points, there is'also
a region of a r b it r a r i y small extent which contains infinitely many
of them. For if the entire domain be divided into an arbitrarily great
finite number of parts, there must still be in at least one of these
parts infinitely many vanishing points. Let s be a point in such a
region; the expression
1) f(s + hi)= f(s) + 1 f'(Z) + Lf () + ' '
must become zero for a value h, whose modulus is arbitrarily small.
Since the coefficients of h, h2... are finite, the amount of the terms
multiplied by h is arbitrarily small; accordingly if the expression
is to vanish, the amount of f(z) must also be smaller than any finite
quantity, i. e. since f(s) is a determinate value, we must have:
f() = 0.
Now considering the product:
hf'(s)+1f(s)+| )h2
there is an arbitrarily small bt nite value for which it is zer
there is an arbitrarily small but finite value 7b for which it is zero,