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.

~ 87 —89.

Separation of factors.

151

these points lies upon the bounding circle. Let the vanishing points
within it be z,, 2... S, and their respective orders;l L2,. * A.
Calling the product:
(z - 1)a1 (2 - 22)'2... (Z - 2)2v = TT(z);
f(z) is divisible by TT(z) and the quotient is a series of powers that
converges for the original domain and is not zero at any point in it.
To see this, let us put h == - zI in the above development, hence:
f(2) = f(z, + z - 1) = f (Z) + 2/- ( + ('f' +
but since zI is a vanishing point of the order (nnllituce) A,, we have:
f(Z l = f'(1) = "(ZC)..=. / =f (l ) = o,
therefore:
( - f 2)) + + (j ) + etc..
This absolutely convergent series proceeding by powers of - 2j can
be rearranged by powers of S and resumes the original circle of
convergence. For, then again the circle of convergence of this series
must include that for the development by powers of 2 - iv. Since the
new resulting series vanishes in the point z2 in the order;, it is
divisible by ( - 2)22; in this way we ultimately obtain:
f ()= - (z), or: f () - T(z). ) (z),
where p (S) is a series of powers that is not zero at any point in the
domain.
The propositions hitherto proved apply in particular to the integer
rational algebraic function:
f(z) = ao + ai 2 + a,2 +. * * + anZ.
The convergency of this function, in which a,... a,, mean determinate
finite complex values, is the entire plane, i.e. to each finite value
of z belongs a determinate finite value of f(z); this function cannot
vanish for infinitely many values of z without vanishing identically;
and further if 1, 22... 4 be vanishing points, f(z) is divisible by
TT(z), the quotient being again an integer rational algebraic function;
in this case, if A; + A2 + ' ' ' -v === n i, () is constant and equal to a,,.
Therefore an integer rational algebraic function of the order n can
certainly not have m ore than n vanishing points in the entire plane.
89. Forming the logarithm of f(z) we have:
zf/() = z1(2) + lP() = AlZ(I - z) + 1i (- 2) + -.
+ i1(2 - Sv) + l9(z).
For each of the logarithms on the right we take one of its infinitely
many values, and let 2 describe the circumference of the bounding
circle from any point A, keeping the inside of the circle on the