Colloquium publications.

LECTURE IV

IMPLICIT FUNCTIONS
~ 1. WEIERSTRASS'S THEOREM OF FACTORIZATION
The following theorem is due to Weierstrass.*
Theorem of Factorization. Let F(u; xi,.., xn) be a function
of the n + 1 variables u, x1, *.*, xn, analytic in the origin
(0; 0, ~ ~, 0) and vanishing there. Let
(1) F(u; 0,..., 0) 0.
Then, throughout a certain neighborhood of the origin,
T: |ul < h, Xk < 7', k = 1,..., n,
the following equation holds:
(2) F(u; 21, * *, Xn) =[Um+ A1Um-l +.-+ Am ]9(u; xl..,Xn),
where Ai is analytic in x1, * -, x, throughout the region Ikl < h'
and vanishes at the origin, and Q is analytic in u, x1,.,,n
throughout T and does not vanish there.
If f(zo, z1, * *, zn) is any function of zo, z, * *, n,, analytic at
the origin and vanishing there, but not vanishing identically,
it is possible by means of a suitable linear transformation of the
n + 1 variables zo, z *, * *, Zn to carry f over into a function
F(u; x1, * *, Xn) satisfying the foregoing conditions.
Irreducible Factors. On the theorem of factorization can be
based a theory of irreducible factors of an analytic function
analogous to the theory in the case of polynomials.t First, as
regards division. If F(zi,. *, zn) and Q(zl, **,,n) are both
analytic in the point (a) = (ai,., a,) and 4 does not vanish
Lithographed, Berlin, 1879; Funktionenlehre, 1886, p. 105 = Werke 2,
p, 135. In a foot note of the page last cited Weierstrass says that he has
repeatedly given the theorem in his university lectures, beginning with 1860.
t Weierstrass, 1. c.
181