Colloquium publications.

112

THE MADISON COLLOQUIUM.

The Factorial Function and Analytic Continuation. One of
the problems with which mathematicians had occupied themselves without obtaining satisfactory results was that of extending
the definition of the function n! to a continuous range of values
for the argument. This question Weierstrass* took up, examining the work of his predecessors and showing that a satisfactory
solution could be reached on the basis of the principle of analytic
continuation, the functions considered being dependent on
several variables. Thus these functions contributed at that
early time to the recognition of the importance of the conception
of the monogenic analytic configuration.
Existence Theorems. Cauchy had established the first existence
theorems for ordinary differential equations and implicit functions.t In his further study of these problems he developed the
method of power series and series majorantes.t
The extension to the case of partial differential equations was
direct, and the results thus obtained were of importance. For,
while much of the theory of these equations appeared plausible
from geometric considerations of a somewhat crude sort or from
analogy with special examples yielding an explicit solution, a
secure foundation had hitherto been lacking.
Weierstrass's Theorem of Factorization. If a mathematical
theory is to gain its independence and take its place among the
powers, it must recognize its own peculiar problems and obtain
methods for dealing with them. One of the earliest distinctive
theorems which became known in the theory of functions of
several complex variables is the theorem of factorization, due to
Weierstrass.~
cf. below, Lecture II, ~ 5. Such citations will be made in the following pages as
IIn, ~ 5.
In order not to interrupt the course of the general account with which we
are now engaged, the consideration of a number of detailed consequences
which follow from the definition will be postponed to a later paragraph; cf. II,
~~ 1, 2.
* Journ. fur Math., 51 (1856), p. 1; Werke, 1, p. 153.
t Cf. Enzyklopadie der math. Wiss., II B 1, p. 103, and ibid. II A 4a, p. 201.
t Turin memoir, 1831; Exercices d'analyse, 1 (1840), p. 327.
~ Cf. IV, ~ 1. The theorem dates from 1860.