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LECTURE II
SOME GENERAL THEOREMS
~ 1. DEFINITIONS AND ELEMENTARY THEOREMS
Let
F(2i, **' zn)
be a complex function of the n complex variables
Zk = Xk + iyk, k = 1, 2, * *, n,
which is defined uniquely at each point of a 2n-dimensional
continuum T. Of the two current definitions mentioned in I,
~ 1, we will choose the second and say: F is analytic in T if, at
every point of T, it admits a derivative with respect to each of
the complex arguments z1, **, Zn and if, furthermore, it is
continuous in T. The latter condition turns out to be a consequence of the former, cf. ~ 5, and may, therefore, be stricken
from the definition. But it is better to retain it for a time, since
it suffices for a simple proof of the integral theorems, and with
the aid of these all the principal theorems are readily established.
The function F(zi, ** *, Zn) is said to be analytic in a point
(a1, * *, an) if it is analytic throughout some region T containing
the point in its interior. Similarly, F is said to be analytic in a
manifold M of one or more dimensions if it is analytic throughout
a region T containing M in its interior. If M is closed, i. e.,
if M contains its boundary points, then, for F to be analytic in
M, it is clearly sufficient that F be analytic in every point of M.
The Cauchiy-Riemann Differential Equations. The differential
equations which the real part u (or the coefficient v of the pure
imaginary part) of an analytic function satisfies are the following:
a2u yu _ u a2u
+ - O. - 0.
XklXIO ayay=Y aXkaYl aykaiD
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