文档介绍：CHAPTER 1
REAL PLEX MANIFOLDS
We shall begin by defining holomorphic functions and the Cauchy-Riemann equa-
tions . In Sections - of this chapter we will review the definitions and
various properties of a smooth real plex manifold. In Section , the Cauchy-
plex is introduced plex manifolds. Section is devoted to
the Frobenius theorem. In the last section, in contrast to the Riemann mapping
theorem in plex variable, we prove the inequivalence between the ball and
the polydisc in several variables.
Holomorphic Functions plex Euclidean Spaces
= C × · · · × C denote the n-plex Euclidean space with
n
product topology. The coordinates of C will be denoted by z = (z1, · · · , zn) with
n 2n
zj = xj + iyj, 1 ≤ j ≤ n. Thus, C can be identified with R in a natural manner,
z 7→(x1, y1, · · · , xn, yn).
Definition . plex-valued C1 function f(z) defined on an open subset D
is called holomorphic, denoted by f ∈ O(D), if f(z) is holomorphic in each
variable zj when the other variables are fixed. In other words, f(z) satisfies
∂f
() = 0,
∂zj
for each j = 1, · · · , n, where
∂ 1 ∂∂
() = + i
∂zj 2 ∂xj ∂yj
is the so-called Cauchy-Riemann operator.
The objective of this book is to study the behavior of holomorphic functions. It
is closely related to the solvability and regularity of the inhomogeneous Cauchy-
Riemann equations
∂u
() = fj, for j = 1, · · · , n,
∂zj
where fj’s are given functions.
1
2 Real plex Manifolds
Some of the properties of holomorphic functions, like power series expansion, do
extend from one variable to several variables. They differ, however, in many impor-
tant aspects. It is therefore, not correct to consider the theory of plex
variables as a straightforward generalization of that of plex variable. For
example, in one variable the zero set of a holomorphic function is a discrete set. The
zero set of a holomorphic function , n ≥ 2, has a real 2n − 2 dimen