Last Updated：2025/12/31
Sentence
However,
it
is
possible
to
characterize
the
real
parts
of
holomorphic
functions
of
several
variables
at
least
locally
as
those
continuous
functions
such
that
their
restrictions
to
all
complex
lines,
not
just
to
complex
lines
parallel
to
the
coordinate
axes,
are
real
parts
of
holomorphic
functions.
The
complex
line
in
C
ⁿ
through
a
point
A∈
C
ⁿ
in
the
direction
of
a
vector
B∈
C
ⁿ
is
the
one-dimensional
complex
submanifold
of
C
ⁿ
described
parametrically
as
A+tB:t∈
C.
Quizzes for review
However, it is possible to characterize the real parts of holomorphic functions of several variables at least locally as those continuous functions such that their restrictions to all complex lines, not just to complex lines parallel to the coordinate axes, are real parts of holomorphic functions. The complex line in C ⁿ through a point A∈ C ⁿ in the direction of a vector B∈ C ⁿ is the one-dimensional complex submanifold of C ⁿ described parametrically as A+tB:t∈ C.
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