Complex Integrals
Why
We extend integrability to complex functions.
Definition
A measurable
complex-valued function on a measurable space is
one whose real and imaginary parts are both
measurable.
An integrable
complex-valued function on a measurable space is
one whose real and imaginary parts are both
integrable.
The integral of a
integrable complex-valued function on a measurable
space is the complex number whose real part is
the integral of the real part of the function
and whose imaginary part is the integral
imaginary part of the function.
Notation
Let
be a measure space.
Let denote the set
of complex numbers.
Let
be a function.
is measurable
if and
are measurable.
is integrable if
and
is integrable.
If is integrable,
we denote its integral
by .
We have defined it by:
Results
A linear combination
of two integrable complex-valued
functions is an integrable
complex-valued function.
The integral is a linear
operator on the vector space
of integrable complex-valued
functions.
The absolute value of the integral of a
complex-valued function is smaller than the
integral of the absolute value of the function.
Let be a measure
space.
Let integrable.
There exists with
such that
Since the integral is homogenous,
Since is real, , so
since for all complex
numbers and .