The so-called generalized Fourier integral is a pair of integrals--a "lower Fourier integral" and an "upper Fourier integral"--which
allow certain complex-valued functions 
to be decomposed as the
sum of integral-defined functions,
each of which resembles the usual Fourier integral
associated to 
and maintains several key properties thereof.
Let 
be a real variable, let 
be a complex
variable, and let 
be a function for which 
as 
, for which 
as 
, and for which 
has an analytic
Fourier integral where here, 
are finite real constants. Next, define
the upper and lower generalized Fourier integrals 
and 
associated to 
, respectively, by
 |
(1)
|
and
 |
(2)
|
on the complex regions 
and 
,
respectively. Then, for 
and 
,
 |
(3)
|
where the first integral summand equals 
for 
and is zero for 
while the second integral summand is zero for 
and equals 
for 
. The decomposition () is called the generalized Fourier
integral corresponding to 
.
Note that some literature defines the upper and lower integrals 
and 
with multiplicative constants different from 
, whereby the identity in () may look slightly different.
