The 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 transform
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 regions of the complex plane defined by 
and 
, respectively. At a nonzero point 
where 
is continuous and of bounded
variation in a neighborhood, the symmetric
improper inversion integrals give, for 
and 
,
 |
(3)
|
where, at these continuity points, the first integral addend
equals 
for 
and is zero for 
while the second integral addend is zero for 
and equals 
for 
. The decomposition (3) 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 (3) may look slightly different.
