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)
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and
(2)
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on the complex regions and , respectively. Then, for and ,
(3)
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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.