Statement of Parseval's Theorem
Suppose that A(x) and B(x) are two Riemann integrable, complex-valued functions on R of period 2π with (formal) Fourier series
and
respectively. Then
where i is the imaginary unit and horizontal bars indicate complex conjugation.
Parseval, who apparently had confined himself to real-valued functions, actually presented the theorem without proof, considering it to be self-evident. There are various important special cases of the theorem. First, if A = B one immediately obtains:
from which the unitarity of the Fourier series follows.
Second, one often considers only the Fourier series for real-valued functions A and B, which corresponds to the special case: real, real, and . In this case:
where denotes the real part. (In the notation of the Fourier series article, replace and by .)
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