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Parseval’s Theorem

Tags: complex, complex conjugation, Fourier series, function, integrable, Math, Parseval's Theorem, Plancherel theorem, Rayleigh's energy theorem, Rayleigh's Identity, Riemann, unitarity, unitary

September 22, 2009

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How to Solve Parseval’s Theorem

Description

A detailed tutorial on the solving of Parseval’s Theorem. Step by step tutorial including several examples of how to solve Parseval’s Theorem for reference.

Overview

Parseval’s Theorem is sometimes called Rayleigh’s Energy Theorem, or Rayleigh’s Identity. The term is used to describe the unitarity of any Fourier’s transform, but the most general form of this property should be referred to as the Plancherel Theorem.

Parseval’s Theorem states: Suppose that A(X) and B(x) are two Riemann integrable, complex-valued functions on R of period 2pi with Fourier series $A(x)=\sum_{n=-\infty}^\infty a_ne^{inx}$ and $B(x)=\sum_{n=-\infty}^\infty b_ne^{inx}$ respectively. Then $\sum_{n=-\infty}^\infty a_n\overline{b_n} = \frac{1}{2\pi} \int_{-\pi}^\pi A(x)\overline{B(x)} dx,$ where horizontal bars indicate complex conjugation.

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Parseval’s Theorem

How to Solve Parseval’s Theorem

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