Q5Mathematics I

Question

State Parseval's theorem.

Answer 1

In advanced Fourier analysis, Parseval's theorem (frequently referred to as Parseval's identity) is a critically important energy conservation theorem. It mathematically states that if a continuous or piecewise-continuous periodic function $f (x)$ is properly defined over a specific strict interval $(- l, l)$ and possesses valid Fourier coefficients $a_{0}, a_{n}, and b_{n}$ , then the following exact algebraic identity holds true:

\frac{1}{l} \int_{- l}^{l} [f (x)]^{2} d x = \frac{a _{0}^{2}}{2} + n = 1 \sum \infty (a_{n}^{2} + b_{n}^{2})

This profound equation rigorously demonstrates that the total 'energy' of the physical signal or mathematical waveform (represented by the definite integral of the square of the function on the left side) is exactly equal to the sum of the total energies contained within all of its individual harmonic frequency components (represented by the infinite sum of the squared Fourier coefficients on the right side).

Q5Mathematics I

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