Explore BrainMass

Explore BrainMass

    Working with Parseval's Theorem

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    A general form of Parseval's Theorem says that if two functions are expanded in a Fourier Series

    f(x) =1/2 ao + Sigma [(an cos(nx)) + bn sin(nx)]

    g(x) 1/2 ao' + Sigma [(an' cos(nx)) + bn' (sin(nx)]

    Then the average value, < f(x)g(x)>, is:
    1/4 ao = sigma[an an' + bn bn'] prove this and using any two functions

    Please give an example.

    © BrainMass Inc. brainmass.com May 20, 2020, 11:16 pm ad1c9bdddf

    Solution Preview

    Please view the attached document (it is both in Word format and in PDF) to view the solution.

    Thanks for using BrainMass.

    The Fourier eigenfunctions and are orthonormal for any integers n, m in the interval , which means that:
    Of course, for any integers n, m
    So assume we can write the function as a series expansion of the eigenfnctions:
    To find the coefficients we simply multiply both sides of the equation by an eigenfunction.
    For we get:
    Since the summation is over n and not m we can bring the eigenfunction into the sum:

    Solution Summary

    The Parseval's Theorem is analyzed. The average value is provided. Two functions are analyzed.