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The Fourier coefficients of a derivative

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Let f be a 2 pi periodic, differentiable function with Fourier coefficients a_n and b_n.
Let (a_n)*, (b_n)* be the Fourier coefficients of f'.

a) Show that (a_0)*=0

b) Use integration by parts to find a formula for the Fourier coefficients of f' in terms of the Fourier coefficients of f.

(The attachment contains the above question written with clear mathematical notation)

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Solution Summary

The solution is a one page document written in Word and using Mathtype for the equations describing (and proving) how the Fourier coefficients of a periodic function are connected to the Fourier coefficients of the derivative of the function. This method uses purely real integrals as opposed to the alternative method using complex integration.

Full explanation of all calculations is given.

See Also This Related BrainMass Solution

Fourier Series and Gibbs Phenomenon

Please see the attached file for the fully formatted problems.

Consider the ODE

y" - a^2 * y = H(x-Pi/2)


Where H is the step function.

solve for y using Fourier series
The Fourier series for the step function exhibits the Gibss phenomenon. Will the solution y(x) exhibit it too? explain why or why not.

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