# The Fourier coefficients of a derivative

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.