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## Homework Statement

Hi there.

The question is:

if f(x) is a periodic function with period 2pi and the derivative is f'(x) continuous on [-pi, pi]

show that the fourier coefficients of f' are kbk and -kak if the fourier coefficients of f are ak and bk respectively.

## Homework Equations

## The Attempt at a Solution

This is my attempt. Doesn't quite get there:

so ak' = 1/2pi int from -pi to pi of f'(x) cos kx dx

use parts:

f'(x) = du

f(x) = u

cos kx = v

-1/k sin kx = dv

i get

1/2pi[f(x) cos kx/k + 1/k int from -pi to pi f(x) sin kx]

of course the integral in the parts equation is just bk, and the f(x) cos kx/k evaluated from pi to -pi just becomes 2f(pi)cos kpi.

So then I'm left with.

1/pi[f(pi)cos k pi + (1/k)bk]

I guess I can write it as

[(-1)^k f(pi) + (1/k) bk]/pi

What did i do wrong??

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