# Solve the Sine and Cosine Fourier Series and Determine the Sum of Each

Write the cosine and sine Fourier Series. Determine the sum of each.

{ 4x 0≤x≤2

f(x) ={ -3 2<x<4

{ 1 4≤x≤7

Please see the attached file for the fully formatted problems.

Â© BrainMass Inc. brainmass.com December 15, 2022, 5:05 pm ad1c9bdddfhttps://brainmass.com/math/fourier-analysis/sine-cosine-fourier-series-determine-sum-each-60232

#### Solution Preview

Hi again;

(1) This part I'll repeat from last time, so that you have it with you:

Two important functions that we'll have to integrate when we try to find the fourier coefficients are:

xsin(kx) and xcos(kx), for some constant k.

We can integrate these by parts, or note that:

xsin(kx) = d/dx ( (-1/k)xcos(kx) + (1/k^2)sin(kx) )

and

xcos(kx) = d/dx ( (1/k)xsin(kx) + (1/k^2)cos(kx) )

Now, for a function defined on the interval [0,2L], recall that the Fourier ...

#### Solution Summary

Sine and Cosine Fourier Series and sums are found. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.