a. Represent f(x)=|x| in -2<x<2 with a complex Fourier series
b. Show that the complex Fourier Series can be rearranged into a cosine series
c. Take the derivative of that cosine series. What function does the resulting series represent?
The Attempt at a Solution
Ok, so the first part seems pretty straight forward, I plugged the variables in, and due to the absolute value broke the integral up from -2 to 0 and then from 0 to -2 and added them together.
Providing I made no errors (I hope...), I get
Using Euler's, I simplify this to:
which, since for each n the sin (n pi) will become 0, my assumption is the above simply becomes (...?):
I am completely befuddled as to how to move into part b though... I also don’t know what to do with the n’s of the summation. So far, all the cosine series I've done so far have been summed from 0 to ∞ - I don’t know what to do with the -∞ to 0 part of the summation.
Any help would be greatly appreciated. Thanks for looking...
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