Part II: Fourier series Write the Fourier series for the following function: f(x

Question

Part II: Fourier series

Write the Fourier series for the following function: f(x) ={(-2x -??x<0)/(2x 0<x??)}

1. Find a_0 using the formula a_0=1/2? ?_(-?)^? f(x)dx

?_(-?)^0 -2x dx=

?_0^? 2x dx=

a_0=1/2? [?_(-?)^0 -2x dx+?_0^? 2x dx] =

2.   Find a_k using the formula a_k=1/? ?_(-?)^? f(x) cos(kx)dx.  

Note: We can use the integration formula ?ucos(u)du=cos(u)+usin(u). To do this we will set u = kx, which makes du = kdx.   To complete the substitution we need to have a k attached to both the f(x) and dx portion of the integral. Thus, we are multiplying in two k

Explanation / Answer

7.   Write a Fourier Series for

cos(pik)=cos(-pik)=1 if k is even-

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