Science:Math Exam Resources/Courses/MATH257/December 2011/Question 03 (a)
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Question 03 (a) 

(a) Find both the Fourier sine series and the Fourier cosine series of the function on the interval . 
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Hint 

Recall that the Fourier coefficients of a sine series of a function are given by, and the coefficients of a cosine series in the form are given by 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We will start by computing the Fourier sine series. The Fourier sine series of is given by
Here, we have and . So Integrate by parts using we get that Since and , we get that
With and , When we have that =1 and so we have two distinct cases for evaluating the integral; when and when . For the case where , integrate by parts using we get that
Now for the case where , we get Therefore, the Fourier cosine series of is 