Fourier Series of sinx

Mar 2010
13
0
Hi there,

I'm slightly confused about something in a question on a past exam paper for my Fourier analysis module.

I have a question were f(x) is sinx on the interval [0, pi], and f bar denotes the periodic extension with period pi of f.

It asks me to sketch the graph of f bar, and show on the interval [-pi, pi] that f bar is even. I can do both of these. It then asks me to calculate the trigonometric Fourier series of f. I'm just not sure which interval it wants me to use, that with the extention or without. Also, can I ignore the an cosnx part of the Fourier Series formula as I know I am calculating for sinnx? The first couple of lines of the calculation would be appreciated.
 

chisigma

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Mar 2009
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The period of function is \(\displaystyle \pi\) and You have to compute the coefficients \(\displaystyle a_{n}\) and \(\displaystyle b_{n}\) as...

\(\displaystyle a_{n} = \frac{2}{\pi} \int_{0}^{\pi} \sin x\cdot \cos 2 n x \cdot dx\)

\(\displaystyle b_{n} = \frac{2}{\pi} \int_{0}^{\pi} \sin x\cdot \sin 2 n x \cdot dx\) (1)

The periodoc function will be expressed as...

\(\displaystyle f(x) = \frac{a_{0}}{2} + \sum_{n=1}^{\infty} (a_{n} \cos 2 n x + b_{n} \sin 2 n x)\) (2)

Kind regards

\(\displaystyle \chi\) \(\displaystyle \sigma\)
 
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Mar 2010
13
0
So, when it asks me for the trigonometric fourier series, is this a different question to if it asked for the Fourier cosine series or the Fourier sine series? Does the fact that the periodic extension on [0, pi] is even mean I just need the Fourier cosine series?
 
May 2010
2
0
ok, I'm only a student myself but so I could be wrong, I don't understand why you are saying sin is even, sin is an odd function, such that f(-x)=-f(x). However your bn term would be even, it's an odd function by an odd function. I don't know if that helps you. this way you can use the interval of integration 0 to pi. if you multiply by 2...
I hope this helps.
and yes, "Also, can I ignore the an cosnx part of the Fourier Series formula as I know I am calculating for sinnx?" you can 'ignore' the cosnx part.... this is because when the function is odd An is equal to Zero.

I have a question though...
"Show on the interval [-pi, pi] that f bar is even." why is this so?
 
May 2010
2
0
Sorry, ignore my question I didn't read it right! and I'm sorry I can't help you with which interval to use....(Itwasntme)