I was asked to compute the Fourier series for $\sin^2(x)$ on $[0,\pi]$. Now this is what I did and I'd like to know if I'm right. $\sin^2(x)=\frac12-\frac12\cos(2x)$ . I got the right hand side using trig identities. I'm wondering If I can do this without using the formulas. Thanks.
Fourier series for $\sin^2(x)$
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fourier-series
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4That's a perfectly valid derivation and probably the quickest one. Alternatively, you can take inner products with the sin and cos functions with different periods to determine their coefficients in the expansion. That is the more general method. But in this particular case, what you did is the more clever way to go about it. – 2010-12-15
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1To add to Alex's comment: if you go the long way and do the inner products anyway, you'll find that you'll need to convert $\sin^2(x)$ to that form you derived to do the required integrations, and you'll easily see that all the expected higher terms zero out. – 2010-12-15