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The wording of the following problem confuses me:

Find the Fourier sine series for $f(x)=x$, on $0\leq x<L$. To which value does the series converge at $x=\frac32 L$?

Can someone clarify what is meant by "the Fourier sine series"? From what I know, Fourier series are defined for functions with a domain $[-\pi, \pi]$. To get to this, we let $$ \theta = \left(\frac{2x}{L}-1\right)\pi $$ However, now the function $f(\theta)$ is not even in $\theta$ so the cosine terms won't dissapear.

Can someone clarify the situation for me?

Christian E. Ramirez
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Maths Wizzard
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  • Can someone explain the downvote? – Maths Wizzard Dec 30 '22 at 07:08
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    It was probably in part due to the fact that you used an image when you could have just written the problem in LaTeX. You also had a few typos. I've fixed that for you. – Christian E. Ramirez Dec 30 '22 at 07:18
  • The sine series is just the Fourier series without the cosine bits. – Sean Roberson Dec 30 '22 at 07:23
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    Extend the domain to $(-L,L)$ $x$ is anti-symmetric, so it's cosine coefficients are zero. $\theta=\frac{nx\pi}{2L}$ – ryaron Dec 30 '22 at 08:04
  • The "rationale" of a Fourier sine series is to take a "piece" of a function $f$ and replicate it in order to get another function $g$ which is 1) periodic 2) odd (with $g$ identical with $f$ on the concerned "piece"). For Fourier cosine series replace "odd" by "even". – Jean Marie Dec 30 '22 at 09:32
  • @ryaron how is this not arbitrary? Can't I by this logic expand to $[-2L,2L]$ and get a different series? – Maths Wizzard Dec 31 '22 at 21:28
  • Does it converge to the same function on $(0,L)$? If so and it is more tractable than yes. – ryaron Jan 02 '23 at 05:05
  • Hi, I think it must converge to it by construction. Although I am not sure what tractable means. Does it mean for the smallest extension in domain? Also, do you know where I can find a source for this terminology? – Maths Wizzard Jan 02 '23 at 06:57
  • Tractable means computable, if it is easier for me to compute a function by extension, then i should extend. – ryaron Jan 02 '23 at 14:30
  • How would you know how to extend it? – Maths Wizzard Jan 02 '23 at 14:31

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