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Suppose I want to write the function $x \sin(t)$ as the series over the interval $x \in (0,\pi)$

$$x\sin(t) = \sum_{n=1}^{\infty}(a_n \cos(t) + b_n \sin(t) )\sin(nx)$$

Then would the coefficients $a_n$ and $b_n$ be just simply

$$a_n =\frac{1}{\pi \cos(t)} \int_{0}^{\pi}x\sin(nx) \;\mathrm{d}x = \frac{\tan(t)}{\pi} \int_{0}^{\pi}x\sin(nx) \;\mathrm{d}x $$

$$b_n =\frac{1}{\pi \sin(t)} \int_{0}^{\pi}x\sin(t)\sin(nx) \;\mathrm{d}x = \frac{1}{\pi} \int_{0}^{\pi}x\sin(nx) \;\mathrm{d}x$$

Could i just treat the trig functions as parameters?

Thanks

EDIT: The bounty is supposed to read "not enough attention". I may have misselcted the wrong item when I set the bounty

  • 1
    Expand $x$ as usual, and then multiply by $\sin\,t$.2012-08-03
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    Right, I could cancel out that sine for $b_n$. But is my theory right?2012-08-03
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    I'm afraid a lot went wrong in your question as it stands on 08/06/12.2012-08-06
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    @ChristianBlatter, what do you mean by that?2012-08-06
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    Do you mean the function $x \sin(x)$? It is not clear what $t$ is doing here. If it is not the independent variable, $\sin(t)$ and $\cos (t)$ are just constants.2012-08-07
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    @RossMillikan, I do mean what I mean in the OP. This was on a PDE final I wrote. That's why I wasn't sure if the question was trying to trick me.2012-08-07
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    How does $1/\cos t$ becomes $\tan t$?2012-08-10

1 Answers 1

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(Cf. my comment above)

The only sense one can give the formula $$x\sin(t) = \sum_{n=1}^{\infty}(a_n \cos(t) + b_n \sin(t) )\sin(x)$$ is the following: Assume that the (constant) series $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ both converge with sum $a$ and $b$ respectively. Then one would have the equation $$x\sin t=(a\cos t + b\sin t)\sin x$$ connecting the variables $t$ and $x$. Nothing about Fourier series here.

The formula $$a_n =\frac{1}{\pi \cos(t)} \int_{0}^{\pi}x\sin(x) \;\mathrm{d}x = \frac{\tan(t)}{\pi} \int_{0}^{\pi}x\sin(x) \;\mathrm{d}x $$ and the similar formula for $b_n$ doesn't make any sense whatsoever. Note that the letter $n$ isn't even appearing on the RHS.

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    Does it improve anything if I add back "n"? I'll do it. I may have forgotten about that.2012-08-07