How to Expand cx(x-l) into a Fourier Series?

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Discussion Overview

The discussion revolves around the expansion of the function $$cx(x-l)$$ into a Fourier series. Participants are exploring the form of the series and the role of the constant $\alpha$ in the exponential terms.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant seeks guidance on expanding the function $$cx(x-l)$$ into a series of the form $$\sum_{n=-\infty}^{\infty}a_{n}e^{-\alpha x}$$.
  • Another participant questions the formulation, noting that the exponential term does not appear to depend on $n$ and expresses confusion regarding the meaning of $\alpha$.
  • A later reply reiterates the concern about the exponential term and clarifies that $\alpha$ is intended to be a constant.
  • Another participant suggests that the transformation resembles a Fourier series, providing an example of the expansion involving complex exponentials.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the formulation of the series, particularly about the dependence of the exponential on $n$ and the definition of $\alpha$. There is no consensus on the correct approach to the expansion.

Contextual Notes

There are unresolved questions about the assumptions underlying the expansion, particularly regarding the nature of the constant $\alpha$ and the correct form of the series.

Another1
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I have a function $$cx(x-l)$$ where c is constant

I want to expansion this function $$cx(x-l)$$ to $$\sum_{n=-\infty}^{\infty}a_{n}e^{-\alpha x}$$

how can i do it? you have a idea
 
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Is there perhaps a typo in the summand? The exponential does not seem to depend on $n$ and it is not clear to me what $\alpha$ is.
 
Janssens said:
Is there perhaps a typo in the summand? The exponential does not seem to depend on $n$ and it is not clear to me what $\alpha$ is.


$$cx(x-l)$$ to $$\sum_{n=-\infty}^{\infty}a_{n}e^{-\alpha_{n} x}$$ $\alpha$ is constant any constant
 
Another said:
$$cx(x-l)$$ to $$\sum_{n=-\infty}^{\infty}a_{n}e^{-\alpha_{n} x}$$ $\alpha$ is constant any constant

That looks like a Fourier series transform.

We have:
$$cx(x-1) = c\left(\frac{\pi^2}3 + (-2-i)e^{-ix} - (2-i)e^{ix} + ...\right)$$
 

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