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- #1

\[

f(x) = \begin{cases}

1, & 0\leq x\leq 1\\

-1, & -1\leq x\leq 0

\end{cases}

\]

Then

\[

c_n = \frac{2n + 1}{2}\int_{0}^1\mathcal{P}_n(x)dx -

\frac{2n + 1}{2}\int_{-1}^0\mathcal{P}_n(x)dx

\]

where \(\mathcal{P}_n(x)\) is the Legendre Polynomial of order n.

Our first few \(c_n\) are \(0, 3/2, 0, -7/8, 0, 11/16, 0, -75/128, 0, ...\).

Is there a pattern to this? I know \(n\) even is 0 but can I obtain a nice solution?

By this I mean, if I had a Fourier series, I could get a solution of the form

\[

A_n = \begin{cases}

0, & \text{if n is even}\\

\frac{4}{n\pi}, & \text{if n is odd}

\end{cases}

\]

If I can obtain such a solution, how? Is it by simply noticing a geometric pattern in the terms or can I integrate \(\mathcal{P}_n(x)\)?

Does the Rodrigues's formula need to be used in the integral?