Legendre Polynomials: Pattern Analysis & Integration

[Dustinsfl]

Consider
\[
f(x) = \begin{cases}
1, & 0\leq x\leq 1\\
-1, & -1\leq x\leq 0
\end{cases}
\]
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?

 

[DreamWeaver]

For reference - sorry! - but I think you should define Legendre polynimials...

 

[topsquark]

@ Dreamweaver: Here's everything you want to know about Legendre polynomials, and more.

I'm kinda curious about the series solution myself.

-Dan

 

[Dustinsfl]

Consider
\[
f(x) = \begin{cases}
1, & 0\leq x\leq 1\\
-1, & -1\leq x\leq 0
\end{cases}
\]
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?

\begin{align}
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\\
&= \frac{2n + 1}{2}\left[\int_{0}^1\mathcal{P}_n(x)dx +
\int_{0}^1\mathcal{P}_n(-x)dx\right]
\end{align}
Since \(\mathcal{P}_{\ell}(-x) = (-1)^{\ell}\mathcal{P}_{\ell}(x)\), we now have
\[
c_n = \frac{2n + 1}{2}\int_{0}^1\mathcal{P}_n(x)(1 - (-1)^{\ell})dx\\
\]
If \(\ell\) is even, then the integral is 0, but if \(\ell\) is odd, we gain a factor of 2.
\[
c_n =
\begin{cases}
(2n + 1)\int_{0}^1\mathcal{P}_n(x), & \text{\(\ell\) odd}\\
0, & \text{if \(\ell\) is even}
\end{cases}
\]
We can now use the recursive relation
\[
\mathcal{P}_{\ell}(x) = \frac{1}{2\ell + 1}\left(\frac{d\mathcal{P}_{\ell + 1}(x)}{dx} - \frac{\mathcal{P}_{\ell - 1}(x)}{dx}\right).
\]
Then
\begin{align}
I_{\ell} &= \frac{1}{2\ell + 1}\int_0^1\left(\frac{d\mathcal{P}_{\ell + 1}(x)}{dx} - \frac{\mathcal{P}_{\ell - 1}(x)}{dx}\right)dx\\
&= \frac{1}{2\ell + 1}\left[\mathcal{P}_{\ell + 1}(1) - \mathcal{P}_{\ell + 1}(0) - \mathcal{P}_{\ell - 1}(1) + \mathcal{P}_{\ell - 1}(0)\right]\\
&= \frac{1}{2\ell + 1}\left[\mathcal{P}_{\ell - 1}(0) - \mathcal{P}_{\ell + 1}(0)\right]\quad\text{for }\ell\geq 1
\end{align}
We have that Rodrigues's formula is
\[
\mathcal{P}_{\ell}(x) = \frac{1}{2^{\ell}\ell !}\frac{d^{\ell}}{dx^{\ell}}(x^2 - 1)^{\ell}
\]
so
\[
\mathcal{P}_{\ell}(0) = \left.\frac{1}{2^{\ell}\ell !}\frac{d^{\ell}}{dx^{\ell}} \sum_{n=0}^{\ell}\binom{\ell}{n}(x^2)^n(-1)^{\ell - n}\right|_{x=0}
\]
Since we need even terms, we now have
\[
\mathcal{P}_{\ell}(0) = \frac{1}{2^{\ell}\ell !}\binom{\ell}{\frac{\ell}{2}}
\ell!(-1)^{\ell/2}
\]
Then
\begin{align}
I_{\ell} &= \frac{1}{2\ell + 1}\left[\frac{1}{2^{\ell - 1}}\binom{\ell - 1}{\frac{\ell - 1}{2}}(-1)^{(\ell - 1)/2} - \frac{1}{2^{\ell + 1}}\binom{\ell + 1}{\frac{\ell + 1}{2}}(-1)^{(\ell + 1)/2}\right]\\
&= \frac{(-1)^{(\ell - 1)/2}}{2^{\ell - 1}}\frac{(\ell - 1)!}{\left(\frac{\ell - 1}{2}\right)!\left(\frac{\ell - 1}{2}\right)!}\frac{1}{\ell + 1}
\end{align}
Thus,
\[
c_n =
\begin{cases}
\frac{2n + 1}{n + 1}\frac{(-1)^{(n - 1)/2}}{2^{n - 1}}\frac{(n - 1)!}{\left[\left(\frac{n - 1}{2}\right)!\right]^2}, & \text{for } n \text{ odd}\\
0, & \text{for } n \text{ even}
\end{cases}
\]