# Math Help - Legendre-Fourier series question

1. ## Legendre-Fourier series question

Consider the equation:

$\sum_{n=0}^\infty d_{n}P_{n}(x) \hspace{1mm} = \hspace{1mm} sin(x)$

where $P_{n}(x)$ is the Legende polynomial of degree n.

How can I find out the value of each $d_{1}, d_{2}, d_{3},...$?

2. Originally Posted by garunas
Consider the equation:

$\sum_{n=0}^\infty d_{n}P_{n}(x) \hspace{1mm} = \hspace{1mm} sin(x)$

where $P_{n}(x)$ is the Legende polynomial of degree n.

How can I find out the value of each $d_{1}, d_{2}, d_{3},...$?
Note that

$\int_{-1}^{1}P_n(x)P_m(x)dx=\frac{2}{2n+1}\delta_{n,m}$

Where the right hand side is the Kronecker delta.

So multiply both sides by the nth Legendre polynomial and integrate

$\frac{2 d_n}{2n+1}=\int_{-1}^{1}P_n(x)\sin(x)dx$

Can you finish from here?

3. Sure can! Thanks mate