Problem #2
UCI. MAE200B, winter 2006. by nasser Abbasi.
Problem: expand the function
in terms of infinite series of , by using the orthogonality of legendra polynomials Find the coefficients of the first 6 terms and plot your results with terms respectively.
Solution
Hence
Now, We know what is for different , (we can use Rodrigues formula to find them) so we substitute these in the above integral and evaluate the coefficients for each by performing the above integration. I show the first 2 coefficients calculations, then a Mathematica output for the rest since it is the same process repeated.
Hence, For
For
Hence based on the above, we can write
The above is the approximation to using only the first 6 terms in the series. If we write to mean approximation to using terms, then we obtain
Since when is even, I will show only the odd valued terms, and will show this for up to