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