Problem #2

UCI. MAE200B, winter 2006. by nasser Abbasi.

Problem: expand the function MATH

in terms of infinite series of MATH,$n=1,2,3,\cdots $ by using the orthogonality of legendra polynomials MATH Find the coefficients of the first 6 terms and plot your results with $1,2,\cdots ,6$ terms respectively.

Solution


Figure

MATH

Hence MATH

Now, We know what MATH is for different $n$, (we can use Rodrigues formula to find them) so we substitute these in the above integral and evaluate the coefficients for each $n$ 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.


Figure

Hence, For $n=1$

MATH

For $n=2$

MATH


Figure

Hence based on the above, we can write

MATH

The above is the approximation to $f\left( x\right) $ using only the first 6 terms in the series. If we write MATH to mean approximation to $f\left( x\right) $ using $n$ terms, then we obtain

MATH

Since $f_{n}=f_{n-1}$ when $n$ is even, I will show only the odd valued terms, and will show this for up to $n=15$


Figure