# Thread: Finding coefficients using Lengendre Polynomials

1. ## Finding coefficients using Lengendre Polynomials

f(x, y) = 3 x^2 + x y - x + y^2

f(x, y) = Sigma an(y)Pn(x)

f(x, y) = Sigma bn(x)Pn(y)

Find a and b.

2. Hey, aren't those just the generalized Fourier coefficients? That is:

$\displaystyle f(x,y)=\sum_{n=0}^{\infty}c_n x^n$

where:

$\displaystyle c_n=\langle P_n,f(x,y)\rangle=\int_{-1}^{1} f(x,y) P_n(x)dx$

Although may have to normalize them first. Not sure though. It's a start however.

3. Ok I've being doing some reading about the Fourier-Legendre series, but I don't know what to do when its a function of two variables. Also the question asks me to look at the first three Legendre polynomials, why?

4. Hey I made a mistake up there: should have formed the sum with the Legendre polynomials:

$\displaystyle f(x,y)=\sum_{n=0}^{\infty}c_n(y) P_n(x)$

and:

$\displaystyle c_n(y)=\langle P_n,f(x,y)\rangle=\frac{2n+1}{2}\int_{-1}^{1} f(x,y) P_n(x)dx$

Need to review orthognal polynomials and orthonormal basis. I'm not sure why the $\displaystyle \frac{2n+1}{2}$ is needed above. Perhaps to normalize the set of functions. Anyway, just to check this, I plugged it into Mathematica:

Code:
In[26]:=
f[x_, y_] := 3*x^2 + x*y - x + y^2;
clist = Table[Subscript[c, n] = ((2*n + 1)/2)*
Integrate[f[x, y]*LegendreP[n, x],
{x, -1, 1}], {n, 0, 10}]
FullSimplify[Sum[clist[[n + 1]]*LegendreP[n, x],
{n, 0, 10}]]

Out[27]=
{(1/2)*(2 + 2*y^2), (3/2)*(-(2/3) + (2*y)/3), 2,
0, 0, 0, 0, 0, 0, 0, 0}

Out[28]=
3*x^2 + x*(-1 + y) + y^2