Hi, I don't even now where to start with this question.

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.

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- Nov 25th 2008, 09:13 AMHappy DancerFinding coefficients using Lengendre Polynomials
Hi, I don't even now where to start with this question.

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. - Nov 25th 2008, 11:16 AMshawsend
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. :) - Nov 25th 2008, 01:39 PMHappy Dancer
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?

- Nov 25th 2008, 03:03 PMshawsend
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