Results 1 to 4 of 4

Math Help - Finding coefficients using Lengendre Polynomials

  1. #1
    Junior Member
    Joined
    Aug 2008
    Posts
    27

    Finding 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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    Hey, aren't those just the generalized Fourier coefficients? That is:

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

    where:

    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Aug 2008
    Posts
    27
    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?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Aug 2008
    Posts
    903
    Hey I made a mistake up there: should have formed the sum with the Legendre polynomials:


    f(x,y)=\sum_{n=0}^{\infty}c_n(y) P_n(x)<br />

    and:

    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 \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

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 10
    Last Post: February 12th 2011, 10:50 AM
  2. Equating Coefficients of Polynomials Proof
    Posted in the Algebra Forum
    Replies: 3
    Last Post: November 21st 2010, 01:03 PM
  3. Finding Unknown Coefficients of X
    Posted in the Algebra Forum
    Replies: 3
    Last Post: September 22nd 2010, 06:16 PM
  4. Finding Polynomials based on coefficients
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: July 6th 2009, 12:49 AM
  5. Finding a function from Fourier coefficients
    Posted in the Calculus Forum
    Replies: 7
    Last Post: July 5th 2006, 03:50 AM

Search Tags


/mathhelpforum @mathhelpforum