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Pade approximation

  1. Aug 28, 2012 #1
    1. The problem statement, all variables and given/known data
    Pade approximation
    [tex][N/D]=\frac{a_0+a_1x+...+a_Nx^N}{1+b_1x+...+b_Dx^D}[/tex]
    With this approximation we approximate Maclaurin series
    [tex]f(x)=\sum^{\infty}_{i=0}c_ix^i=[N/D]+O(x^{N+D+1})[/tex]
    How to calculate [tex][1/1][/tex] for [tex]f(x)=1-\frac{1}{2}x+\frac{1}{3}x^2-...[/tex] ?


    2. Relevant equations
    [tex][N/D]=\frac{a_0+a_1x+...+a_Nx^N}{1+b_1x+...+b_Dx^D}[/tex]
    [tex]\sum^{\infty}_{i=0}c_ix^i=[N/D]+O(x^{N+D+1})[/tex]

    3. The attempt at a solution
    [tex](1+b_1x)(1-\frac{1}{2}x)=a_0+a_1x[/tex]
    [tex]a_0=1[/tex]
    [tex]b_1-\frac{1}{2}=a_1[/tex]

    How to calculate [tex]a_1,b_1[/tex]

    In solution
    [tex][1/1]=\frac{1+\frac{1}{6}x}{1+\frac{2}{3}x}[/tex]
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 28, 2012 #2
    Write more terms on both sides
    [tex](1+b_1 x)(1-x/2+x^2/3) = a_0 + a_1 x + \mathcal{O}(x^3) [/tex]
    So what happens to the term proportional to [itex] x^2 [/itex]?
     
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