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Pade approximation

  • #1

Homework Statement


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] ?


Homework 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]

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]

Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
938
9
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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