# An approximation of rational function with polynomials

To compute some asymptotic expression I need to approximate $$\frac{(x-1)^{r+u}\left((x-1)^{p-r+1}+x^{p-r+1}\right)\left(x^{p-u+1}+(x-1)^{p+u+1}\right)}{\left(x^{2p+2}+(x-1)^{2p+2}\right)}$$ by some polynomial. Here $x\in(0,0.5)$,$p\geq r,u \geq 0$ some integers. Somehow Taylor and Lagrange methods don't work to me. What other methods should I try?

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Have you tried multiplying out the numerator and using the binomial theorem? –  Mose Wintner Apr 23 '11 at 15:19

## 1 Answer

If you just tabulate the values for fixed $r,u$ at the proper points in (0,0.5) you can use a Chebyshev series, which is handy to give a bounded error. A description is available at Wikipedia (also see Chebyshev nodes) and at Numerical Recipes page 190, which includes C code.

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