Harken43

2022-02-17

If I have an analytic function of a complex variable, I can write a Taylor series and in some cases can truncate the high powers to obtain a good approximation over some part of the function's domain. I would like to be able to generate rational functions (quotients of polynomials, as in $\frac{n\left(x\right)}{d\left(x\right)}$ where n and d are polynomial functions) that play a similar role. Is there a general way to do this?

Cicolinif73

To see how it arises, first note that in a rational function of degree N in the numerator and degree M in the denominator,
$R\left(x\right)=\frac{\sum _{N}^{n=0}{a}_{n}{x}^{n}}{1+\sum _{M}^{m=1}{b}_{m}{x}^{m}}$,
where a 1 has been taken out of the sum in the denominator to eliminate a redundancy in the coefficients, there are M+N+1 free parameters: ${a}_{0},{a}_{1},\dots {a}_{N}$, along with ${b}_{1},\dots ,{b}_{M}$. So, we assert that the rational function should match the function we are approximating all the way up to M+Nth order around some point ${x}_{0}$:
$f\left({x}_{0}\right)=R\left({x}_{0}\right)$
$\frac{df}{dx}\left({x}_{0}\right)=\frac{dR}{dx}\left({x}_{0}\right)$
$\frac{{d}^{2}f}{{dx}^{2}}\left({x}_{0}\right)=\frac{{d}^{2}R}{{dx}^{2}}\left({x}_{0}\right)$
$\frac{{d}^{M+N}}{d{x}^{M+N}}\left({x}_{0}\right)=\frac{{d}^{M+N}R}{d{x}^{M+N}}\left({x}_{0}\right)$
It seems natural that this would uniquely specify what R must be, and indeed it does.

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