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Is there any function $f(x)$ which can not be approximated by a Padé rational approximant?

$$f(x) \approx \frac{a_0+a_1 x+ \ldots +a_nx^n}{b_0+b_1 x+ \ldots +b_m x^m} $$

What happens with $f(x)= \tan(x)$ or $f(x)=\log^{a}(x)$

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There are conditions, the simplest being that you cannot guarantee approximation unless the function is continuous on a closed bounded interval. As you noticed, logarithm fails at $\infty.$ For that purpose, you could just ask about $\sqrt x.$ Again, you can do $\tan x$ on $- \pi / 4 \leq x \leq \pi / 4.$ – Will Jagy Mar 9 '13 at 22:46
Also take a look at CORDIC for practical ways of computing several functions. – vonbrand Mar 9 '13 at 23:20
Do you know how to compute Pade approximation? – Mhenni Benghorbal Mar 14 '13 at 1:34

Here is Padé approximation for $\tan(x)$

$$ \tan(x)\approx \frac{-{\frac {1}{135135}}\,{x}^{7}+{\frac {2}{715}}\,{x}^{5}-{ \frac {5}{39}}\,{x}^{3}+x }{ 1-{\frac {6}{13}}\,{x}^{2}+{ \frac {10}{429}}\,{x}^{4}-{\frac {4}{19305}}\,{x}^{6} } . $$

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