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## Main Question or Discussion Point

I want to have the following function approximated (in elementary functions) within a error of 1e-6:

[tex] (x^2+a^2)^{1/4}\ \forall x\ \in [-1,1],\ a\in \mathbb{R} [/tex]

If I use Legendre polynomials I have to use a lot to get convergence (since if I use an estimate of the error the error decreases with n^3/2 with n the number of polynomials). So I suppose that I can use taylor polynomials, right? But taylored around 0 the polynomials tends to have a large error moving away from zero. So my idea is to use several taylor polynomials for certain intervals. Is this an idea or do I have to do something else?

[tex] (x^2+a^2)^{1/4}\ \forall x\ \in [-1,1],\ a\in \mathbb{R} [/tex]

If I use Legendre polynomials I have to use a lot to get convergence (since if I use an estimate of the error the error decreases with n^3/2 with n the number of polynomials). So I suppose that I can use taylor polynomials, right? But taylored around 0 the polynomials tends to have a large error moving away from zero. So my idea is to use several taylor polynomials for certain intervals. Is this an idea or do I have to do something else?