Table 1 Padé approximants for arcsin x and arctan x
From: Padé approximants for inverse trigonometric functions and their applications
Function | Padé approximant | Associate Taylor polynomials |
|---|---|---|
arcsinx | \(\arcsin _{[1/2]}(x)=\frac{6}{6-x^{2}}\) | \(x+\frac{x^{3}}{6}\) |
arcsinx | \(\arcsin_{[5/2]}(x)=\frac{61x^{5}+1\text{,}080x^{3}-2\text{,}520x}{1\text{,}500x^{2}-2\text{,}520}\) | \(x+\frac{x^{3}}{6}+\frac{3x^{5}}{40}+\frac{5x^{7}}{112} \) |
arctanx | \(\arctan_{[1/2]}(x)=\frac{3x}{x^{2}+3}\) | \(x-\frac{x^{3}}{3} \) |
arctanx | \(\arctan_{[3/2]}(x)=\frac{4x^{3}+15x}{9x^{2}+15}\) | \(x-\frac{x^{3}}{3}+\frac{x^{5}}{5}\) |
arctanx | \(\arctan _{[5/2]}(x)=\frac{-4x^{5}+40x^{3}+105x}{75x^{2}+105}\) | \(x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}\) |
arctanx | \(\arctan _{[3/4]}(x)=\frac{55x^{3}+105x}{9x^{4}+90x^{2}+105}\) | \(x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}\) |