On approximating the error function - Zhen-Hang Yang; Yu-Ming Chu (Journal of Inequalities and Applications, 2016)
Abstract In the article, we present the necessary and sufficient condition for the parameter p on the interval ( 7 / 5 , ∞ ) $(7/5, \infty)$ such that the function x → erf ( x ) / B p ( x ) $x\rightarrow\operatorname{erf}(x)/B_{p}(x)$ is strictly increasing (decreasing) on ( 0 , ∞ ) $(0, \infty)$ , and find the best possible parameters p, q on the interval ( 7 / 5 , ∞ ) $(7/5, \infty)$ such that the double inequality B p ( x ) < erf ( x ) < B q ( x ) $B_{p}(x)<\operatorname{erf}(x)<B_{q}(x)$ holds for all x > 0 $x>0$ , where erf ( x ) = 2 ∫ 0 x e − t 2 d t / π $\operatorname{erf}(x)=2\int_{0}^{x}e^{-t^{2}}\,dt/\sqrt{\pi}$ is the error function, B p ( x ) = 1 − λ ( p ) e − p x 2 − [ 1 − λ ( p ) ] e − μ ( p ) x 2 $B_{p}(x)=\sqrt{1-\lambda(p)e^{-px^{2}}-[1-\lambda(p)]e^{-\mu(p)x^{2}}}$ , λ ( p ) = 16 ( 5 p − 7 ) / [ ( 15 p 2 − 40 p + 28 ) ( 45 p 2 − 60 p − 4 ) ] $\lambda(p)=16(5p-7)/[(15p^{2}-40p+28)(45p^{2}-60p-4)]$ and μ ( p ) = 4 ( 5 p − 7 ) / [ 5 ( 3 p − 4 ) ] $\mu(p)=4(5p-7)/[5(3p-4)]$ .