On approximating the error function

By Zhen-Hang Yang, Y. Chu. 2016

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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)/Bp(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 Bp(x)0$x>0$, where erf(x)=2∫0xe−t2dt/π$\operatorname{erf}(x)=2\int_{0}^{x}e^{-t^{2}}\,dt/\sqrt{\pi}$ is the error function, Bp(x)=1−λ(p)e−px2−[1−λ(p)]e−μ(p)x2$B_{p}(x)=\sqrt{1-\lambda(p)e^{-px^{2}}-[1-\lambda(p)]e^{-\mu(p)x^{2}}}$, λ(p)=16(5p−7)/[(15p2−40p+28)(45p2−60p−4)]$\lambda(p)=16(5p-7)/[(15p^{2}-40p+28)(45p^{2}-60p-4)]$ and μ(p)=4(5p−7)/[5(3p−4)]$\mu(p)=4(5p-7)/[5(3p-4)]$.

Subfields: Applied Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/09877b3e3f697dfcf4b79afdb2b21087f9d3b1d0