On approximating the error function
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 , ∞ ) $(0, \infty)$ ,...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2016-11-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-016-1261-3 |
| _version_ | 1819090889661743104 |
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| author | Zhen-Hang Yang Yu-Ming Chu |
| author_facet | Zhen-Hang Yang Yu-Ming Chu |
| author_sort | Zhen-Hang Yang |
| collection | DOAJ |
| description | 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 ) ] $\mu(p)=4(5p-7)/[5(3p-4)]$ . |
| first_indexed | 2024-12-21T22:31:00Z |
| format | Article |
| id | doaj.art-f4631d0b36264ef3865a17355b4f918d |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-21T22:31:00Z |
| publishDate | 2016-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-f4631d0b36264ef3865a17355b4f918d2022-12-21T18:48:06ZengSpringerOpenJournal of Inequalities and Applications1029-242X2016-11-012016111710.1186/s13660-016-1261-3On approximating the error functionZhen-Hang Yang0Yu-Ming Chu1School of Mathematics and Computation Sciences, Hunan City UniversitySchool of Mathematics and Computation Sciences, Hunan City UniversityAbstract 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 ) ] $\mu(p)=4(5p-7)/[5(3p-4)]$ .http://link.springer.com/article/10.1186/s13660-016-1261-3error functionmonotonicitybound |
| spellingShingle | Zhen-Hang Yang Yu-Ming Chu On approximating the error function Journal of Inequalities and Applications error function monotonicity bound |
| title | On approximating the error function |
| title_full | On approximating the error function |
| title_fullStr | On approximating the error function |
| title_full_unstemmed | On approximating the error function |
| title_short | On approximating the error function |
| title_sort | on approximating the error function |
| topic | error function monotonicity bound |
| url | http://link.springer.com/article/10.1186/s13660-016-1261-3 |
| work_keys_str_mv | AT zhenhangyang onapproximatingtheerrorfunction AT yumingchu onapproximatingtheerrorfunction |