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A245967
Decimal expansion of a constant appearing in a theorem by Árpád Baricz about Mills' ratio of the standard normal distribution.
1
1, 1, 6, 1, 5, 2, 7, 8, 8, 9, 2, 7, 4, 4, 7, 7, 3, 6, 7, 0, 0, 4, 5, 9, 5, 0, 3, 1, 0, 9, 6, 3, 1, 7, 9, 0, 5, 5, 2, 4, 8, 1, 7, 3, 4, 3, 0, 0, 9, 6, 3, 2, 3, 4, 2, 4, 7, 6, 9, 9, 5, 5, 4, 1, 4, 3, 8, 2, 2, 6, 9, 1, 7, 1, 6, 0, 8, 7, 3, 1, 9, 4, 6, 0, 7, 7, 5, 5, 6, 1, 2, 7, 2, 9, 8, 3, 5, 7, 1, 6, 8, 4
OFFSET
1,3
LINKS
Árpád Baricz, Mills' ratio: Monotonicity patterns and functional inequalities, Journal of Mathematical Analysis and Applications, vol.340, no.2, Apr 15 2008, pp.1362-1370
Iosif Pinelis, Monotonicity properties of the relative error of a Padé approximation for Mills’ ratio, Journal of Inequalities in Pure and Applied Mathematics 01/2002; 3.
Eric Weisstein's MathWorld, Mills ratio
FORMULA
The unique positive root of the transcendent equation x*(x^2+2)*r(x) = x^2+1, where r(x) is Mills' ratio exp(x^2/2)*sqrt(Pi/2)*erfc(x/sqrt(2)).
r(0) = sqrt(Pi/2).
Asymptotic expansion: r(x) ~ 1/x - 1/x^3 + 3/x^5 - 3*5/x^7 + ... + (-1)^k*(2k-1)!!/x^(2k+1) + ...
EXAMPLE
1.16152788927447736700459503109631790552481734300963234247699554...
MATHEMATICA
r[x_] := Exp[x^2/2]*Sqrt[Pi/2]*Erfc[x/Sqrt[2]]; x0 = x /. FindRoot[x*(x^2+2)*r[x] == (x^2+1), {x, 1}, WorkingPrecision -> 102]; RealDigits[x0] // First
PROG
(PARI) solve(x=1, 2, x*(x^2+2)*exp(x^2/2)*sqrt(Pi/2)*erfc(x/sqrt(2))-x^2-1) \\ Charles R Greathouse IV, Apr 30 2015
(Python)
from mpmath import *
mp.dps=103
print([int(n) for n in list(str(findroot(lambda x: x*(x**2+2)*exp(x**2/2)*sqrt(pi/2)*erfc(x/sqrt(2))-x**2-1, (1, 2))).replace('.', ''))]) # Indranil Ghosh, Jul 07 2017
CROSSREFS
Cf. A001147.
Sequence in context: A160135 A178645 A010137 * A212006 A245725 A011096
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved