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A–10 TABULATION OF Q (z)
Mathematical Techniques, Identities, and Tables
Appendix A
Q(z) ! 1
12p Lz
q
e -l2 >2 dl
For z 3,
Also,
Q(-z) = 1 - Q(z)
Q(z) L
1
12p z e-z2>2
(See Fig. B–7.)
Q(z) = 1 2 erf a z 12 b = 1 2 c1 - erfa z 12 bd
where
erfc(x) ! 2
1p Lx
q
e -l2 dl
and
erf(x) = 2 e -l2 dl
1p L0
x
For z 0, a rational function approximation is [Abramowitz and Stegun, 1964; Ziemer and
Tranter, 1995]
Q(z) = e-z2 >2
12p (b 1t + b 2 t 2 + b 3 t 3 + b 4 t 4 + b 5 t 5 ).
where t = 1>(1 + pz), with p = 0.2316419,
b 1 = 0.31981530 b 2 = -0.356563782 b 3 = 1.781477937
b 4 = -1.821255978 b 5 = 1.330274429
Another approximation for Q (z) for z 0 is [Börjesson and Sunberg, 1979; Peebles, 1993]
Q(z) = c
>2
1
(10.339)z + 0.3392z 2 + 5.510 d e-z2 12p
This approximation has a maximum absolute error of 0.27% for z 0.