Complementary Error Function
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that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ 0 x e − complementary error function excel t 2 d t . {\displaystyle {\begin − 2\operatorname − 1 (x)&={\frac − 0{\sqrt
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{\pi }}}\int _{-x}^ 9e^{-t^ 8}\,\mathrm 7 t\\&={\frac 6{\sqrt {\pi }}}\int _ 5^ 4e^{-t^ 3}\,\mathrm 2 complementary error function table t.\end 1}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t
Complimentary Error Function
= e − x 2 erfcx ( x ) , {\displaystyle {\begin Φ 8\operatorname Φ 7 (x)&=1-\operatorname Φ 6 (x)\\&={\frac Φ 5{\sqrt {\pi }}}\int _ Φ 4^{\infty }e^{-t^ Φ 3}\,\mathrm Φ 2 t\\&=e^{-x^ Φ 1}\operatorname Φ 0 (x),\end 9}} which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname Φ 8 (x)} for inverse complementary error function non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname Φ 6 (x|x\geq 0)={\frac Φ 5{\pi }}\int _ Φ 4^{\pi /2}\exp \left(-{\frac Φ 3}{\sin ^ Φ 2\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin − 6\operatorname − 5 (x)&=-i\operatorname − 4 (ix)\\&={\frac − 3{\sqrt {\pi }}}\int _ − 2^ − 1e^ − 0}\,\mathrm − 9 t\\&={\frac − 8{\sqrt {\pi }}}e^ − 7}D(x),\end − 6}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 4 (x)} is real when x is real. When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: w ( z ) = e − z 2 erfc ( − i z ) = erfcx
that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ 0
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x e − t 2 d t . {\displaystyle {\begin − 2\operatorname − 1 (x)&={\frac
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− 0{\sqrt {\pi }}}\int _{-x}^ 9e^{-t^ 8}\,\mathrm 7 t\\&={\frac 6{\sqrt {\pi }}}\int _ 5^ 4e^{-t^ complementary error function ti 89 3}\,\mathrm 2 t.\end 1}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − https://en.wikipedia.org/wiki/Error_function t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin Φ 8\operatorname Φ 7 (x)&=1-\operatorname Φ 6 (x)\\&={\frac Φ 5{\sqrt {\pi }}}\int _ Φ 4^{\infty }e^{-t^ Φ 3}\,\mathrm Φ 2 t\\&=e^{-x^ Φ 1}\operatorname Φ 0 (x),\end 9}} which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x https://en.wikipedia.org/wiki/Error_function ) {\displaystyle \operatorname Φ 8 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname Φ 6 (x|x\geq 0)={\frac Φ 5{\pi }}\int _ Φ 4^{\pi /2}\exp \left(-{\frac Φ 3}{\sin ^ Φ 2\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin − 6\operatorname − 5 (x)&=-i\operatorname − 4 (ix)\\&={\frac − 3{\sqrt {\pi }}}\int _ − 2^ − 1e^ − 0}\,\mathrm − 9 t\\&={\frac − 8{\sqrt {\pi }}}e^ − 7}D(x),\end − 6}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 4 (x)} is real when x is real. When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function: w ( z ) = e
function of a given number. Complementary Error Function In mathematics, the complementary error function (also complementary error function known as Gauss complementary error function) is defined as: Complementary Error Function Table The following is the error function and complementary error function table that shows the values of erf(x) and erfc(x) for x ranging from 0 to 3.5 with increment of 0.01. xerf(x)erfc(x)0.00.01.00.010.0112834160.9887165840.020.0225645750.9774354250.030.0338412220.9661587780.040.0451111060.9548888940.050.0563719780.9436280220.060.0676215940.9323784060.070.078857720.921142280.080.0900781260.9099218740.090.1012805940.8987194060.10.1124629160.8875370840.110.1236228960.8763771040.120.1347583520.8652416480.130.1458671150.8541328850.140.1569470330.8430529670.150.1679959710.8320040290.160.1790118130.8209881870.170.1899924610.8100075390.180.2009358390.7990641610.190.2118398920.7881601080.20.2227025890.7772974110.210.2335219230.7664780770.220.2442959120.7557040880.230.25502260.74497740.240.2657000590.7342999410.250.276326390.723673610.260.2868997230.7131002770.270.2974182190.7025817810.280.3078800680.6921199320.290.3182834960.6817165040.30.3286267590.6713732410.310.338908150.661091850.320.3491259950.6508740050.330.3592786550.6407213450.340.3693645290.6306354710.350.3793820540.6206179460.360.3893297010.6106702990.370.3992059840.6007940160.380.4090094530.5909905470.390.41873870.58126130.40.4283923550.5716076450.410.437969090.562030910.420.4474676180.5525323820.430.4568866950.5431133050.440.4662251150.5337748850.450.475481720.524518280.460.484655390.515344610.470.4937450510.5062549490.480.5027496710.4972503290.490.5116682610.4883317390.50.5204998780.4795001220.510.529243620.470756380.520.537898630.462101370.530.5464640970.4535359030.540.554939250.445060750.550.5633233660.4366766340.560.5716157640.4283842360.570.5798158060.4201841940.580.58792290.41207710.590.5959364970.4040635030.60.6038560910.3961439090.610.6116812190.3883187810.620.6194114620.3805885380.630.6270464430
of the Complementary Error Function Keywords: error functions, repeated integrals of the complementary error function Referenced by: §12.7(ii) Permalink: http://dlmf.nist.gov/7.18 See also: info for 7 Contents §7.18(i) Definition §7.18(ii) Graphics §7.18(iii) Properties §7.18(iv) Relations to Other Functions §7.18(v) Continued Fraction §7.18(vi) Asymptotic Expansion §7.18(i) Definition Keywords: repeated integrals of the complementary error function Permalink: http://dlmf.nist.gov/7.18.i See also: info for 7.18 7.18.1 i-1erfc(z) =2πe-z2, i0erfc(z) =erfcz, Symbols: erfcz: complementary error function, e: base of exponential function, inerfc(z): repeated integrals of the complementary error function and z: complex variable A&S Ref: 7.2.1 (in modified form) Referenced by: §7.22(iii) Permalink: http://dlmf.nist.gov/7.18.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: info for 7.18(i) and for n=0,1,2,…, 7.18.2 inerfc(z)=∫z∞in-1erfc(t)dt=2π∫z∞(t-z)nn!e-t2dt. Defines: inerfc(z): repeated integrals of the complementary error function Symbols: dx: differential of x, e: base of exponential function, !: factorial (as in n!), ∫: integral, z: complex variable and n: nonnegative integer A&S Ref: 7.2.1 (in modified form) 7.2.3 Permalink: http://dlmf.nist.gov/7.18.E2 Encodings: TeX, pMML, png See also: info for 7.18(i) §7.18(ii) Graphics Notes: These graphs were produced at NIST. Keywords: repeated integrals of the complementary error function Permalink: http://dlmf.nist.gov/7.18.ii See also: info for 7.18 Figure 7.18.1: Repeated integrals of the scaled complementary error function 2nΓ(12n+1)inerfc(x), n=0,1,2,4,8,16. Symbols: Γ(z): gamma function, inerfc(z): repeated integrals of the complementary error function, x: real variable and n: nonnegative integer Keywords: repeated integrals of the complementary error function Permalink: http://dlmf.nist.gov/7.18.F1 Encodings: pdf, png See also: info for 7.18(ii) §7.18(iii) Properties Notes: See Hartree (1936). Keywords: derivatives, repeated integrals of the complementary error function Permalink: http://dlmf.nist.gov/7.18.iii See also: info for 7.18 7.18.3 ddzinerfc(z)=-in-1erfc(z), n=0,1,2,…, Symbols: dfdx: derivative of f with respect to x, inerfc(z): repeated integrals of the complementary error function, z: complex variable and n: nonnegative integer A&S Ref: 7.2.8 Permalink: http://dlmf.nist.gov/7.