Complementry Error Function
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Examples> History and Terminology>Wolfram Language Commands> Less... Erfc Erfc is the complementary error function, commonly denoted , is an entire complementary error function table function defined by (1) (2) It is implemented in the Wolfram Language as Erfc[z]. Note that some authors (e.g., Whittaker and Watson 1990, p.341) define without the leading factor of . For ,
Complimentary Error Function
(3) where is the incomplete gamma function. The derivative is given by (4) and the indefinite integral by (5) It has the special values (6) (7) (8) It satisfies the identity (9) It has definite integrals (10) (11) (12) For , is bounded by (13) Min Max Re Im Erfc can also be extended to the complex plane, as illustrated above. A generalization is obtained from inverse complementary error function the erfc differential equation (14) (Abramowitz and Stegun 1972, p.299; Zwillinger 1997, p.122). The general solution is then (15) where is the repeated erfc integral. For integer , (16) (17) (18) (19) (Abramowitz and Stegun 1972, p.299), where is a confluent hypergeometric function of the first kind and is a gamma function. The first few values, extended by the definition for and 0, are given by (20) (21) (22) SEE ALSO: Erf, Erfc Differential Equation, Erfi, Inverse Erfc RELATED WOLFRAM SITES: http://functions.wolfram.com/GammaBetaErf/Erfc/ REFERENCES: Abramowitz, M. and Stegun, I.A. (Eds.). "Repeated Integrals of the Error Function." §7.2 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp.299-300, 1972. Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp.568-569, 1985. Press, W.H.; Flannery, B.P.; Teukolsky, S.A.; and Vetterling, W.T. "Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp.209-214, 1992. Spanier, J. and Oldham, K.B. "The Error Function and Its Complement " and "The and and Related Functions." Chs.40 and 41 in An Atlas of Functions. Washington, DC: Hemi
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 =
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2 π ∫ 0 x e − t 2 d t . {\displaystyle
Complementary Error Function Mathematica
{\begin − 5\operatorname − 4 (x)&={\frac − 3{\sqrt {\pi }}}\int _{-x}^ − 2e^{-t^ − 1}\,\mathrm − 0 t\\&={\frac 9{\sqrt {\pi complementary error function ti 89 }}}\int _ 8^ 7e^{-t^ 6}\,\mathrm 5 t.\end 4}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( http://mathworld.wolfram.com/Erfc.html x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 1\operatorname 0 (x)&=1-\operatorname Φ 9 (x)\\&={\frac Φ 8{\sqrt {\pi }}}\int _ Φ 7^{\infty }e^{-t^ Φ 6}\,\mathrm Φ 5 t\\&=e^{-x^ Φ 4}\operatorname Φ 3 (x),\end Φ 2}} which also defines erfcx, the scaled complementary error function[3] (which can be used https://en.wikipedia.org/wiki/Error_function instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname 1 (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 9 (x|x\geq 0)={\frac Φ 8{\pi }}\int _ Φ 7^{\pi /2}\exp \left(-{\frac Φ 6}{\sin ^ Φ 5\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 − 9\operatorname − 8 (x)&=-i\operatorname − 7 (ix)\\&={\frac − 6{\sqrt {\pi }}}\int _ − 5^ − 4e^ − 3}\,\mathrm − 2 t\\&={\frac − 1{\sqrt {\pi }}}e^ − 0}D(x),\end − 9}} 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 7 (x)} is real when x is real. When the error fun
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)0.00.01.00.010.0112834160.9887165840.020.0225645750.9774354250.030.0338412220.9661587780.040.0451111060.9548888940.050.0563719780.9436280220.060.0676215940.9323784060.070.078857720.921142280.080.0900781260.9099218740.090.1012805940.8987194060.10.1124629160.8875370840.110.1236228960.8763771040.120.1347583520.8652416480.130.1458671150.8541328850.140.1569470330.8430529670.150.1679959710.8320040290.160.1790118130.8209881870.170.1899924610.8100075390.180.2009358390.7990641610.190.2118398920.7881601080.20.2227025890.7772974110.210.2335219230.7664780770.220.2442959120.7557040880.230.25502260.74497740.240.2657000590.7342999410.250.276326390.723673610.260.2868997230.7131002770.270.2974182190.7025817810.280.3078800680.6921199320.290.3182834960.6817165040.30.3286267590.6713732410.310.338908150.661091850.320.3491259950.6508740050.330.3592786550.6407213450.340.3693645290.6306354710.350.3793820540.6206179460.360.3893297010.6106702990.370.3992059840.6007940160.380.4090094530.5909905470.390.41873870.58126130.40.4283923550.5716076450.410.437969090.562030910.420.4474676180.5525323820.430.4568866950.5431133050.440.4662251150.5337748850.450.475481720.524518280.460.484655390.515344610.470.4937450510.5062549490.480.5027496710.4972503290.490.5116682610.4883317390.50.5204998780.4795001220.510.529243620.470756380.520.537898630.462101370.530.5464640970.4535359030.540.554939250.445060750.550.5633233660.4366766340.560.5716157640.4283842360.570.5798158060.4201841940.580.58792290.41207710.590.5959364970.4040635030.60.6038560910.3961439090.610.6116812190.3883187810.620.6194114620.3805885380.630.6270464430.3729535570.640.6345858290.3654141710.650.6420293270.3579706730.660.6493766880.3506233120.670.6566277020.3433722980.680.6637822030.3362177970.690.6708400620.3291599380.70.6778011940.3221988060.710.684665550.315334450.720.6914331230.3085668770.730.6981039430.3018960570.740.7046780780.2953219220.750.7111556340.2888443660.760.7175367530.2824632470.770.7238216140.2761783860.780.7300104310.2699895690.790.7361034540.2638965460.80.7421009650.2578990350.810.7480032810.2519967190.820.7538107510.2461892490.830.7595237570.2404762430.840.7651427110.2348572890.850.7706680580.2
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