Error Function Wikipedia
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of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf
Gamma Function Wikipedia
( x ) = 1 π ∫ − x x gaussian function wikipedia e − t 2 d t = 2 π ∫ 0 x e − t 2
Normal Distribution Wikipedia
d t . {\displaystyle {\begin{2m+1}8\operatorname {2m+1}7 (x)&={\frac {2m+1}6{\sqrt {\pi }}}\int _{-x}^{2m+1}5e^{-t^{2m+1}4}\,\mathrm {2m+1}3 t\\&={\frac {2m+1}2{\sqrt {\pi }}}\int _{2m+1}1^{2m+1}0e^{-t^{erfc}9}\,\mathrm {erfc}8 t.\end{erfc}7}} The complementary error function, denoted erfc, is defined error function values as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin{q=0}4\operatorname {q=0}3 (x)&=1-\operatorname {q=0}2 (x)\\&={\frac {q=0}1{\sqrt {\pi }}}\int _{q=0}0^{\infty }e^{-t^{m}9}\,\mathrm {m}8 t\\&=e^{-x^{m}7}\operatorname {m}6 (x),\end{m}5}} which why is it called the error function 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 {z^{2q+1}4 (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 {z^{2q+1}2 (x|x\geq 0)={\frac {z^{2q+1}1{\pi }}\int _{z^{2q+1}0^{\pi /2}\exp \left(-{\frac {2^{2(m-q)-1}9}{\sin ^{2^{2(m-q)-1}8\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{2^{2(m-q)-1}2\operatorname {2^{2(m-q)-1}1 (x)&=-i\operatorname {2^{2(m-q)-1}0 (ix)\\&={\frac {i}9{\sqrt {\pi }}}\int _{i}8^{i}7e^{i}6}\,\mathrm {i}5 t\\&={\frac {i}4{\sqrt {\pi }}}e^{i}3}D(x),\end{i}2}} where D(x) is the Dawson function (which can be used instead of
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Gamma Function Wiki
Sep 27 2016 Created, developed, and nurturedbyEricWeisstein at WolframResearch Calculus and error function calculator Analysis>Special Functions>Erf> Calculus and Analysis>Complex Analysis>Entire Functions> Interactive Entries>webMathematica Examples> More... History and Terminology>Wolfram Language Commands>
Inverse Error Function
MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It is an entire https://en.m.wikipedia.org/wiki/Inverse_error_function function defined by (1) Note that some authors (e.g., Whittaker and Watson 1990, p.341) define without the leading factor of . Erf is implemented in the Wolfram Language as Erf[z]. A two-argument form giving is also implemented as Erf[z0, z1]. Erf satisfies the identities (2) (3) (4) where is erfc, the complementary error function, and http://mathworld.wolfram.com/Erf.html is a confluent hypergeometric function of the first kind. For , (5) where is the incomplete gamma function. Erf can also be defined as a Maclaurin series (6) (7) (OEIS A007680). Similarly, (8) (OEIS A103979 and A103980). For , may be computed from (9) (10) (OEIS A000079 and A001147; Acton 1990). For , (11) (12) Using integration by parts gives (13) (14) (15) (16) so (17) and continuing the procedure gives the asymptotic series (18) (19) (20) (OEIS A001147 and A000079). Erf has the values (21) (22) It is an odd function (23) and satisfies (24) Erf may be expressed in terms of a confluent hypergeometric function of the first kind as (25) (26) Its derivative is (27) where is a Hermite polynomial. The first derivative is (28) and the integral is (29) Min Max Re Im Erf can also be extended to the complex plane, as illustrated above. A simple integral involving erf that Wolfram Language cannot do is given
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undefined value of lowValue(x) defined (highValue(x) + highValue(x)) / 2 lowValue(x) undefined highValue(x) undefined Note: That this is not equivalent to mean(lowValue(x), highValue(x)). point mean(point x, point y) evaluates the arithmetic mean of x and y. If either x or y is undefined, this function evaluates to undefined. This function is equivalent to (x + y) / 2. geometric_mean point geometric_mean(range x) point geometric_mean(point x) point geometric_mean(point x, point y) Evaluates the geometric mean of x and y. If either x or y is undefined, this function evaluates to undefined. Otherwise this function is evaluated as follows: x < 0 x = 0 x > 0 y < 0 -sqrt(x * y) undefined undefined y = 0 undefined undefined undefined y > 0 undefined undefined sqrt(x * y) erf point erf(point x) Evaluates erf(x), the error function of x. If x is undefined, this function evaluates to undefined. See the error function Wikipedia article for further details. erfc point erfc(point x) Evaluates erfc(x), the complementary error function of x. If x is undefined, this function evaluates to undefined. This function is equivalent to 1 - erf(x) See the error function Wikipedia article for further details. inverf point inverf(point x) Evaluates erf-1(x), the inverse error function of x. If x lies outside the interval -1 to 1, or x is undefined, this function evaluates to undefined. See the error function Wikipedia article for further details. inverfc point inverfc(point x) Evaluates erfc-1(x), the inverse complementary error function of x. If x lies outside the interval -1 to 1, or x is undefined, this function evaluates to undefined. See the error function Wikipedia article for further details. phi point phi(point x) Evaluates Φ(x), the normal cumulative distribution function of x. If x is undefined, this function evaluates to undefined. This function is equivalent to (erf(x / sqrt(2)) + 1) / 2 See the normal cumulative distribution function Wikipedia article for further details. probit point probit(point x) Evaluates probit(x), the inverse normal cumulative distribution function of x. If x is undefined, this function evaluates to undefined. This function is equivalent to sqrt(2) * inverf(2*x - 1) See the probit function Wikipedia article for further details.