Error Function & Table Of Integrals
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Table Of Integrals Of Exponential Functions
Erf is the "error function" encountered in integrating the normal distribution (which is a normalized form of the Gaussian function). It is table of integrals hyperbolic functions an entire 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
Integral Complementary Error Function
giving is also implemented as Erf[z0, z1]. Erf satisfies the identities (2) (3) (4) where is erfc, the complementary error function, and 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 integral of error function with gaussian density function 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 by (30) (M.R.D'Orsogna, pers. comm., May 9, 2004). More complicated integrals include (31) (M.R.D'Orsogna, pers. comm., Dec.15, 2005). Erf has the continued fraction (32) (33) (Wall 1948, p.357), first stated by Laplace in 1805 and Legendre in 1826 (Olds 1963, p.139), proved by Jacobi, and rediscovered by Ramanujan (Watson 1928; Hardy 1999, pp.8-9). Definite integrals involving include Definite integrals involving include (34) (35) (36) (37) (38) The first two of these appear in Prudnikov et al. (1990, p.123, eqns. 2.8.19.8 and 2.8.19.11), with , . A complex generalization of is defined as (39) (40) Integral representations vali
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Error Function Integral Calculation
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Integral Gamma Function
(Learn how and when to remove this template message) Part of a series of articles about trig integral tables Calculus Fundamental theorem Limits of functions Continuity Mean value theorem Rolle's theorem Differential Definitions Derivative(generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative http://mathworld.wolfram.com/Erf.html Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient General Leibniz Faà di Bruno's formula Integral Lists of integrals Definitions Antiderivative Integral(improper) Riemann integral Lebesgue integration Contour integration Integration by Parts Discs Cylindrical shells Substitution(trigonometric) Partial fractions Order Reduction formulae Series Geometric(arithmetico-geometric) https://en.wikipedia.org/wiki/Lists_of_integrals Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Divergence Gradient Green's Kelvin–Stokes Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian matrix Specialized Fractional Malliavin Stochastic Variations v t e Integration is the basic operation in integral calculus. While differentiation has easy rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives. Contents 1 Historical development of integrals 2 Lists of integrals 3 Integrals of simple functions 3.1 Integrals with a singularity 3.2 Rational functions 3.3 Exponential functions 3.4 Logarithms 3.5 Trigonometric functions 3.6 Inverse trigonometric functions 3.7 Hyperbolic functions 3.8 Inv
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