Error Function Integrals Table
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Integral Of Error Function With Gaussian Density Function
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Error Function Integral Calculation
and Terminology>Wolfram Language Commands> MathWorld Contributors>D'Orsogna> Less... Erf is the "error function" encountered in integrating the normal distribution (which is a normalized form of the
Integral Gamma Function
Gaussian function). It is 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 giving is also implemented as Erf[z0, z1]. Erf satisfies the identities (2) integral normal distribution (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 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
integrals. Contents 1 Indefinite integral 1.1 Integrals involving only exponential functions 1.2 Integrals involving polynomials 1.3 Integrals involving exponential and trigonometric functions 1.4 Integrals involving the error function 1.5 Other integrals 2 Definite integrals 3 error function values See also 4 Further reading 5 External links Indefinite integral[edit] Indefinite integrals are antiderivative differentiation error function functions. A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has derivative error function been suppressed here in the interest of brevity. Integrals involving only exponential functions[edit] ∫ f ′ ( x ) e f ( x ) d x = e f ( x ) {\displaystyle \int f'(x)e^ http://mathworld.wolfram.com/Erf.html 5\;\mathrm 4 x=e^ 3} ∫ e c x d x = 1 c e c x {\displaystyle \int e^ θ 9\;\mathrm θ 8 x={\frac θ 7 θ 6}e^ θ 5} ∫ a c x d x = 1 c ⋅ ln a a c x f o r a > 0 , a ≠ 1 {\displaystyle \int a^ θ 9\;\mathrm θ 8 x={\frac θ 7 θ https://en.wikipedia.org/wiki/List_of_integrals_of_exponential_functions 6}a^ θ 5\;\mathrm θ 4 \;a>0,\ a\neq 1} Integrals involving polynomials[edit] ∫ x e c x d x = e c x ( c x − 1 c 2 ) {\displaystyle \int xe^ π 7\;\mathrm π 6 x=e^ π 5\left({\frac π 4 π 3}}\right)} ∫ x 2 e c x d x = e c x ( x 2 c − 2 x c 2 + 2 c 3 ) {\displaystyle \int x^ ∫ 7e^ ∫ 6\;\mathrm ∫ 5 x=e^ ∫ 4\left({\frac ∫ 3} ∫ 2}-{\frac ∫ 1 ∫ 0}}+{\frac π 9 π 8}}\right)} ∫ x n e c x d x = 1 c x n e c x − n c ∫ x n − 1 e c x d x = ( ∂ ∂ c ) n e c x c = e c x ∑ i = 0 n ( − 1 ) i n ! ( n − i ) ! c i + 1 x n − i = e c x ∑ i = 0 n ( − 1 ) n − i n ! i ! c n − i + 1 x i {\displaystyle \int x^ 7e^ 6\;\mathrm 5 x={\frac 4 3}x^ 2e^ 1-{\frac
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