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Fungrim entry: fae9d3

erf(n)(z)=2π(1)n+1Hn1 ⁣(z)ez2{\operatorname{erf}}^{(n)}(z) = \frac{2}{\sqrt{\pi}} {\left(-1\right)}^{n + 1} H_{n - 1}\!\left(z\right) {e}^{-{z}^{2}}
Assumptions:zC  and  nZ1z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}
{\operatorname{erf}}^{(n)}(z) = \frac{2}{\sqrt{\pi}} {\left(-1\right)}^{n + 1} H_{n - 1}\!\left(z\right) {e}^{-{z}^{2}}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}
Fungrim symbol Notation Short description
ComplexDerivativeddzf ⁣(z)\frac{d}{d z}\, f\!\left(z\right) Complex derivative
Erferf(z)\operatorname{erf}(z) Error function
Sqrtz\sqrt{z} Principal square root
Piπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
HermitePolynomialHn ⁣(z)H_{n}\!\left(z\right) Hermite polynomial
Expez{e}^{z} Exponential function
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(Equal(ComplexDerivative(Erf(z), For(z, z, n)), Mul(Mul(Mul(Div(2, Sqrt(Pi)), Pow(-1, Add(n, 1))), HermitePolynomial(Sub(n, 1), z)), Exp(Neg(Pow(z, 2)))))),
    Variables(z, n),
    Assumptions(And(Element(z, CC), Element(n, ZZGreaterEqual(1)))))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC