# Fungrim entry: fae9d3

${\operatorname{erf}}^{(n)}(z) = \frac{2}{\sqrt{\pi}} {\left(-1\right)}^{n + 1} H_{n - 1}\!\left(z\right) {e}^{-{z}^{2}}$
Assumptions:$z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}$
TeX:
{\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}
Definitions:
Fungrim symbol Notation Short description
ComplexDerivative$\frac{d}{d z}\, f\!\left(z\right)$ Complex derivative
Erf$\operatorname{erf}(z)$ Error function
Sqrt$\sqrt{z}$ Principal square root
Pi$\pi$ The constant pi (3.14...)
Pow${a}^{b}$ Power
HermitePolynomial$H_{n}\!\left(z\right)$ Hermite polynomial
Exp${e}^{z}$ Exponential function
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("fae9d3"),
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)))))

## Topics using this entry

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