Fungrim entry: cb93ea

\operatorname{erf}(z) = \frac{z}{\sqrt{{z}^{2}}} - \frac{{e}^{-{z}^{2}}}{z \sqrt{\pi}} U^{*}\!\left(\frac{1}{2}, \frac{1}{2}, {z}^{2}\right)

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0

TeX:

\operatorname{erf}(z) = \frac{z}{\sqrt{{z}^{2}}} - \frac{{e}^{-{z}^{2}}}{z \sqrt{\pi}} U^{*}\!\left(\frac{1}{2}, \frac{1}{2}, {z}^{2}\right)

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Erf`	$\operatorname{erf}(z)$	Error function
`Sqrt`	$\sqrt{z}$	Principal square root
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Pi`	$\pi$	The constant pi (3.14...)
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("cb93ea"),
    Formula(Equal(Erf(z), Sub(Div(z, Sqrt(Pow(z, 2))), Mul(Div(Exp(Neg(Pow(z, 2))), Mul(z, Sqrt(Pi))), HypergeometricUStar(Div(1, 2), Div(1, 2), Pow(z, 2)))))),
    Variables(z),
    Assumptions(And(Element(z, CC), NotEqual(z, 0))))

Topics using this entry

Error functions