Fungrim entry: ae3110

\operatorname{erfc}\!\left(z\right) = \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}}\, \operatorname{Re}\!\left(z\right) \gt 0

TeX:

\operatorname{erfc}\!\left(z\right) = \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}}\, \operatorname{Re}\!\left(z\right) \gt 0

Definitions:

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

Source code for this entry:

Entry(ID("ae3110"),
    Formula(Equal(Erfc(z), Mul(Div(Exp(Neg(Pow(z, 2))), Mul(z, Sqrt(ConstPi))), HypergeometricUStar(Div(1, 2), Div(1, 2), Pow(z, 2))))),
    Variables(z),
    Assumptions(And(Element(z, CC), Greater(Re(z), 0))))

Topics using this entry

Error functions