Fungrim entry: f8de2e

\int_{z}^{\infty} {e}^{-a {x}^{2} + b} \, dx = \frac{{e}^{b}}{2} \sqrt{\frac{\pi}{a}} \operatorname{erfc}\!\left(\sqrt{a} z\right)

Assumptions:

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(a\right) \gt 0

TeX:

\int_{z}^{\infty} {e}^{-a {x}^{2} + b} \, dx = \frac{{e}^{b}}{2} \sqrt{\frac{\pi}{a}} \operatorname{erfc}\!\left(\sqrt{a} z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(a\right) \gt 0

Definitions:

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

Source code for this entry:

Entry(ID("f8de2e"),
    Formula(Equal(Integral(Exp(Add(Neg(Mul(a, Pow(x, 2))), b)), Tuple(x, z, Infinity)), Mul(Mul(Div(Exp(b), 2), Sqrt(Div(ConstPi, a))), Erfc(Mul(Sqrt(a), z))))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Greater(Re(a), 0))))

Topics using this entry

Definite integrals