Fungrim entry: 16a1f4

\int_{z}^{\infty} {e}^{-a {x}^{c} + b} \, dx = \frac{{e}^{b}}{c {a}^{1 / c}} \Gamma\!\left(\frac{1}{c}, a {z}^{c}\right)

Assumptions:

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; b \in \mathbb{R} \;\mathbin{\operatorname{and}}\; c \in \mathbb{R} \;\mathbin{\operatorname{and}}\; z \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a > 0 \;\mathbin{\operatorname{and}}\; c > 0 \;\mathbin{\operatorname{and}}\; z > 0

TeX:

\int_{z}^{\infty} {e}^{-a {x}^{c} + b} \, dx = \frac{{e}^{b}}{c {a}^{1 / c}} \Gamma\!\left(\frac{1}{c}, a {z}^{c}\right)

a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; b \in \mathbb{R} \;\mathbin{\operatorname{and}}\; c \in \mathbb{R} \;\mathbin{\operatorname{and}}\; z \in \mathbb{R} \;\mathbin{\operatorname{and}}\; a > 0 \;\mathbin{\operatorname{and}}\; c > 0 \;\mathbin{\operatorname{and}}\; z > 0

Definitions:

Fungrim symbol	Notation	Short description
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Exp`	${e}^{z}$	Exponential function
`Pow`	${a}^{b}$	Power
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("16a1f4"),
    Formula(Equal(Integral(Exp(Add(Neg(Mul(a, Pow(x, c))), b)), For(x, z, Infinity)), Mul(Div(Exp(b), Mul(c, Pow(a, Div(1, c)))), UpperGamma(Div(1, c), Mul(a, Pow(z, c)))))),
    Variables(a, b, c, z),
    Assumptions(And(Element(a, RR), Element(b, RR), Element(c, RR), Element(z, RR), Greater(a, 0), Greater(c, 0), Greater(z, 0))))

Topics using this entry

Definite integrals