Fungrim entry: 9a06fb

\int_{z}^{\infty} {x}^{c} {e}^{-a x + b} \, dx = \frac{{e}^{b}}{{a}^{c + 1}} \Gamma\!\left(c + 1, a z\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 \gt 0 \,\mathbin{\operatorname{and}}\, c \gt 0 \,\mathbin{\operatorname{and}}\, z \gt 0

TeX:

\int_{z}^{\infty} {x}^{c} {e}^{-a x + b} \, dx = \frac{{e}^{b}}{{a}^{c + 1}} \Gamma\!\left(c + 1, a z\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 \gt 0 \,\mathbin{\operatorname{and}}\, c \gt 0 \,\mathbin{\operatorname{and}}\, z \gt 0

Definitions:

Fungrim symbol	Notation	Short description
`Pow`	${a}^{b}$	Power
`Exp`	${e}^{z}$	Exponential function
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("9a06fb"),
    Formula(Equal(Integral(Mul(Pow(x, c), Exp(Add(Neg(Mul(a, x)), b))), Tuple(x, z, Infinity)), Mul(Div(Exp(b), Pow(a, Add(c, 1))), UpperGamma(Add(c, 1), Mul(a, z))))),
    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