# 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

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC