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Fungrim entry: 16a1f4

zeaxc+bdx=ebca1/cΓ ⁣(1c,azc)\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:aRandbRandcRandzRanda>0andc>0andz>0a \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
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Expez{e}^{z} Exponential function
Powab{a}^{b} Power
Infinity\infty Positive infinity
RRR\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))))

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

2019-11-19 15:10:20.037976 UTC