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Fungrim entry: 9a06fb

zxceax+bdx=ebac+1Γ ⁣(c+1,az)\int_{z}^{\infty} {x}^{c} {e}^{-a x + b} \, dx = \frac{{e}^{b}}{{a}^{c + 1}} \Gamma\!\left(c + 1, a z\right)
Assumptions:aR  and  bR  and  cR  and  zR  and  a>0  and  c>0  and  z>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
\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 > 0 \;\mathbin{\operatorname{and}}\; c > 0 \;\mathbin{\operatorname{and}}\; z > 0
Fungrim symbol Notation Short description
Integralabf(x)dx\int_{a}^{b} f(x) \, dx Integral
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
Infinity\infty Positive infinity
RRR\mathbb{R} Real numbers
Source code for this entry:
    Formula(Equal(Integral(Mul(Pow(x, c), Exp(Add(Neg(Mul(a, x)), b))), For(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))))

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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