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Fungrim entry: f8de2e

zeax2+bdx=eb2πaerfc ⁣(az)\int_{z}^{\infty} {e}^{-a {x}^{2} + b} \, dx = \frac{{e}^{b}}{2} \sqrt{\frac{\pi}{a}} \operatorname{erfc}\!\left(\sqrt{a} z\right)
Assumptions:aCandbCandzCandRe(a)>0a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(a) > 0
TeX:
\int_{z}^{\infty} {e}^{-a {x}^{2} + b} \, dx = \frac{{e}^{b}}{2} \sqrt{\frac{\pi}{a}} \operatorname{erfc}\!\left(\sqrt{a} z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Re}(a) > 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
Sqrtz\sqrt{z} Principal square root
Piπ\pi The constant pi (3.14...)
Erfcerfc(z)\operatorname{erfc}(z) Complementary error function
CCC\mathbb{C} Complex numbers
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
Entry(ID("f8de2e"),
    Formula(Equal(Integral(Exp(Add(Neg(Mul(a, Pow(x, 2))), b)), For(x, z, Infinity)), Mul(Mul(Div(Exp(b), 2), Sqrt(Div(Pi, a))), Erfc(Mul(Sqrt(a), z))))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Greater(Re(a), 0))))

Topics using this entry

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2020-01-31 18:09:28.494564 UTC