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Fungrim entry: 6cf802

U ⁣(a,b,z)=Γ ⁣(1b)Γ ⁣(ab+1)1F1 ⁣(a,b,z)+Γ ⁣(b1)Γ(a)z1b1F1 ⁣(ab+1,2b,z)U\!\left(a, b, z\right) = \frac{\Gamma\!\left(1 - b\right)}{\Gamma\!\left(a - b + 1\right)} \,{}_1F_1\!\left(a, b, z\right) + \frac{\Gamma\!\left(b - 1\right)}{\Gamma(a)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)
Assumptions:aCandbCandzCandz0andbZa \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, b \notin \mathbb{Z}
TeX:
U\!\left(a, b, z\right) = \frac{\Gamma\!\left(1 - b\right)}{\Gamma\!\left(a - b + 1\right)} \,{}_1F_1\!\left(a, b, z\right) + \frac{\Gamma\!\left(b - 1\right)}{\Gamma(a)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0 \,\mathbin{\operatorname{and}}\, b \notin \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
HypergeometricUU ⁣(a,b,z)U\!\left(a, b, z\right) Tricomi confluent hypergeometric function
GammaFunctionΓ(z)\Gamma(z) Gamma function
Hypergeometric1F11F1 ⁣(a,b,z)\,{}_1F_1\!\left(a, b, z\right) Kummer confluent hypergeometric function
Powab{a}^{b} Power
CCC\mathbb{C} Complex numbers
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("6cf802"),
    Formula(Equal(HypergeometricU(a, b, z), Add(Mul(Div(GammaFunction(Sub(1, b)), GammaFunction(Add(Sub(a, b), 1))), Hypergeometric1F1(a, b, z)), Mul(Mul(Div(GammaFunction(Sub(b, 1)), GammaFunction(a)), Pow(z, Sub(1, b))), Hypergeometric1F1(Add(Sub(a, b), 1), Sub(2, b), z))))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), Unequal(z, 0), NotElement(b, ZZ))))

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

2019-10-05 13:11:19.856591 UTC