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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:aC  and  bC  and  zC  and  z0  and  bZa \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
GammaΓ(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(Gamma(Sub(1, b)), Gamma(Add(Sub(a, b), 1))), Hypergeometric1F1(a, b, z)), Mul(Mul(Div(Gamma(Sub(b, 1)), Gamma(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), NotEqual(z, 0), NotElement(b, ZZ))))

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2020-04-08 16:14:44.404316 UTC