Fungrim home page

Fungrim entry: 18ef23

U ⁣(a,n,z)=limbnΓ ⁣(1b)Γ ⁣(ab+1)1F1 ⁣(a,b,z)+Γ ⁣(b1)Γ(a)z1b1F1 ⁣(ab+1,2b,z)U\!\left(a, n, z\right) = \lim_{b \to n} \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:aCandnZandzCandz0a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0
TeX:
U\!\left(a, n, z\right) = \lim_{b \to n} \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}}\, n \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, z \in \mathbb{C} \,\mathbin{\operatorname{and}}\, z \ne 0
Definitions:
Fungrim symbol Notation Short description
HypergeometricUU ⁣(a,b,z)U\!\left(a, b, z\right) Tricomi confluent hypergeometric function
ComplexLimitlimzaf(z)\lim_{z \to a} f(z) Limiting value, complex variable
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("18ef23"),
    Formula(Equal(HypergeometricU(a, n, z), ComplexLimit(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))), For(b, n)))),
    Variables(a, n, z),
    Assumptions(And(Element(a, CC), Element(n, ZZ), Element(z, CC), Unequal(z, 0))))

Topics using this entry

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