Fungrim entry: 18ef23

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\!\left(a\right)} {z}^{1 - b} \,{}_1F_1\!\left(a - b + 1, 2 - b, z\right)

Assumptions:

a \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\!\left(a\right)} {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
`HypergeometricU`	$U\!\left(a, b, z\right)$	Tricomi confluent hypergeometric function
`ComplexLimit`	$\lim_{z \to a} f\!\left(z\right)$	Limiting value, complex variable
`GammaFunction`	$\Gamma\!\left(z\right)$	Gamma function
`Hypergeometric1F1`	$\,{}_1F_1\!\left(a, b, z\right)$	Kummer confluent hypergeometric function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	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))), b, n))),
    Variables(a, n, z),
    Assumptions(And(Element(a, CC), Element(n, ZZ), Element(z, CC), Unequal(z, 0))))

Topics using this entry

Confluent hypergeometric functions