Fungrim entry: 6cf802

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

Assumptions:

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}

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\!\left(a\right)} {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
`HypergeometricU`	$U\!\left(a, b, z\right)$	Tricomi confluent hypergeometric function
`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("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))))

Topics using this entry

Confluent hypergeometric functions