Fungrim entry: f7f84e

\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma\!\left(b - a\right)} U^{*}\!\left(a, b, z\right) + \frac{{z}^{a - b} {e}^{z}}{\Gamma(a)} U^{*}\!\left(b - a, 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

TeX:

\,{}_1{\textbf F}_1\!\left(a, b, z\right) = \frac{{\left(-z\right)}^{-a}}{\Gamma\!\left(b - a\right)} U^{*}\!\left(a, b, z\right) + \frac{{z}^{a - b} {e}^{z}}{\Gamma(a)} U^{*}\!\left(b - a, 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

Definitions:

Fungrim symbol	Notation	Short description
`Hypergeometric1F1Regularized`	$\,{}_1{\textbf F}_1\!\left(a, b, z\right)$	Regularized Kummer confluent hypergeometric function
`Pow`	${a}^{b}$	Power
`Gamma`	$\Gamma(z)$	Gamma function
`HypergeometricUStar`	$U^{*}\!\left(a, b, z\right)$	Scaled Tricomi confluent hypergeometric function
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("f7f84e"),
    Formula(Equal(Hypergeometric1F1Regularized(a, b, z), Add(Mul(Div(Pow(Neg(z), Neg(a)), Gamma(Sub(b, a))), HypergeometricUStar(a, b, z)), Mul(Div(Mul(Pow(z, Sub(a, b)), Exp(z)), Gamma(a)), HypergeometricUStar(Sub(b, a), b, Neg(z)))))),
    Variables(a, b, z),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), NotEqual(z, 0))))

Topics using this entry

Confluent hypergeometric functions