Fungrim entry: 876844

\lim_{z \to \infty} \left|R_{n}\!\left(a,b,{e}^{i \theta} z\right)\right| = 0

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \theta \in \mathbb{R} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

TeX:

\lim_{z \to \infty} \left|R_{n}\!\left(a,b,{e}^{i \theta} z\right)\right| = 0

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \theta \in \mathbb{R} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`ComplexLimit`	$\lim_{z \to a} f(z)$	Limiting value, complex variable
`Abs`	$\left\|z\right\|$	Absolute value
`HypergeometricUStarRemainder`	$R_{n}\!\left(a,b,z\right)$	Error term in asymptotic expansion of Tricomi confluent hypergeometric function
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`CC`	$\mathbb{C}$	Complex numbers
`RR`	$\mathbb{R}$	Real numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("876844"),
    Formula(Equal(ComplexLimit(Abs(HypergeometricUStarRemainder(n, a, b, Mul(Exp(Mul(ConstI, theta)), z))), For(z, Infinity)), 0)),
    Variables(a, b, theta, n),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(theta, RR), Element(n, ZZGreaterEqual(1)))))

Topics using this entry

Confluent hypergeometric functions