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Fungrim entry: 876844

limzRn ⁣(a,b,eiθz)=0\lim_{z \to \infty} \left|R_{n}\!\left(a,b,{e}^{i \theta} z\right)\right| = 0
Assumptions:aC  and  bC  and  θR  and  nZ1a \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}
\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}
Fungrim symbol Notation Short description
ComplexLimitlimzaf(z)\lim_{z \to a} f(z) Limiting value, complex variable
Absz\left|z\right| Absolute value
HypergeometricUStarRemainderRn ⁣(a,b,z)R_{n}\!\left(a,b,z\right) Error term in asymptotic expansion of Tricomi confluent hypergeometric function
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
Infinity\infty Positive infinity
CCC\mathbb{C} Complex numbers
RRR\mathbb{R} Real numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    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)))))

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2021-03-15 19:12:00.328586 UTC