# Fungrim entry: 279e4f

$\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2}{1 - \sigma} \exp\!\left(\frac{2 \rho}{\left(1 - \sigma\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\;\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\sigma \left(1 + \frac{\sigma}{4}\right)}{{\left(1 - \sigma\right)}^{2}}$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0 \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > \left|b - 2 a\right|$
References:
• DLMF section 13.7, https://dlmf.nist.gov/13.7
TeX:
\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2}{1 - \sigma} \exp\!\left(\frac{2 \rho}{\left(1 - \sigma\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\;\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\sigma \left(1 + \frac{\sigma}{4}\right)}{{\left(1 - \sigma\right)}^{2}}

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}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(z) > \left|b - 2 a\right|
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
HypergeometricUStarRemainder$R_{n}\!\left(a,b,z\right)$ Error term in asymptotic expansion of Tricomi confluent hypergeometric function
RisingFactorial$\left(z\right)_{k}$ Rising factorial
Factorial$n !$ Factorial
Pow${a}^{b}$ Power
Exp${e}^{z}$ Exponential function
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("279e4f"),
Formula(Where(LessEqual(Abs(HypergeometricUStarRemainder(n, a, b, z)), Mul(Mul(Abs(Div(Mul(RisingFactorial(a, n), RisingFactorial(Add(Sub(a, b), 1), n)), Mul(Factorial(n), Pow(z, n)))), Div(2, Sub(1, sigma))), Exp(Div(Mul(2, rho), Mul(Sub(1, sigma), Abs(z)))))), Equal(sigma, Div(Abs(Sub(b, Mul(2, a))), Abs(z))), Equal(rho, Add(Abs(Add(Sub(Pow(a, 2), Mul(a, b)), Div(b, 2))), Div(Mul(sigma, Add(1, Div(sigma, 4))), Pow(Sub(1, sigma), 2)))))),
Variables(a, b, z, n),
Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), NotEqual(z, 0), Element(n, ZZGreaterEqual(0)), Greater(Re(z), Abs(Sub(b, Mul(2, a)))))),
References("DLMF section 13.7, https://dlmf.nist.gov/13.7"))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC