# Fungrim entry: 7b91b4

$\left|R_{n}\!\left(a,b,z\right)\right| \le \left|\frac{\left(a\right)_{n} \left(a - b + 1\right)_{n}}{n ! {z}^{n}}\right| \frac{2 C(n)}{1 - \tau} \exp\!\left(\frac{2 C(1) \rho}{\left(1 - \tau\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\;\nu = 1 + 2 {\sigma}^{2},\;\tau = \nu \sigma,\;\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\tau \left(1 + \frac{\tau}{4}\right)}{{\left(1 - \tau\right)}^{2}},\;C(m) = \left(\sqrt{1 + \frac{\pi m}{2}} + \sigma {\nu}^{2} m\right) {\nu}^{m}$
Assumptions:$a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \ne 0 \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; \left|z\right| > 2 \left|b - 2 a\right|$
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 C(n)}{1 - \tau} \exp\!\left(\frac{2 C(1) \rho}{\left(1 - \tau\right) \left|z\right|}\right)\; \text{ where } \sigma = \frac{\left|b - 2 a\right|}{\left|z\right|},\;\nu = 1 + 2 {\sigma}^{2},\;\tau = \nu \sigma,\;\rho = \left|{a}^{2} - a b + \frac{b}{2}\right| + \frac{\tau \left(1 + \frac{\tau}{4}\right)}{{\left(1 - \tau\right)}^{2}},\;C(m) = \left(\sqrt{1 + \frac{\pi m}{2}} + \sigma {\nu}^{2} m\right) {\nu}^{m}

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}}\; \left|z\right| > 2 \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
Sqrt$\sqrt{z}$ Principal square root
Pi$\pi$ The constant pi (3.14...)
CC$\mathbb{C}$ Complex numbers
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("7b91b4"),
Formula(Where(LessEqual(Abs(HypergeometricUStarRemainder(n, a, b, z)), Mul(Abs(Div(Mul(RisingFactorial(a, n), RisingFactorial(Add(Sub(a, b), 1), n)), Mul(Factorial(n), Pow(z, n)))), Mul(Div(Mul(2, C(n)), Sub(1, tau)), Exp(Div(Mul(Mul(2, C(1)), rho), Mul(Sub(1, tau), Abs(z))))))), Equal(sigma, Div(Abs(Sub(b, Mul(2, a))), Abs(z))), Equal(nu, Add(1, Mul(2, Pow(sigma, 2)))), Equal(tau, Mul(nu, sigma)), Equal(rho, Add(Abs(Add(Sub(Pow(a, 2), Mul(a, b)), Div(b, 2))), Div(Mul(tau, Add(1, Div(tau, 4))), Pow(Sub(1, tau), 2)))), Equal(C(m), Mul(Add(Sqrt(Add(1, Div(Mul(Pi, m), 2))), Mul(Mul(sigma, Pow(nu, 2)), m)), Pow(nu, m))))),
Variables(a, b, z, n),
Assumptions(And(Element(a, CC), Element(b, CC), Element(z, CC), NotEqual(z, 0), Element(n, ZZGreaterEqual(0)), Greater(Abs(z), Mul(2, 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