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\!\left(n\right)}{1 - \tau} \exp\!\left(\frac{2 C\!\left(1\right) \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\!\left(m\right) = \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| \gt 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\!\left(n\right)}{1 - \tau} \exp\!\left(\frac{2 C\!\left(1\right) \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\!\left(m\right) = \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| \gt 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
`ConstPi`	$\pi$	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(ConstPi, 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), Unequal(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

Confluent hypergeometric functions