Fungrim entry: 461a54

\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 \sqrt{1 + \frac{1}{2} \pi n}}{1 - \sigma} \exp\!\left(\frac{\pi \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}}\; \left(\left|\operatorname{Im}(z)\right| > \left|b - 2 a\right| \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) > \left|b - 2 a\right|\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 \sqrt{1 + \frac{1}{2} \pi n}}{1 - \sigma} \exp\!\left(\frac{\pi \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}}\; \left(\left|\operatorname{Im}(z)\right| > \left|b - 2 a\right| \;\mathbin{\operatorname{or}}\; \operatorname{Re}(z) > \left|b - 2 a\right|\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
`Sqrt`	$\sqrt{z}$	Principal square root
`Pi`	$\pi$	The constant pi (3.14...)
`Exp`	${e}^{z}$	Exponential function
`CC`	$\mathbb{C}$	Complex numbers
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`Im`	$\operatorname{Im}(z)$	Imaginary part
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("461a54"),
    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(Mul(2, Sqrt(Add(1, Mul(Mul(Div(1, 2), Pi), n)))), Sub(1, sigma))), Exp(Div(Mul(Pi, 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)), Or(Greater(Abs(Im(z)), Abs(Sub(b, Mul(2, a)))), Greater(Re(z), Abs(Sub(b, Mul(2, a))))))),
    References("DLMF section 13.7, https://dlmf.nist.gov/13.7"))

Topics using this entry

Confluent hypergeometric functions