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Fungrim entry: 7b91b4

Rn ⁣(a,b,z)(a)n(ab+1)nn!zn2C(n)1τexp ⁣(2C(1)ρ(1τ)z)   where σ=b2az,ν=1+2σ2,τ=νσ,ρ=a2ab+b2+τ(1+τ4)(1τ)2,C(m)=(1+πm2+σν2m)νm\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:aCandbCandzCandz0andnZ0andz>2b2aa \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
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
RisingFactorial(z)k\left(z\right)_{k} Rising factorial
Factorialn!n ! Factorial
Powab{a}^{b} Power
Expez{e}^{z} Exponential function
Sqrtz\sqrt{z} Principal square root
ConstPiπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\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"))

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

2019-10-05 13:11:19.856591 UTC