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Fungrim entry: 461a54

Rn ⁣(a,b,z)(a)n(ab+1)nn!zn21+12πn1σexp ⁣(πρ(1σ)z)   where σ=b2az,  ρ=a2ab+b2+σ(1+σ4)(1σ)2\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:aC  and  bC  and  zC  and  z0  and  nZ0  and  (Im(z)>b2a  or  Re(z)>b2a)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)
  • DLMF section 13.7,
\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)
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
Sqrtz\sqrt{z} Principal square root
Piπ\pi The constant pi (3.14...)
Expez{e}^{z} Exponential function
CCC\mathbb{C} Complex numbers
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ImIm(z)\operatorname{Im}(z) Imaginary part
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
    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,"))

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2021-03-15 19:12:00.328586 UTC