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Fungrim entry: 1d4638

RN ⁣(z)B2N+22N(2N+1)(2N+2)z2N{1,arg(z)π4sec2N+1 ⁣(12arg(z)),arg(z)<π\left|R_{N}\!\left(z\right)\right| \le \frac{\left|B_{2 N + 2}\right|}{2 N \left(2 N + 1\right) \left(2 N + 2\right) {\left|z\right|}^{2 N}} \begin{cases} 1, & \left|\arg(z)\right| \le \frac{\pi}{4}\\\sec^{2 N + 1}\!\left(\frac{1}{2} \arg(z)\right), & \left|\arg(z)\right| < \pi\\ \end{cases}
Assumptions:zC  and  z(,0]  and  NZ1z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}
\left|R_{N}\!\left(z\right)\right| \le \frac{\left|B_{2 N + 2}\right|}{2 N \left(2 N + 1\right) \left(2 N + 2\right) {\left|z\right|}^{2 N}} \begin{cases} 1, & \left|\arg(z)\right| \le \frac{\pi}{4}\\\sec^{2 N + 1}\!\left(\frac{1}{2} \arg(z)\right), & \left|\arg(z)\right| < \pi\\ \end{cases}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
LogBarnesGRemainderRN ⁣(z)R_{N}\!\left(z\right) Remainder term in asymptotic expansion of logarithmic Barnes G-function
BernoulliBBnB_{n} Bernoulli number
Powab{a}^{b} Power
Argarg(z)\arg(z) Complex argument
Piπ\pi The constant pi (3.14...)
CCC\mathbb{C} Complex numbers
OpenClosedInterval(a,b]\left(a, b\right] Open-closed interval
Infinity\infty Positive infinity
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
    Formula(LessEqual(Abs(LogBarnesGRemainder(N, z)), Mul(Div(Abs(BernoulliB(Add(Mul(2, N), 2))), Mul(Mul(Mul(Mul(2, N), Add(Mul(2, N), 1)), Add(Mul(2, N), 2)), Pow(Abs(z), Mul(2, N)))), Cases(Tuple(1, LessEqual(Abs(Arg(z)), Div(Pi, 4))), Tuple(Pow(Sec(Mul(Div(1, 2), Arg(z))), Add(Mul(2, N), 1)), Less(Abs(Arg(z)), Pi)))))),
    Variables(z, N),
    Assumptions(And(Element(z, CC), NotElement(z, OpenClosedInterval(Neg(Infinity), 0)), Element(N, ZZGreaterEqual(1)))),

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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