Fungrim entry: 1d4638

\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:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; z \notin \left(-\infty, 0\right] \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

References:

https://dx.doi.org/10.1098/rspa.2014.0534

TeX:

\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}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`LogBarnesGRemainder`	$R_{N}\!\left(z\right)$	Remainder term in asymptotic expansion of logarithmic Barnes G-function
`BernoulliB`	$B_{n}$	Bernoulli number
`Pow`	${a}^{b}$	Power
`Arg`	$\arg(z)$	Complex argument
`Pi`	$\pi$	The constant pi (3.14...)
`CC`	$\mathbb{C}$	Complex numbers
`OpenClosedInterval`	$\left(a, b\right]$	Open-closed interval
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("1d4638"),
    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)))),
    References("https://dx.doi.org/10.1098/rspa.2014.0534"))

Topics using this entry

Barnes G-function