Fungrim entry: a41c92

\gamma_{n}\!\left(a\right) = -\frac{\pi}{2 \left(n + 1\right)} \int_{0}^{\infty} \frac{\log^{n + 1}\!\left(a - \frac{1}{2} + i x\right) + \log^{n + 1}\!\left(a - \frac{1}{2} - i x\right)}{\cosh^{2}\!\left(\pi x\right)} \, dx

Assumptions:

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > \frac{1}{2}

TeX:

\gamma_{n}\!\left(a\right) = -\frac{\pi}{2 \left(n + 1\right)} \int_{0}^{\infty} \frac{\log^{n + 1}\!\left(a - \frac{1}{2} + i x\right) + \log^{n + 1}\!\left(a - \frac{1}{2} - i x\right)}{\cosh^{2}\!\left(\pi x\right)} \, dx

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) > \frac{1}{2}

Definitions:

Fungrim symbol	Notation	Short description
`StieltjesGamma`	$\gamma_{n}\!\left(a\right)$	Stieltjes constant
`Pi`	$\pi$	The constant pi (3.14...)
`Integral`	$\int_{a}^{b} f(x) \, dx$	Integral
`Pow`	${a}^{b}$	Power
`Log`	$\log(z)$	Natural logarithm
`ConstI`	$i$	Imaginary unit
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`CC`	$\mathbb{C}$	Complex numbers
`Re`	$\operatorname{Re}(z)$	Real part

Source code for this entry:

Entry(ID("a41c92"),
    Formula(Equal(StieltjesGamma(n, a), Neg(Mul(Div(Pi, Mul(2, Add(n, 1))), Integral(Div(Add(Pow(Log(Add(Sub(a, Div(1, 2)), Mul(ConstI, x))), Add(n, 1)), Pow(Log(Sub(Sub(a, Div(1, 2)), Mul(ConstI, x))), Add(n, 1))), Pow(Cosh(Mul(Pi, x)), 2)), For(x, 0, Infinity)))))),
    Variables(n, a),
    Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Greater(Re(a), Div(1, 2)))))

Topics using this entry

Stieltjes constants