# 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"),
Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(a, CC), Greater(Re(a), Div(1, 2)))))