The most frequent appearance of \delta is in the following integrals: : \delta = \int_0^\infty\ln(1+x)e^{-x}dx : \delta = \int_0^1\frac{1}{1-\ln(x)}dx which follow from the definition of by integration of parts and a variable substitution respectively. Applying the Taylor expansion of \operatorname{Ei} we have the series representation : \delta = -e\left(\gamma+\sum_{n=1}^\infty\frac{(-1)^n}{n\cdot n!}\right). Gompertz's constant is connected to the
Gregory coefficients via the 2013 formula of I. Mező: : \delta = \sum_{n=0}^\infty\frac{\ln(n+1)}{n!}-\sum_{n=0}^\infty C_{n+1}\{e\cdot n!\}-\frac{1}{2}. The Gompertz constant also happens to be the regularized value of the summation of alternating factorials of all natural numbers (
1 − 1 + 2 − 6 + 24 − 120 + ⋯), which is defined by
Borel summation: : \delta = \sum_{k=0}^{\infty} (-1)^k k! It is also related to several polynomial
continued fractions: : \frac1\delta = 2-\cfrac{1^2}{4-\cfrac{2^2}{6-\cfrac{3^2}{8-\cfrac{4^2}{\ddots \cfrac{n^2}{2(n+1)-\dots}}}}} : \frac1\delta = 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{2}{1+\cfrac{2}{1+\cfrac{3}{1+\cfrac{3}{1+\cfrac{4}{\dots}}}}}}} : \frac{1}{1-\delta} = 3-\cfrac{2}{5-\cfrac{6}{7-\cfrac{12}{9-\cfrac{20}{\ddots \cfrac{n(n+1)}{2n+3-\dots}}}}} ==Notes==