Stieltjes

Stieltjes refers to several concepts related to the work of Dutch mathematician Thomas Joannes Stieltjes. Most prominently, it is associated with the Stieltjes integral, a generalization of the Riemann integral. In the Stieltjes integral, the increment of the integration variable is replaced by the increment of a second function, often denoted as $\alpha(x)$. This allows for integration with respect to a more general function than just $x$, enabling the integration of functions with discontinuities or with respect to measures.

The Stieltjes integral is denoted as $\int_a^b f(x) \, d\alpha(x)$. If $\alpha(x) = x$, the Stieltjes integral reduces to the Riemann integral. The theory of Stieltjes integration has applications in various fields, including probability theory, where it is used to define cumulative distribution functions, and in the study of distributions and generalized functions.

Another important concept is the Stieltjes constants, denoted by $\gamma_n$. These are a sequence of real numbers that appear in the Laurent series expansion of the Riemann zeta function around its pole at $s=1$. The first Stieltjes constant, $\gamma_0$, is equal to the Euler-Mascheroni constant, $\gamma$. The Stieltjes constants are of interest in number theory and analysis, though their exact values are not known in closed form. Thomas Stieltjes also contributed to continued fractions and the theory of asymptotic expansions.