Stieltjes Integral

Stieltjes integral

[′stēlt·yəs ‚int·ə·grəl] (mathematics) The Stieltjes integral of a real function ƒ(x) relative to a real function g (x) of bounded variation on an interval [a,b ] is defined, analogously to the Riemann integral, as a limit of a sum of terms ƒ(a_i) [g (x_i) - g (x_i-1)] taken as partitions of the interval shrink. Denoted Also known as Riemann-Stieltjes integral.

Stieltjes Integral

a generalization of the definite integral proposed in 1894 by T. Stieltjes. In this generalization, the limit of the Riemann sums ∑f(ξ_i)(x_i – x_i-1) is replaced by the limit of the sums ∑f(ξ_i) [Φ(x_i) – Φ(x_i-1)]. where the integrating function ϕ(x) is a function of bounded variation (seeVARIATION OF A FUNCTION). If ϕ(x) is differentiable, then the Stieltjes integral can be expressed in terms of the Riemann integral (if it exists):

单词	stieltjes integral
释义	Stieltjes Integral Stieltjes integral [′stēlt·yəs ‚int·ə·grəl] (mathematics) The Stieltjes integral of a real function ƒ(x) relative to a real function g (x) of bounded variation on an interval [a,b ] is defined, analogously to the Riemann integral, as a limit of a sum of terms ƒ(a_i) [g (x_i) - g (x_i-1)] taken as partitions of the interval shrink. Denoted Also known as Riemann-Stieltjes integral. Stieltjes Integral a generalization of the definite integral proposed in 1894 by T. Stieltjes. In this generalization, the limit of the Riemann sums ∑f(ξ_i)(x_i – x_i-1) is replaced by the limit of the sums ∑f(ξ_i) [Φ(x_i) – Φ(x_i-1)]. where the integrating function ϕ(x) is a function of bounded variation (seeVARIATION OF A FUNCTION). If ϕ(x) is differentiable, then the Stieltjes integral can be expressed in terms of the Riemann integral (if it exists):
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