Stieltjes Integral
The Stieltjes integral is a generalization of the Riemann integral. Let f(x) and alpha(x) be real-valued bounded functions defined on a closed interval [a,b]. Take a partition of the interval
| a=x_0<x_1<x_2,...<x_(n-1)<x_n=b, |
(1)
|
and consider the Riemann sum
![画像: sum_(i=0)^(n-1)f(xi_i)[alpha(x_(i+1))-alpha(x_i)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/StieltjesIntegral/NumberedEquation2.svg){kind=link}
with xi_i in [x_i,x_(i+1)]. If the sum tends to a fixed number I as max(x_(i+1)-x_i)->0, then I is called the Stieltjes integral, or sometimes the Riemann-Stieltjes integral. The Stieltjes integral of f with respect to alpha is denoted
| [画像: intf(x)dalpha(x) ] |
(3)
|
or sometimes simply
| [画像: intfdalpha. ] |
(4)
|
If f and alpha have a common point of discontinuity, then the integral does not exist. However, if f is continuous and alpha^' is Riemann integrable over the specified interval, then
(Kestelman 1960).
For enumeration of many properties of the Stieltjes integral, see Dresher (1981, p. 105).
See also
Convolution, Riemann IntegralExplore with Wolfram|Alpha
References
Dresher, M. The Mathematics of Games of Strategy: Theory and Applications. New York: Dover, 1981.Hardy, G. H.; Littlewood, J. E.; and Pólya, G. Inequalities, 2nd ed. Cambridge, England: Cambridge University Press, pp. 152-155, 1988.Jeffreys, H. and Jeffreys, B. S. "Integration: Riemann, Stieltjes." §1.10 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 26-36, 1988.Kestelman, H. "Riemann-Stieltjes Integration." Ch. 11 in Modern Theories of Integration, 2nd rev. ed. New York: Dover, pp. 247-269, 1960.Pollard, S. Quart. J. Math. 49, 73-138, 1923.Stieltjes, T. J. "Recherches sur les fractions continues." Ann. d. fac. d. sciences Toulouse 8, No. 4, J1-J122, 1894.Widder, D. V. Ch. 1 in The Laplace Transform. Princeton, NJ: Princeton University Press, 1941.Referenced on Wolfram|Alpha
Stieltjes IntegralCite this as:
Weisstein, Eric W. "Stieltjes Integral." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StieltjesIntegral.html