Darboux integral
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In real analysis, a branch of mathematics, the Darboux integral is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. Darboux integrals have the advantage of being simpler to define than Riemann integrals. Darboux integrals are named after their discoverer: Gaston Darboux.
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Definition
A partition of an interval [a, b] is a finite sequence
- a = x0 < x1 < x2 < ... < xn = b.
Each [xi, xi+1] is called a subinterval of the partition. A refinement of the partition
- x0, ..., xn
is a partition
- y0, ..., ym
such that for every i with
- 0≤i ≤ n,
there is an integer r(i) such that
- xi=yr(i).
In other words, to make a refinement, one cuts the subintervals into smaller pieces and does not remove any cuts.
Let f:[a,b]→R be a bounded function, and let
- x0, ..., xn
be a partition of [a, b]. Let:
- <math>M_i = \sup_{x\in[x_i,x_{i+1}]} f(x)<math>
- <math>m_i = \inf_{x\in[x_i,x_{i+1}]} f(x)<math>
The upper Darboux sum of f with respect to x0, ..., xn is
- <math>U_{f, x_0,\ldots,x_n} = \sum_{i=0}^n M_i (x_{i+1}-x_i)<math>
The lower Darboux sum of f with respect to x0, ..., xn is
- <math>L_{f, x_0,\ldots,x_n} = \sum_{i=0}^n m_i (x_{i+1}-x_i)<math>
The upper Darboux integral of f is
- <math>U_f = \inf_{x_0,\ldots,x_n} U_{f, x_0,\ldots,x_n}<math>
The lower Darboux integral of f is
- <math>L_f = \sup_{x_0,\ldots,x_n} L_{f, x_0,\ldots,x_n}<math>
If Uf = Lf, then we say that f is Darboux-integrable and set ∫f to be the common value of the upper and lower Darboux integrals.
Facts about the Darboux integral
If
- y0, ..., ym
is a refinement of
- x0, ..., xn,
then
- <math>U_{f, x_0,\ldots,x_n} \ge U_{f, y_0,\ldots,y_m}<math>
and
- <math>L_{f, x_0,\ldots,x_n} \le L_{f, y_0,\ldots,y_m}<math>
If
- x0, ..., xn and
- y0, ..., ym
are two partitions (one need not be a refinement of the other), then
- <math>L_{f, x_0,\ldots,x_n} \le U_{f, y_0,\ldots,y_m}<math>.
It follows that
- Lf ≤ Uf.
Riemann sums always lie between the corresponding lower and upper Darboux sums. Formally, if
- x0, ..., xn
and
- t0,...,tm-1
together make a tagged partition (as in the definition of the Riemann integral), and if the Riemann sum of f corresponding to xn and
- t0,...,tm-1
is R, then
- <math>L_{f, x_0,\ldots,x_n} \le R \le U_{f, x_0,\ldots,x_n}<math>.
From the previous fact, Riemann integrals are at least as strong as Darboux integrals: If the Darboux integral exists, then the upper and lower Darboux sums corresponding to a sufficiently fine partition will be close to the value of the integral, so any Riemann sum over the same partition will also be close to the value of the integral. It is not hard to see that there is a tagged partition that comes arbitrarily close to the value of the upper Darboux integral or lower Darboux integral, and consequently, if the Riemann integral exists, then the Darboux integral must exist as well.
See also
External link
- Article on the Darboux integral by the Science Fair Project Encyclopedia (http://www.all-science-fair-projects.com/science_fair_projects_encyclopedia/Darboux_integral)lt:Darbu sumos