Related Directory Categories
- Integrated Circuits (225)
- Systems Integrators (173)
- Distributors and Integrators (149)
- Holistic and Integrated Medicine (138)
- Integrators (101)
- Integrated Development Environments (95)
- Courses and Schools (78)
- System Integration (74)
- Distributors and Integrators (74)
- Structural Integration (65)
Sources
Help build the largest human-edited directory on the web.
Submit a site - Become an editor
Wikipedia. Licensed under the GNU Free Documentation License.
Are you an expert in this subject? Join the discussion and share your knowledge at Wikipedia.org.
Integrable function
From Wikipedia, the free encyclopedia
In mathematics, an integrable function is a function whose integral exists. Unless specifically stated, the integral in question is usually the Lebesgue integral. Otherwise, one can say that the function is "Riemann-integrable" (i.e., its Riemann integral exists), "Henstock-Kurzweil-integrable," etc.
Given a set X with sigma-algebra σ defined on X and a measure μ on σ, a real-valued function f:X → R is integrable if both f + and f - are measurable functions with finite Lebesgue integral. Let
be the "positive" and "negative" part of f. If f is integrable, then its integral is defined as
For a real number p ≥ 0, the function f is p-integrable if the function | f | p is integrable; for p = 1 one says absolutely integrable. The term p-summable is sometimes used as well, especially if the function f is a sequence and μ is discrete.
The L p spaces are one of the main objects of study of functional analysis.
A real- or complex-valued function of a real or complex variable is square-integrable on an interval if the integral of the square of its absolute value, over that interval, is finite. The set of all measurable functions that are square-integrable forms a Hilbert space, the so-called L2 space.
This is especially useful in quantum mechanics as wave functions must be square integrable over all space if a physically possible solution is to be obtained from the theory.