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In mathematics, a locally integrable function is a function which is integrable on any compact set.

Formally, let U be an open set in the Euclidean space Rⁿ and

<math>f\colon U\to\mathbb{C}</math>

be a Lebesgue measurable function. If the Lebesgue integral

<math> \int_K | f| dx \,</math>

is finite for all compact subsets K in U, then f is called locally integrable. The set of all such functions is denoted by

<math>L^1_{loc}(U).</math>

Examples

• Every (globally) integrable function on U is locally integrable, that is,

<math>L^1(U)\subset L^1_{loc}(U)</math> (see Lp space).

• More generally, every p-power integrable function (1 ≤ p ≤ ∞) on U is locally integrable:

<math>L^p(U)\subset L^1_{loc}(U)</math>.

• The constant function 1 defined on the real line is locally integrable but not globally integrable. More generally, continuous functions are locally integrable.
• The function <math>f(x)=1/x</math> for <math>x\neq 0</math> and <math>f(0)=0</math> is not locally integrable.

Uses

Locally integrable functions play a prominent role in distribution theory.

References

• Robert S Strichartz, A Guide to Distribution Theory and Fourier Transforms, World Scientific, 2003. ISBN 981-238-430-8.

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In microeconomics, the expenditure function describes the minimum amount of money an individual needs to achieve some level of utility, given a utility function and prices.

Formally, if there is a utility function <math>u</math> that describes preferences over L commodities, the expenditure function

<math>e(p, u^*) : \textbf R^L_+ \times \textbf R

\rightarrow \textbf R</math>

says what amount of money is needed to achieve a utility <math>u^*</math> if prices are set by <math>p</math>.
This function is defined by

<math>e(p, u^*) = \min_{x \in \geq(u^*)} p \cdot x</math>

where

<math>\geq(u^*) = \{x \in \textbf R^L_+ : u(x) \geq u^*\}</math>

is the set of all packages that give utility at least as good as <math>u^*</math>.

See also

• Expenditure minimization problem
• Hicksian demand function
• Utility maximization problem

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In microeconomics, a consumer’s Hicksian demand function <math>h(p, u)</math> gives the cheapest bundle under a price level <math>p</math> for which the consumer derives a utility level of at least <math>u</math>. The function is named after John Hicks.

Hicksian demand functions are often convenient for mathematical manipulation because they don’t require income or wealth to be represented. However, Marshallian demand functions of the form <math>x(p, w)</math> that describe demand given prices <math>p</math> and income <math>w</math> are easier to observe directly. The two are trivially related by

<math>h(p, u) = x(p, e(p, u)), \ </math>

where <math>e(p, u)</math> is the expenditure function (the function that gives the minimum wealth required to get to a given utility level), and by

<math>h(p, v(p, w)) = x(p, w), \ </math>

where <math>v(p, w)</math> is the indirect utility function (which gives the utility level of having a given wealth under a fixed price regime). Their derivatives are more fundamentally related by the Slutsky equation.

The Hicksian demand function is intimately related to the expenditure function. If the consumer’s utility function <math>u(x)</math> is locally nonsatiated and strictly convex, then
<math>h(p, u) = \nabla_p e(p, u).</math>

See also

• Marshallian demand function
• Convex preferences
• Expenditure minimization problem

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Bidirectional texture function (BTF) is a 6-dimensional function depending on planar texture coordinates (x,y) as well as on view and illumination spherical angles. In practice this function is obtained as a set of several thousands images of material sample taken during different camera and light positions. Such images are subsequently rectified and compressed using various methods.

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In mathematics, a function f(x) between two ordered sets is unimodal if for some value m (the mode), it is monotonically increasing for x ≤ m and monotonically decreasing for x ≥ m. In that case, the maximum value of f(x) is f(m) and there are no other local maxima.

Examples of unimodal function:

• Quadratic polynomial
• Logistic map
• Tent map

Function <math>\ f(x)</math> is S-unimodal if its Schwartzian derivative is negative for all <math>\ x \ne 0</math> <ref>http://web.udl.es/usuaris/y4370980/abstracts/abstracts/vol-42/de_melo.pdf W. De Melo : Bifurcation of Unimodal Maps Qualitative Theory of Dynamical Systems VOLUME 4 - Number 2 (pages 413-424) </ref>.

In probability and statistics, a unimodal probability distribution is a probability distribution whose probability density function is a unimodal function, or more generally, whose cumulative distribution function is convex up to m and concave thereafter (this allows for the possibility of a non-zero probability for x=m). For a unimodal probability distribution of a continuous random variable, the Vysochanskii-Petunin inequality provides a refinement of the Chebyshev inequality. Compare multimodal distribution.

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In mathematics, characteristic function can refer to any of several distinct concepts:

• The most common and universal usage is as a synonym for indicator function, that is the function

<math>\mathbf{1}_A: X \to \{0, 1\}</math>

which for every subset A of X, has value 1 at points of A and 0 at points of X − A.

• When applied to a natural number an effective procedure determines correctly if a natural number is or is not in the procedure’s “set”: “The characteristic function is the function that takes the value 1 for numbers in the set, and the value 0 for numbers not in the set” (cf Boolos-Burgess-Jeffrey (2002) p. 73).

• The characteristic function in convex analysis:

<math>\chi_{A} (x) := \begin{cases} 0, & x \in A; \\ + \infty, & x \not \in A. \end{cases}</math>

• The characteristic state function in statistical mechanics.

• In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question:

<math>\varphi_X(t) = \operatorname{E}\left(e^{itX}\right)\,</math>

where “E” means expected value. See characteristic function (probability theory).

• The characteristic polynomial in linear algebra.

• The Euler characteristic, a topological invariant.

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In mathematics, by sigma function one can mean one of the following:

• The sum-of-divisors function σ_a(n), an arithmetic functions.
• Weierstrass sigma function, related to elliptic functions
• Rado’s sigma function, see busy beaver.

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In mathematics, by sigma function one can mean one of the following:

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• Weierstrass sigma function, related to elliptic functions
• Rado’s sigma function, see busy beaver.

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In mathematics, the psi function can refer to either

• Dedekind psi function
• Digamma function

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A single-valued function is an emphatic term for a mathematical function in the usual sense. That is, each element of the function’s domain maps to a single, well-defined element of its range. This contrasts with a general binary relation, which can be viewed as being a multi-valued function.

Any function is a single-valued if it is continuously differentiable.

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In mathematics, the q-theta function is a type of q-series. It is given by

<math>\theta(z;q)=\prod_{n=0}^\infty (1-q^nz)(1-q^{n+1}/z)</math>

where one takes <math>0\le|q|<1.</math> It obeys the identities

<math>\theta(z;q)=\theta(q/z;q)=-z\theta(1/z;q).\,</math>

See also

• Jacobi theta function
• Ramanujan theta function

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