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A necessary and sufficient condition that there should exist at least one nondecreasing function such thatfor , 1, 2, ..., with all the integrals converging, is that sequence is positive definite (Widder 1941, p. 129).

A function or transformation in which does not overlap .In modular function theory, a function is called univalent on a subgroup if it is automorphic under and valence 1 (Apostol 1997).

Any symmetric polynomial (respectively, symmetric rational function) can be expressed as a polynomial (respectively, rational function) in the elementary symmetric polynomials on those variables.There is a generalization of this theorem to polynomial invariants of permutation groups , which states that any polynomial invariant can be represented as a finite linear combination of special -invariant orbit polynomials with symmetric functions as coefficients, i.e.,where ,and , ..., are elementary symmetric functions, and , ..., are special terms. Furthermore, any special term has a total degree , and a maximal variable degree .

A pair of functions and are orthonormal if they are orthogonal and each normalized so that(1)(2)These two conditions can be succinctly written as(3)where is a weighting function and is the Kronecker delta.

Two functions and are orthogonal over the interval with weighting function if(1)If, in addition,(2)(3)the functions and are said to be orthonormal.

is the collection of all real-valued continuous functions defined on some interval . is the collection of all functions with continuous th derivatives. A function space is a topological vector space whose "points" are functions.

Let be a set. An operation on is a function from a power of into . More precisely, given an ordinal number , a function from into is an -ary operation on . If is a finite ordinal, then the -ary operation is a finitary operation on .

A univariate function is said to be odd provided that . Geometrically, such functions are symmetric about the origin. Examples of odd functions include , , the sine , hyperbolic sine , tangent , hyperbolic tangent , error function erf , inverse erf , and the Fresnel integrals , and .An even function times an odd function is odd, and the product of two odd functions is even while the sum or difference of two nonzero functions is odd if and only if each summand function is odd. The product and quotient of two odd functions is an even function.If an even function is differentiable, then its derivative is an odd function; what's more, if an odd function is integrable, then its integral over a symmetric interval , , is identically zero. Similarly, if an even function is differentiable, then its derivative is an odd function while the integral of such a function over a symmetric interval is twice the value of its integral over the interval .Ostensibly, one can define..

By analogy with the geometric centroid, the centroid of an arbitrary function is defined as(1)where the integrals are taken over the domain of . For example, for the Gaussian function , the centroid is(2)If is normalized so that(3)then its centroid is equivalent to its mean.

A symmetric function on variables , ..., is a function that is unchanged by any permutation of its variables. In most contexts, the term "symmetric function" refers to a polynomial on variables with this feature (more properly called a "symmetric polynomial"). Another type of symmetric functions is symmetric rational functions, which are the rational functions that are unchanged by permutation of variables.The symmetric polynomials (respectively, symmetric rational functions) can be expressed as polynomials (respectively, rational functions) in the elementary symmetric polynomials. This is called the fundamental theorem of symmetric functions.A function is sometimes said to be symmetric about the y-axis if . Examples of such functions include (the absolute value) and (the parabola)...

A function is a relation that uniquely associates members of one set with members of another set. More formally, a function from to is an object such that every is uniquely associated with an object . A function is therefore a many-to-one (or sometimes one-to-one) relation. The set of values at which a function is defined is called its domain, while the set of values that the function can produce is called its range. Here, the set is called the codomain of .In the context of univariate, real-valued functions , the fact that domain elements are mapped to unique range elements can be expressed graphically by way of the vertical line test.In some literature, the term "map" is synonymous with function. Some caution must be exhibited, however, as it is not uncommon for the term map to denote a function with some sort of unspoken regularity assumption, e.g., in point-set topology, where "map" sometimes refers to a function which is continuous..

Let be a function defined on a set and taking values in a set . Then is said to be a surjection (or surjective map) if, for any , there exists an for which . A surjection is sometimes referred to as being "onto."Let the function be an operator which maps points in the domain to every point in the range and let be a vector space with . Then a transformation defined on is a surjection if there is an such that for all .In the categories of sets, groups, modules, etc., an epimorphism is the same as a surjection, and is used synonymously with "surjection" outside of category theory.

A function is said to be strictly increasing on an interval if for all , where . On the other hand, if for all , the function is said to be (nonstrictly) increasing.

A function is said to be strictly decreasing on an interval if for all , where . On the other hand, if for all , the function is said to be (nonstrictly) decreasing.

Let , then the negative part of is the function defined byNote that the negative part is itself a nonnegative function. The negative part satisfies the identitywhere is the positive part of .

Consider a power series in a complex variable (1)that is convergent within the open disk . Convergence is limited to within by the presence of at least one singularity on the boundary of . If the singularities on are so densely packed that analytic continuation cannot be carried out on a path that crosses , then is said to form a natural boundary (or "natural boundary of analyticity") for the function .As an example, consider the function(2)Then formally satisfies the functional equation(3)The series (◇) clearly converges within . Now consider . Equation (◇) tells us that which can only be satisfied if . Considering now , equation (◇) becomes and hence . Substituting for in equation (◇) then gives(4)from which it follows that(5)Now consider equal to any of the fourth roots of unity, , , for example . Then . Applying this procedure recursively shows that is infinite for any such that with , 1, 2, .... In any arc of..

A univariate function is said to be even provided that . Geometrically, such functions are symmetric about the -axis. Examples of even functions include 1 (or, in general, any constant function), , , , and .An even function times an odd function is odd, while the sum or difference of two nonzero functions is even if and only if each summand function is even. The product or quotient of two even functions is again even.If a univariate even function is differentiable, then its derivative is an odd function; what's more, if an even function is integrable, then its integral over a symmetric interval , , is precisely the same as twice the integral over the interval . Similarly, if an odd function is differentiable, then its derivative is an even function while the integral of such a function over a symmetric interval is identically zero.Ostensibly, one can define a similar notion for multivariate functions by saying that such a function is even if and only ifEven..

Let be a homogeneous function of order so that(1)Then define and . Then(2)(3)(4)Let , then(5)This can be generalized to an arbitrary number of variables(6)where Einstein summation has been used.

A multivalued function, also known as a multiple-valued function (Knopp 1996, part 1 p. 103), is a "function" that assumes two or more distinct values in its range for at least one point in its domain. While these "functions" are not functions in the normal sense of being one-to-one or many-to-one, the usage is so common that there is no way to dislodge it. When considering multivalued functions, it is therefore necessary to refer to usual "functions" as single-valued functions.While the trigonometric, hyperbolic, exponential, and integer power functions are all single-valued functions, their inverses are multivalued. For example, the function maps each complex number to a well-defined number , while its inverse function maps, for example, the value to . While a unique principal value can be chosen for such functions (in this case, the principal square root is the positive one), the choices cannot..

A function built up of a finite combination of constant functions, field operations (addition, multiplication, division, and root extractions--the elementary operations)--and algebraic, exponential, and logarithmic functions and their inverses under repeated compositions (Shanks 1993, p. 145; Chow 1999). Among the simplest elementary functions are the logarithm, exponential function (including the hyperbolic functions), power function, and trigonometric functions.Following Liouville (1837, 1838, 1839), Watson (1966, p. 111) defines the elementarytranscendental functions as(1)(2)(3)and lets , etc.Not all functions are elementary. For example, the normaldistribution function(4)(5)is a notorious example of a nonelementary function, where is erf (sometimes known as the error function). The elliptic integral(6)is another, where is an elliptic integral of the first kind...

A positive definite function on a group is a function for which the matrix is always positive semidefinite Hermitian.

The function from a given nonempty set to the power set that maps every element of to the set .

A function is called multiplicative if (i.e., the statement that and are relatively prime) implies(Wilf 1994, p. 58).Examples of multiplicative functions are the Möbiusfunction and totient function.

A function is said to be doubly periodic if it has two periods and whose ratio is not real. A doubly periodic function that is analytic (except at poles) and that has no singularities other than poles in the finite part of the plane is called an elliptic function (Whittaker and Watson 1990, p. 429). The periods and play the same part in the theory of elliptic functions as does the single period in the case of the trigonometric functions.Jacobi (1835) proved that if a univariate single-valued function is doubly periodic, then the ratio of periods cannot be real, as well as the impossibility for a single-valued univariate function to have more than two distinct periods (Boyer and Merzbach 1991, p. 525).

Order the natural numbers as follows:Now let be a continuous function from the reals to the reals and suppose in the above ordering. Then if has a point of least period , then also has a point of least period .A special case of this general result, also known as Sharkovsky's theorem, states that if a continuous real function has a periodic point with period 3, then there is a periodic point of period for every integer .A converse to Sharkovsky's theorem says that if in the above ordering, then we can find a continuous function which has a point of least period , but does not have any points of least period (Elaydi 1996). For example, there is a continuous function with no points of least period 3 but having points of all other least periods.Sharkovsky's theorem includes the period threetheorem as a special case (Borwein and Bailey 2003, p. 79)...

is semianalytic if, for all , there is an open neighborhood of such that is a finite Boolean combination of sets and , where are analytic.

A function decreases on an interval if for all , where . If for all , the function is said to be strictly decreasing.Conversely, a function increases on an interval if for all with . If for all , the function is said to be strictly increasing.If the derivative of a continuous function satisfies on an open interval , then is decreasing on . However, a function may decrease on an interval without having a derivative defined at all points. For example, the function is decreasing everywhere, including the origin , despite the fact that the derivative is not defined at that point.

Let and be any two real integrable functions in , then Schwarz's inequality is given by(1)Written out explicitly(2)with equality iff with a constant. Schwarz's inequality is sometimes also called the Cauchy-Schwarz inequality (Gradshteyn and Ryzhik 2000, p. 1099) or Buniakowsky inequality (Hardy et al. 1952, p. 16).To derive the inequality, let be a complex function and a complex constant such that for some and . Since , where is the complex conjugate,(3)with equality when . Writing this in compact notation,(4)Now define(5)(6)Multiply (4) by and then plug in (5) and (6) to obtain(7)which simplifies to(8)so(9)Bessel's inequality follows from Schwarz'sinequality.

The conditional intensity associated to a temporal point process is defined to be the expected infinitesimal rate at which events are expected to occur around time given the history of at times prior to time . Algebraically,provided the limit exists where here, is the history of over all times strictly prior to time .

A function is called a Schwartz function if it goes to zero as faster than any inverse power of , as do all its derivatives. That is, a function is a Schwartz function if there exist real constants such thatwhere multi-index notation has been used for and .The set of all Schwartz functions is called a Schwartz space and is denoted by . It can also be proven that the Fourier transform gives a one-to-one and onto correspondence between and , where the pointwise product is taken into the convolution product and vice versa. The Fourier transform has a fixed point in , which is the function , the Gaussian function. Its image under the Fourier transform is the function (times some factors of ).Instead of , one can also consider . It consists of functions that go to zero, as , faster than any inverse power of (). It is well known that the Fourier transform carries onto , where is the -torus, defined as the direct product of copies of the circle ...

Consider the local behavior of a map by choosing a point and an open neighborhood such that . Now consider the set of all mappings It is possible to put these mappings into categories by introducing an equivalence relation. Given two mappings and , write provided there exists a neighborhood of such that and and the restriction coincides with These equivalence classes are called map germs and members are called representatives of the germ. It follows from this that , hence it is common to write for the germ where .Consider a map germ , which is an equivalence class of maps agreeing in a small neighborhood of the origin. The group of germs of diffeomorphisms is denoted , whereas the is denoted . These give coordinate changes in the source and target respectively.Denote the space of all analytic map germs by The group acts on in a standard way. Let , , and . Then .The orbit of under this action isIf and , then for some and . This is the same as and means the above diagram..

A map is a way of associating unique objects to every element in a given set. So a map from to is a function such that for every , there is a unique object . The terms function and mapping are synonymous for map.

Min Max Min Max Re Im The modulus of a complex number , also called the complex norm, is denoted and defined by(1)If is expressed as a complex exponential (i.e., a phasor), then(2)The complex modulus is implemented in the Wolfram Language as Abs[z], or as Norm[z].The square of is sometimes called the absolute square.Let and be two complex numbers. Then(3)(4)so(5)Also,(6)(7)so(8)and, by extension,(9)The only functions satisfying identities of the form(10)are , , and (Robinson 1957).

A complex map is a map . The following table lists several common types of complex maps.mapformuladomaincomplex magnification, complex rotationcomplex translationinversionlinear fractional transformationmagnification+rotation

A completely multiplicative function, sometimes known as linear or totally multiplicative function, is an arithmetic function such thatholds for each pair of positive integers .

A function is logarithmically convex on the interval if and is convex on . If and are logarithmically convex on the interval , then the functions and are also logarithmically convex on . The definition can also be extended to functions (Dharmadhikari and Joag-Dev 1988, p. 18).

A function is logarithmically concave on the interval if and is concave on . The definition can also be extended to functions (Dharmadhikari and Joag-Dev 1988, p. 18).

A set of orthogonal functions is termed complete in the closed interval if, for every piecewise continuous function in the interval, the minimum square error(where denotes the L2-norm with respect to a weighting function ) converges to zero as becomes infinite. Symbolically, a set of functions is complete ifwhere the above integral is a Lebesgue integral.Examples of complete orthogonal systems include over (which actually form a slightly more special type of system known as a complete biorthogonal system), the Legendre polynomials over (Kaplan 1992, p. 512), and on , where is a Bessel function of the first kind and is its th root (Kaplan 1992, p. 514). These systems lead to the Fourier series, Fourier-Legendre series, and Fourier-Bessel series, respectively.

Let there be two particularly well-behaved functions and . If the limitexists, then is a regular sequence of particularly well-behaved functions.

A function is completely convex in an open interval if it has derivatives of all orders there and iffor , 1, 2, ... in that interval (Widder 1941, p. 177). For example, the functions and are completely convex in the intervals and respectively.

A reflection relation is a functional equation relating to , or more generally, to .Perhaps the best known example of a reflection formula is the gammafunction identity(1)originally discovered by Euler (Havil 2003, pp. 58-59).The reflection relation for the Riemann zeta function is given by(2)where(3)and is the gamma function, as first suggested by Euler in 1761 (Havil 2003, p. 193).The xi-function has the reflection relation(4)(Havil 2003, p. 203).The Barnes G-function satisfies(5)The Rogers L-function satisfies(6)The tau Dirichlet series satisfies the reflection relation(7)(Hardy 1999, p. 173).

Let be a differential field with constant field . For , suppose that the equation (i.e., ) has a solution , where is an elementary extension of having the same constant field . Then there exist , , ..., and constants , ..., such thatIn other words, such that

A set of functions is termed a complete biorthogonal system in the closed interval if, they are biorthogonal, i.e.,(1)(2)(3)(4)(5)and complete.A complete biorthogonal system has a very special type of generalized Fourier series. The prototypical example of a complete biorthogonal system is over , which can be used as a basis for constructing "the" Fourier series of an arbitrary function.

The functions , , ..., are linearly dependent if, for some , , ..., not all zero,(1)for all in some interval . If the functions are not linearly dependent, they are said to be linearly independent. Now, if the functions and in (the space of functions with continuous derivatives), we can differentiate (1) up to times. Therefore, linear dependence also requires(2)(3)(4)where the sums are over , ..., . These equations have a nontrivial solution iff the determinant(5)where the determinant is conventionally called the Wronskian and is denoted .If the Wronskian for any value in the interval , then the only solution possible for (2) is (, ..., ), and the functions are linearly independent. If, on the other hand, over some range, then the functions are linearly dependent somewhere in the range. This is equivalent to stating that if the vectors , ..., defined by(6)are linearly independent for at least one , then the functions are linearly independent in ...

A map between topological spaces that maps closed sets to closed sets. If is bijective, thenwhere denotes the inverse map. In particular, a homeomorphism can be characterized as a continuous bijection which is open (or, equivalently, closed).

The smallest for which a point is a periodic point of a function so that . For example, for the function , all points have period 2 (including ). However, has a least period of 1. The analogous concept exists for a periodic sequence, but not for a periodic function. The least period is also called the exact period.

If is a map (a.k.a. function, transformation, etc.) over a domain , then the range of , also called the image of under , is defined as the set of all values that can take as its argument varies over , i.e.,Note that among mathematicians, the word "image"is used more commonly than "range."The range is a subset of and does not have to be all of .Unfortunately, term "range" is often used to mean domain--its precise opposite--in probability theory, with Feller (1968, p. 200) and Evans et al. (2000, p. 5) calling the set of values that a variate can assume (i.e., the set of values that a probability density function is defined over) the "range", denoted by (Evans et al. 2000, p. 5).Even worse, statistics most commonly uses "range" to refer to the completely different statistical quantity as the difference between the largest and smallest order statistics. In this work, this form..

A radial function is a function satisfying for points in some subset . Here, denotes the standard Euclidean norm in and is a discrete subset of whose elements are called centers.A collection of such functions which independently span a space is usually called a radial basis of . In this case, the functions are known as radial basis functions. Radial bases and radial basis functions play an important role in many areas of mathematics and approximation theory including statistics and partial differential equations.

Kepler's equation gives the relation between the polar coordinates of a celestial body (such as a planet) and the time elapsed from a given initial point. Kepler's equation is of fundamental importance in celestial mechanics, but cannot be directly inverted in terms of simple functions in order to determine where the planet will be at a given time.Let be the mean anomaly (a parameterization of time) and the eccentric anomaly (a parameterization of polar angle) of a body orbiting on an ellipse with eccentricity , then(1)For not a multiple of , Kepler's equation has a unique solution, but is a transcendental equation and so cannot be inverted and solved directly for given an arbitrary . However, many algorithms have been derived for solving the equation as a result of its importance in celestial mechanics.Writing a as a power series in gives(2)where the coefficients are given by the Lagrangeinversion theorem as(3)(Wintner 1941, Moulton 1970,..

Given a smooth function , if the Jacobian is invertible at 0, then there is a neighborhood containing 0 such that is a diffeomorphism. That is, there is a smooth inverse .

A function is said to be antiperiodic with antiperiod iffor , 3, .... For example, the sine function is antiperiodic with period (as well as with antiperiods , , etc.).It can be easily shown that if is antiperiodic with period , then it is periodic with period . But if is periodic with period , may or may not be antiperiodic with period .The constant function has the interesting property of being periodic with any period and antiperiodic with any antiperiod for all nonzero real numbers .

A function representable as a generalized Fourier series. Let be a metric space with metric . Following Bohr (1947), a continuous function for with values in is called an almost periodic function if, for every , there exists such that every interval contains at least one number for whichfor . Another formal description can be found in Krasnosel'skii et al. (1973).Every almost periodic function is bounded and uniformly continuous on the entirereal line.

An additive function is an arithmetic function such that whenever positive integers and are relatively prime,An example of an additive function is , since

Let be the set of real analytic functions on . Then is a subalgebra of . A necessary and sufficient condition for a function to belong to is thatfor , 1, ... for a suitable constant .

Let be a function defined on a set and taking values in a set . Then is said to be an injection (or injective map, or embedding) if, whenever , it must be the case that . Equivalently, implies . In other words, is an injection if it maps distinct objects to distinct objects. An injection is sometimes also called one-to-one.A linear transformation is injective if the kernel of the function is zero, i.e., a function is injective iff .A function which is both an injection and a surjectionis said to be a bijection.In the categories of sets, groups, modules, etc., a monomorphism is the same as an injection, and is used synonymously with "injection" outside of category theory.

Min Max The absolute value of a real number is denoted and defined as the "unsigned" portion of ,(1)(2)where is the sign function. The absolute value is therefore always greater than or equal to 0. The absolute value of for real is plotted above. Min Max Re Im The absolute value of a complex number , also called the complex modulus, is defined as(3)This form is implemented in the Wolfram Language as Abs[z] and is illustrated above for complex .Note that the derivative (read: complex derivative) does not exist because at every point in the complex plane, the value of the derivative of depends on the direction in which the derivative is taken (so the Cauchy-Riemann equations cannot and do not hold). However, the real derivative (i.e., restricting the derivative to directions along the real axis) can be defined for points other than as(4)As a result of the fact that computer algebra languages such as the Wolfram Language generically deal with..

Let , then the positive part of is the function defined byThe positive part satisfies the identitywhere is the negative part of .

A function increases on an interval if for all , where . If for all , the function is said to be strictly increasing.Conversely, a function decreases on an interval if for all with . If for all , the function is said to be strictly decreasing.If the derivative of a continuous function satisfies on an open interval , then is increasing on . However, a function may increase on an interval without having a derivative defined at all points. For example, the function is increasing everywhere, including the origin , despite the fact that the derivative is not defined at that point.

Given(1)(2)(3)if the determinantof the Jacobian(4)then , , and can be solved for in terms of , , and and partial derivatives of , , with respect to , , and can be found by differentiating implicitly.More generally, let be an open set in and let be a function. Write in the form , where and are elements of and . Suppose that (, ) is a point in such that and the determinant of the matrix whose elements are the derivatives of the component functions of with respect to the variables, written as , evaluated at , is not equal to zero. The latter may be rewritten as(5)Then there exists a neighborhood of in and a unique function such that and for all .

A function which is not defined explicitly, but rather is defined in terms of an algebraic relationship (which can not, in general, be "solved" for the function in question). For example, the eccentric anomaly of a body orbiting on an ellipse with eccentricity is defined implicitly in terms of the mean anomaly by Kepler's equation

The vertical line test is a graphical method of determining whether a curve in the plane represents the graph of a function by visually examining the number of intersections of the curve with vertical lines.The motivation for the vertical line test is as follows: A relation is a function precisely when each element is matched to at most one value and, as a result, any vertical line in the plane can intersect the graph of a function at most once. Therefore, the vertical line test concludes that a curve in the plane represents the graph of a function if and only if no vertical line intersects it more than once.A plane curve which doesn't represent the graph of a function is sometimes said to have failed the vertical line test.The figure above shows two curves in the plane. The leftmost curve fails the vertical line test due to the fact that the single vertical line drawn intersects the curve in two points. On the other hand, the vertical line test shows that the..

A point is said to be a periodic point of a function of period if , where and is defined recursively by .

If is a map (a.k.a. function, transformation, etc.) over a domain , then the image of , also called the range of under , is defined as the set of all values that can take as its argument varies over , i.e.,"Image" is a synonym for "range," but"image" is the term preferred in formal mathematical writing.The notation denotes the image of the interval under the function . Formally,

A function is said to be periodic (or, when emphasizing the presence of a single period instead of multiple periods, singly periodic) with period iffor , 2, .... For example, the sine function , illustrated above, is periodic with least period (often simply called "the" period) (as well as with period , , , etc.).The constant function is periodic with any period for all nonzero real numbers , so there is no concept analogous to the least period for constant functions. The following table summarizes the names given to periodic functions based on the number of independent periods they posses.number of periodsname1singly periodic function2doubly periodic function3triply periodic function

A function which satisfiesfor a fixed . Means, the Weierstrass elliptic function, and triangle center functions are homogeneous functions. A transformation of the variables of a tensor changes the tensor into another whose components are linear homogeneous functions of the components of the original tensor.

A composition of a function with itself gives a nested function , which gives , etc. Function nesting is implemented in the Wolfram Language as Nest[f, expr, n], and a listing of incremental nestings is implemented as NestList[f, expr, n].Integrals of nested functions rapidly become difficult to find in exact form. For example, the first two nestings of the exponential function gives(1)(2)where is the exponential integral, the first two nestings of the natural logarithm give(3)(4)where is the logarithmic integral.

There are at least three theorems known as Jensen's theorem.The first states that, for a fixed vector , the functionis a decreasing function of (Cheney 1999).The second states that if is a real polynomial not identically constant, then all nonreal zeros of lie inside the Jensen disks determined by all pairs of conjugate nonreal zeros of (Walsh 1955, 1961; Householder 1970; Trott 2004, p. 22). This theorem is a sharpening of Lucas's root theorem.The third theorem considers a function defined and analytic throughout a disk and supposes that has no zeros on the bounding circle , that inside the disk it has zeros , , ..., (where a zero of order is included times in the list, and that . Then(Edwards 2001, p. 40).

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