If we are discussing differentiable complex-valued functions, then Af and V are complex vector spaces, and Af may be any complex numbers.

* Given any topological space X, the continuous real-or complex-valued functions on X form a real or complex unitary associative algebra ; here the functions are added and multiplied pointwise.

In topology, differential geometry and algebraic geometry, several structures defined on a topological space ( e. g., a differentiable manifold ) can be naturally localised or restricted to open subsets of the space: typical examples include continuous real or complex-valued functions, n times differentiable ( real or complex-valued ) functions, bounded real-valued functions, vector fields, and sections of any vector bundle on the space.

An arbitrary topological space X can be considered a locally ringed space by taking O < sub > X </ sub > to be the sheaf of real-valued ( or complex-valued ) continuous functions on open subsets of X ( there may exist continuous functions over open subsets of X which are not the restriction of any continuous function over X ).

A continuous-time real ( or complex ) signal is any real-valued ( or complex-valued ) function which is defined at every time t in an interval, most commonly an infinite interval.

If M is a differentiable manifold, a real or complex-valued function ƒ on M is said to be differentiable at a point p if it is differentiable with respect to some ( or any ) coordinate chart defined around p. More generally, if M and N are differentiable manifolds, a function ƒ: M → N is said to be differentiable at a point p if it is differentiable with respect to some ( or any ) coordinate charts defined around p and ƒ ( p ).

For any locally compact Hausdorff topological space X, the space C < sub > 0 </ sub >( X ) of continuous complex-valued functions on X which vanish at infinity is in a natural way a commutative C *- algebra:

Using the Haar measure, one can define a convolution operation on the space C < sub > c </ sub >( G ) of complex-valued continuous functions on G with compact support ; C < sub > c </ sub >( G ) can then be given any of various norms and the completion will be a group algebra.

A positive-definite function of a real variable x is a complex-valued function f: R → C such that for any real numbers x < sub > 1 </ sub >, ..., x < sub > n </ sub > the n × n matrix

for general complex-valued s and t and any non-negative integer n. This identity may be re-written in terms of the falling Pochhammer symbols as

Suppose U is an open set in the Euclidean space R < sup > n </ sup >, and suppose that f < sub > 0 </ sub >, f < sub > 1 </ sub > ... is a sequence of smooth, complex-valued functions on U. If I is an any open interval in R containing 0 ( possibly I

A set ( also called a family ) U of real-valued or complex-valued functions defined on some topological space X is called locally bounded if for any x < sub > 0 </ sub > in X there exists a neighborhood A of x < sub > 0 </ sub > and a positive number M such that

In mathematical analysis, a Banach limit is a continuous linear functional defined on the Banach space of all bounded complex-valued sequences such that for any sequences and, the following conditions are satisfied:

The following theorem represents positive linear functionals on C < sub > c </ sub >( X ), the space of continuous compactly supported complex-valued functions on a locally compact Hausdorff space X.

The model can be a linear or non-linear, time-continuous or time-discrete ( sampled ), memoryless or dynamic ( resulting in burst errors ), time-invariant or time-variant ( also resulting in burst errors ), baseband, passband ( RF signal model ), real-valued or complex-valued signal model.

The Gelfand representation or Gelfand isomorphism for a commutative C *- algebra with unit is an isometric *- isomorphism from to the algebra of continuous complex-valued functions on the space of multiplicative linear functionals, which in the commutative case are precisely the pure states, of A with the weak * topology.

The shift operator acting on real-or complex-valued functions or sequences is a linear operator which preserves most of the standard norms which appear in functional analysis.

A useful criterion for a function f: U → X to be holomorphic is that T o f: U → C is a holomorphic complex-valued function for every continuous linear functional T: X → C. Such an f is weakly holomorphic.

An almost periodic function is a complex-valued function of a real variable that has the properties expected of a function on a phase space describing the time evolution of such a system.

Then there is a local coordinate system consisting of n complex-valued functions z < sup > 1 </ sup >,..., z < sup > n </ sup > such that the coordinate transitions from one patch to another are holomorphic functions of these variables.

though the coordinates and vectors are now all complex-valued.

Typically u and v are taken to be the real and imaginary parts respectively of a complex-valued function of a single complex variable z = x + iy, f ( x + iy ) = u ( x, y ) + iv ( x, y ).

In other words, if u and v are real-differentiable functions of two real variables, obviously u + iv is a ( complex-valued ) real-differentiable function, but u + iv is complex-differentiable if and only if the Cauchy – Riemann equations hold.

Equations for coefficients corresponding to ratios of the electric field complex-valued amplitudes of the waves ( not necessarily real-valued magnitudes ) are also called " Fresnel equations ".

A related inversion formula more useful in combinatorics is as follows: suppose F ( x ) and G ( x ) are complex-valued functions defined on the interval < nowiki >

More specifically, instead of representing spinors as complex-valued 2D column vectors as Pauli had done, they represented them as complex-valued 2 × 2 matrices in which only the elements of the left column are non-zero.

In a conducting medium, permittivity and index of refraction are complex-valued.

* The sum and product of two complex-valued measurable functions are measurable.

* The idea that quantum states are vectors in a Hilbert space is completely general in all aspects of quantum mechanics and quantum field theory, whereas the idea that quantum states are complex-valued " wave " functions of space is only true in certain situations.

In this case, the principal axes are complex-valued vectors, corresponding to elliptically polarized light, and time-reversal symmetry can be broken.

As above, let f and g denote measurable real-or complex-valued functions defined on S. If || fg ||< sub > 1 </ sup > is finite, then the products of f with g and its complex conjugate function, respectively, are μ-integrable, the estimates

where is the Laplace operator, and f and φ are real or complex-valued functions on a manifold.

If g ( x, y ) is a known, complex-valued function of two real variables, and g is periodic in x and y ( that is, g ( x, y )= g ( x + 2π, y )= g ( x, y + 2π )) then we are interested in finding a function f ( x, y ) so that

The first part states that the matrix coefficients of irreducible representations of G are dense in the space C ( G ) of continuous complex-valued functions on G, and thus also in the space L < sup > 2 </ sup >( G ) of square-integrable functions.

More generally, for n real-or complex-valued functions f < sub > 1 </ sub >, ..., f < sub > n </ sub >, which are n − 1 times differentiable on an interval I, the Wronskian W ( f < sub > 1 </ sub >, ..., f < sub > n </ sub >) as a function on I is defined by

* Suitably regular complex-valued functions on the real line have Fourier transforms that are also functions on the real line and, just as for periodic functions, these functions can be recovered from their Fourier transforms ; and

* Test functions in Fourier analysis, which are complex-valued functions on the real line, that are 0 everywhere outside of a given limited interval ( hence all derivatives will also be 0 outside of the interval ) and inside of the interval, but are still infinitely differentiable everywhere.

An infinite matrix with complex-valued entries defines a regular summability method if and only if it satisfies all of the following properties

A is a symmetric operator without any eigenvalues and eigenfunctions.

The statement that any wavefunction for the particle on a ring can be written as a superposition of energy eigenfunctions is exactly identical to the Fourier theorem about the development of any periodic function in a Fourier series.

Can any function be expressed in terms of the eigenfunctions ( are they a complete set ) and under what circumstances does a point spectrum or a continuous spectrum arise?