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Mathematical series
calculus, a function series is a series where each of its terms is a function, not just a real or complex number. Examples of function series include ordinary
Function_series
Formal power series
generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often
Generating_function
Analytic function in mathematics
Riemann zeta function, or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ {\displaystyle \zeta } (zeta), is a function of a complex
Riemann_zeta_function
Mathematical approximation of a function
analysis, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single
Taylor_series
Type of function in mathematics
an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at
Analytic_function
Mathematical function having a characteristic S-shaped curve or sigmoid curve
sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the
Sigmoid_function
Special functions of several complex variables
absolutely convergent series. At fixed τ, this is a Fourier series for a 1-periodic entire function of z. Accordingly, the theta function is 1-periodic in
Theta_function
Decomposition of periodic functions
A Fourier series (/ˈfʊrieɪ, -iər/) is a series expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example
Fourier_series
Complex-differentiable (mathematical) function
that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is analytic). Holomorphic functions are the central
Holomorphic_function
Mathematical function
the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial
Beta_function
Family of power series in mathematics
hypergeometric series is a power series in which the ratio of successive coefficients indexed by n {\displaystyle n} is a rational function of n {\displaystyle
Generalized hypergeometric function
Generalized_hypergeometric_function
Association of one output to each input
mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the
Function_(mathematics)
Family of solutions to related differential equations
functions. Anger function Bessel polynomials Bessel–Clifford function Bessel–Maitland function Fourier–Bessel series Hahn–Exton q-Bessel function Hankel transform
Bessel_function
Function with a repeating pattern
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe waves
Periodic_function
Meromorphic function on the complex plane
Dirichlet series, usually convergent on a half-plane, that may give rise to an L-function via analytic continuation, is called an L-series. Fundamental
L-function
Functions such that f(–x) equals f(x) or –f(x)
In mathematics, an even function is a real function such that f ( − x ) = f ( x ) {\displaystyle f(-x)=f(x)} for every x {\displaystyle x} in its domain
Even_and_odd_functions
Functions of an angle
mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of
Trigonometric_functions
Solution of a confluent hypergeometric equation
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric
Confluent hypergeometric function
Confluent_hypergeometric_function
Mathematical function, denoted exp(x) or e^x
In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted
Exponential_function
Function that is holomorphic on the whole complex plane
entire functions. Entire functions correspond exactly to Taylor series converging on the whole complex plane. A transcendental entire function is an entire
Entire_function
Special mathematical function defined as sin(x)/x
In mathematics, physics and engineering, the sinc function (/ˈsɪŋk/), denoted by sinc(x), is defined as either sinc ( x ) = sin x x {\displaystyle
Sinc_function
Analytic function that does not satisfy a polynomial equation
mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable
Transcendental_function
Special mathematical functions defined on the surface of a sphere
similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines
Spherical_harmonics
Function studied by Ramanujan
In mathematics, the Ramanujan tau function, studied by Srinivasa Ramanujan, is the function τ : N → Z {\displaystyle \tau :\mathbb {N} \to \mathbb {Z}
Ramanujan_tau_function
Type of mathematical function
In mathematics, a Dirichlet L-series is a function of the form L(s,χ)=∑n=1∞χ(n)ns, where χ is a Dirichlet character and s a complex variable with real
Dirichlet_L-function
Function that is continuous everywhere but differentiable nowhere
infinitely jagged functions (nowadays known as fractal curves). In Weierstrass's original paper, the function was defined as a Fourier series: f ( x ) =
Weierstrass function (nowhere-differentiable function)
Weierstrass_function_(nowhere-differentiable_function)
Branch of mathematics studying functions of a complex variable
complex function exists. In particular, if a complex function has a derivative, it has derivatives of every order and equals the sum of its Taylor series in
Complex_analysis
Extension of the factorial function
gamma function, denoted by Γ {\displaystyle \Gamma } (capital Greek letter gamma), is the most common extension of the factorial function to complex
Gamma_function
Mathematical function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z )
Digamma_function
Arithmetic function related to the divisors of an integer
including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number
Divisor_function
Order-preserving mathematical function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept
Monotonic_function
Partial correlation of a time series with its lagged values
In time series analysis, the partial autocorrelation function (PACF) gives the partial correlation of a stationary time series with its own lagged values
Partial autocorrelation function
Partial_autocorrelation_function
Power series derived from a discrete probability distribution
generating function of a discrete random variable is a power series representation (the generating function) of the probability mass function of the random
Probability generating function
Probability_generating_function
Mathematical transform that expresses a function of time as a function of frequency
takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output
Fourier_transform
Q-analog of hypergeometric series
elliptic hypergeometric series. A series xn is called hypergeometric if the ratio of successive terms xn+1/xn is a rational function of n. If the ratio of
Basic_hypergeometric_series
Ratio of polynomial functions
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator
Rational_function
Sigmoid shape special function
mathematics, the error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2
Error_function
Mathematical theorem
{\textstyle \Gamma (s)} is the gamma function. It was widely used by Ramanujan to calculate definite integrals and infinite series. Higher-dimensional versions
Ramanujan's_master_theorem
Fundamental trigonometric functions
In mathematics, sine and cosine are trigonometric functions. The sine and cosine of an acute angle are defined in the context of a right triangle: the
Sine_and_cosine
Function defined by a hypergeometric series
hypergeometric function 2F1(a, b; c; z) is a special function represented by the hypergeometric series, that includes many other special functions as specific
Hypergeometric_function
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
Analytic function in mathematics
analysis, a lacunary function or series is an analytic function containing many zero coefficients in its basis decomposition. A series can be lacunary in
Lacunary_function
Asymmetric sigmoid function
or Gompertz function is a type of mathematical model for a time series, named after Benjamin Gompertz (1779–1865). It is a sigmoid function which describes
Gompertz_function
Logarithm to the base of the mathematical constant e
ln(x) is undefined at 0, the function itself does not have a Maclaurin series, unlike many other elementary functions. Instead, one looks for Taylor
Natural_logarithm
Approximation of a function by a polynomial
the order k of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor
Taylor's_theorem
Functions in mathematics
cosines, functions which are thus referred to as "harmonics." Fourier analysis involves expanding functions on the unit circle in terms of a series of these
Harmonic_function
Mathematical series
Dirichlet series, as are the Dirichlet L-functions. Specifically, the Riemann zeta function ζ(s) is the Dirichlet series of the constant unit function u(n)
Dirichlet_series
Concept in the theory of integral equations
In mathematics, the Liouville–Neumann series is a function series that results from applying the resolvent formalism to solve Fredholm integral equations
Liouville–Neumann_series
In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman
Bateman_function
Smooth and compactly supported function
analysis, a bump function is a localized auxiliary function, usually chosen to be smooth and to have compact support. Bump functions are commonly used
Bump_function
Infinite sum of monomials
power series is the Taylor series of some smooth function. In many situations, the center c is equal to zero, for instance for Maclaurin series. In such
Power_series
Description of limiting behavior of a function
asymptotic analysis for computing function approximations, implicit functions, integrals, iterated functions, series summation, partial sums, solutions
Asymptotic_analysis
Power series with negative powers
mathematics, the Laurent series of a complex function f ( z ) {\displaystyle f(z)} is a representation of that function as a power series which includes terms
Laurent_series
Negative of a convex function
In mathematics, a concave function is one for which the function value at any convex combination of elements in the domain is greater than or equal to
Concave_function
Transcendental single-variable function
trigonometric series, and various other forms. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta
Clausen_function
Class of mathematical function
In complex analysis, a meromorphic function on an open subset D {\displaystyle D} of the complex plane is a function that is holomorphic on all of D {\displaystyle
Meromorphic_function
Statistical function that defines the quantiles of a probability distribution
the quantile function of a probability distribution is the inverse of its cumulative distribution function. That is, the quantile function of a distribution
Quantile_function
Number, approximately 3.14
consecutive zeroes of the sine function. The cosine and sine can be defined independently of geometry as a power series, or as the solution of a differential
Pi
Multivalued function in mathematics
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse
Lambert_W_function
Oscillatory error in Fourier series
Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The N {\textstyle N} th partial Fourier series of
Gibbs_phenomenon
Mathematical functions which are smooth but not analytic
convergent power series in a neighbourhood of that point. All real analytic functions are smooth, but there exist smooth real functions that are not real
Non-analytic_smooth_function
Types of special mathematical functions
defined similarly to the gamma function but with different or "incomplete" integral limits. The usual gamma function is defined as an integral from zero
Incomplete_gamma_function
Base of natural logarithms
sum of the infinite series e=∑n=0∞1n!=1+11+11⋅2+11⋅2⋅3+⋯. It is the unique positive number a such that the graph of the function y = ax has a slope of
E_(mathematical_constant)
Meromorphic function
In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers C {\displaystyle \mathbb {C} } defined as the (m +
Polygamma_function
Mathematical relation assigning a probability event to a cost
optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one
Loss_function
Inverse of the gamma function
mathematics, the inverse gamma function Γ − 1 ( x ) {\displaystyle \Gamma ^{-1}(x)} is the inverse function of the gamma function. In other words, y = Γ − 1
Inverse_gamma_function
Mathematical function with convex lower level sets
In mathematics, a quasiconvex function is a real-valued function defined on a convex subset of a real vector space, such that for any real number y, the
Quasiconvex_function
Function whose domain is the positive integers
Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum. An arithmetic function a is
Arithmetic_function
Element of a basis for a function space
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as
Basis_function
Function that returns its argument unchanged
mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value
Identity_function
American health technology company
Function Health, often stylized as simply Function, is an American company and platform headquartered in Austin, Texas. The venture capital-backed company
Function_Health
Expression of a function as an infinite sum of simpler functions
In mathematics, a series expansion is a technique that expresses a function as an infinite sum, or series, of simpler functions. It is a method for calculating
Series_expansion
In mathematics, E-functions are a type of power series that satisfy particular arithmetic conditions on the coefficients. They are of interest in transcendental
E-function
Theorem in mathematics
mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that if
Inverse_function_theorem
American science fiction television series
television series created by Blake Crouch, based on his 2016 novel. The series premiered on Apple TV+ on May 8, 2024. In August 2024, the series was renewed
Dark_Matter_(2024_TV_series)
Hyperbolic analogues of trigonometric functions
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just
Hyperbolic_functions
Elementary functions and their finitely iterated integrals
sums. Liouvillian functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841. All elementary functions are Liouvillian. Examples
Liouvillian_function
Mathematical functions having established names and notations
series or asymptotic series are available. In addition, sometimes there exist relations with other special functions; a complicated special function can
Special_functions
Mathematical function
_{1}(z)={\frac {d}{dz}}\psi (z)} where ψ(z) is the digamma function. It may also be defined as the sum of the series ψ 1 ( z ) = ∑ n = 0 ∞ 1 ( z + n ) 2 , {\displaystyle
Trigamma_function
Indicator function of rational numbers
In mathematics, the Dirichlet function is the indicator function 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} of the set of rational numbers Q {\displaystyle
Dirichlet_function
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
Theorem in complex analysis
since power series are infinitely differentiable, so are holomorphic functions (this is in contrast to the case of real differentiable functions), and the
Analyticity of holomorphic functions
Analyticity_of_holomorphic_functions
S-shaped curve
various generalizations, depending on the field. The logistic function was introduced in a series of three papers by Pierre François Verhulst between 1838
Logistic_function
Mathematical function, inverse of an exponential function
elementary calculus, the series is said to converge to the function ln(z), and the function is the limit of the series. It is the Taylor series of the natural logarithm
Logarithm
Multiplicative function in number theory
The Möbius function μ ( n ) {\displaystyle \mu (n)} is a multiplicative function in number theory introduced by the German mathematician August Ferdinand
Möbius_function
quasisymmetric function is any element in the ring of quasisymmetric functions which is in turn a subring of the formal power series ring with a countable
Quasisymmetric_function
Mathematical function with no sudden changes
In mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies
Continuous_function
Function family in complex analysis
antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions. A function of
Antiholomorphic_function
Index of lists with the same name
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle
List_of_zeta_functions
Number of integers coprime to and less than n
( x ) {\displaystyle \log _{e}(x)} . In number theory, Euler's totient function counts the positive integers up to a given integer n {\displaystyle n}
Euler's_totient_function
Continuous function that is not absolutely continuous
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in
Cantor_function
Decompositions of inner product spaces into orthonormal bases
Fourier series is the expansion of a square integrable function into a sum of square integrable orthogonal basis functions. The standard Fourier series uses
Generalized_Fourier_series
Function on an integer n which is log(p) if n equals p^k and zero otherwise
Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that
Von_Mangoldt_function
Mathematical description of quantum state
In quantum mechanics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common
Wave_function
Limiting a position to an area
languages and libraries provide functions for fast and vectorized clamping. In Python, the pandas library offers the Series.clip and DataFrame.clip methods
Clamp_(function)
British period crime drama television series (2013–2022)
characters" that serve various narrative functions. This situates the protagonists on the margins of society, allowing the series to address themes of intersectionality
Peaky_Blinders_(TV_series)
Set of four hypergeometric series
Gauss's hypergeometric series 2F1 of one variable. Appell established the set of partial differential equations of which these functions are solutions, and
Appell_series
Point to which functions converge in analysis
mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which
Limit_of_a_function
Divergent sum of positive unit fractions
infinite sum of values of a monotone decreasing positive function of n (like the harmonic series) has partial sums that are within a bounded distance of
Harmonic_series_(mathematics)
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FUNCTION SERIES
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FUNCTION SERIES
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