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Concept in mathematics
In mathematics, the K-function, typically denoted K(z), is a generalization of the hyperfactorial to complex numbers, similar to the generalization of
K-function
Degree of differentiability of a function or map
integer k such that a function has all derivatives up to order k, and such that all of these derivatives are continuous. One says that such a function has
Smoothness
S-shaped curve
A logistic function or logistic curve is a common S-shaped curve (sigmoid curve) with the equation f ( x ) = L 1 + e − k ( x − x 0 ) {\displaystyle f(x)={\frac
Logistic_function
Smooth approximation of one-hot arg max
softmax function, also known as softargmax or normalized exponential function, converts a tuple of K real numbers into a probability distribution over K possible
Softmax_function
Multivalued function in mathematics
described the W function per se in 1783. For each integer k {\displaystyle k} there is one branch, denoted by W k ( z ) {\displaystyle W_{k}\left(z\right)}
Lambert_W_function
Extension of the factorial function
only analytic function that extends the factorial, as one may add any analytic function that is zero on the positive integers, such as k sin ( m π x
Gamma_function
Function with a multiplicative scaling behaviour
homogeneous function. For example, a homogeneous polynomial of degree k defines a homogeneous function of degree k. The above definition extends to functions whose
Homogeneous_function
Family of solutions to related differential equations
+1)}}\sum _{k=0}^{\infty }{\frac {L_{k}^{(\alpha )}\left({\frac {x^{2}}{4t}}\right)}{\binom {k+\alpha }{k}}}{\frac {t^{k}}{k!}}.} The Bessel functions of the
Bessel_function
Ratio of polynomial functions
set of rational functions over a field K is a field, the field of fractions of the ring of the polynomial functions over K. A function f is called a rational
Rational_function
Mathematical function
K-convex functions, first introduced by Scarf, are a special weakening of the concept of convex function which is crucial in the proof of the optimality
K-convex_function
Methods used in statistics
{\widehat {K}}(t)} should be approximately equal to πt2. For data analysis, the variance stabilized Ripley K function called the L function is generally
Spatial descriptive statistics
Spatial_descriptive_statistics
Extension of superfactorials to the complex numbers
superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician
Barnes_G-function
Number of integers coprime to and less than n
called Euler's phi function. In other words, it is the number of integers k {\displaystyle k} in the range 1 ≤ k ≤ n {\displaystyle 1\leq k\leq n} for which
Euler's_totient_function
Mathematical function whose derivative exists
integer k, f is said to be of class Ck if its first k derivatives f′,f″,…,f(k) exist and are continuous over the domain of f. For a multivariable function, as
Differentiable_function
Special mathematical function defined as sin(x)/x
sinc(k) = 0 for nonzero integer k. The functions xk(t) = sinc(t − k) (k integer) form an orthonormal basis for bandlimited functions in the function space
Sinc_function
Function returning one of only two values
2 k {\displaystyle 2^{k}} entries. Every k {\displaystyle k} -ary Boolean function can be expressed as a propositional formula in k {\displaystyle k} variables
Boolean_function
Mathematical function, denoted exp(x) or e^x
complex exponential function is the sum of the series e z = ∑ k = 0 ∞ z k k ! . {\displaystyle e^{z}=\sum _{k=0}^{\infty }{\frac {z^{k}}{k!}}.} This series
Exponential_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
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
Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman defined it by k ν ( x
Bateman_function
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
Finitely generated extension field of positive transcendence degree
extension K / k {\displaystyle K/k} which has transcendence degree n {\displaystyle n} over k {\displaystyle k} . Equivalently, an algebraic function field
Algebraic_function_field
Method of solution to differential equations
− cos k s k ; c 3 = − sin k s k {\displaystyle c_{2}=-{\frac {\cos ks}{k}}\quad ;\quad c_{3}=-{\frac {\sin ks}{k}}} So Green's function for this
Green's_function
Indicator function of positive numbers
to the step function. Among the possibilities are: H ( x ) = lim k → ∞ ( 1 2 + 1 π arctan k x ) H ( x ) = lim k → ∞ ( 1 2 + 1 2 erf k x ) {\displaystyle
Heaviside_step_function
Function used in signal processing
processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside
Window_function
Special function in the physical sciences
mathematics, the Airy function (or Airy function of the first kind) A i ( x ) {\displaystyle \mathbf {Ai({\boldsymbol {x}})} } is a special function named after
Airy_function
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
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
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
Mapping arbitrary data to fixed-size values
A hash function is any function that can be used to map data of arbitrary size to fixed-size values, though there are some hash functions that support
Hash_function
Mathematical function
\left({\frac {k}{c}}\right)^{2}\right)=c\cdot \sum _{k\in \mathbb {Z} }\exp \left(-\pi \cdot (kc)^{2}\right).} The integral of an arbitrary Gaussian function is
Gaussian_function
Multiplicative function in number theory
the product of }}k{\text{ distinct primes}}\\0&{\text{if }}n{\text{ is divisible by a square}}>1.\end{cases}}} The Möbius function can alternatively
Möbius_function
Formal power series
[ k 0 , 1 k 1 , 1 0 0 ⋯ k 0 , 2 k 1 , 2 k 2 , 2 0 ⋯ k 0 , 3 k 1 , 3 k 2 , 3 k 3 , 3 ⋯ ⋮ ⋮ ⋮ ⋮ ] = [ k 0 , 0 0 0 0 ⋯ k 0 , 1 k 1 , 1 0 0 ⋯ k 0 , 2 k 1
Generating_function
Product of numbers from 1 to n
most notably in the series for the exponential function, e x = 1 + x 1 + x 2 2 + x 3 6 + ⋯ = ∑ k = 0 ∞ x k k ! , {\displaystyle e^{x}=1+{\frac {x}{1}}+{\frac
Factorial
Fat-soluble vitamers
(menadione), a synthetic form of vitamin K, was used to treat vitamin K deficiency, but because it interferes with the function of glutathione, it is no longer
Vitamin_K
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
Continuous function that is not absolutely continuous
the function C z ( y ) = ∑ k = 1 ∞ b k z k . {\displaystyle C_{z}(y)=\sum _{k=1}^{\infty }b_{k}z^{k}.} For z = 1/3, the inverse of the function x = 2 C1/3(y)
Cantor_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
Mathematical constant
is a mathematical constant, related to special functions like the K-function and the Barnes G-function. The constant also appears in a number of sums
Glaisher–Kinkelin_constant
special comparison functions. Class K {\displaystyle {\mathcal {K}}} functions belong to this family: Definition: a continuous function α : [ 0 , a ) → [
Class_kappa_function
Mathematical function
2 sn [ 4 5 K ( k ) ; k ] 2 = sn [ 4 5 K ( k ) ; k ] 2 − sn [ 2 5 K ( k ) ; k ] 2 2 sn [ 2 5 K ( k ) ; k ] sn [ 4 5 K ( k ) ; k ] {\displaystyle
Jacobi_elliptic_functions
Fast-growing function
Friedman's SSCG function is a mathematical function defined by Harvey Friedman. It is defined by SSCG ( k ) {\displaystyle {\text{SSCG}}(k)} as the largest
Friedman's_SSCG_function
Nearest integers from a number
Floor and ceiling functions In mathematics, the floor function is the function that takes a real number x as input and returns the greatest integer less
Floor_and_ceiling_functions
Bessel function or this type generalized-version of incomplete gamma function: K v ( x , y ) = ∫ 1 ∞ e − x t − y t t v + 1 d t {\displaystyle K_{v}(x
Incomplete Bessel K function/generalized incomplete gamma function
Incomplete_Bessel_K_function/generalized_incomplete_gamma_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
Mathematical function
positive integers n as H n = ∑ k = 1 n 1 k , {\displaystyle H_{n}=\sum _{k=1}^{n}{\frac {1}{k}},} the digamma function is related to them by ψ ( n ) =
Digamma_function
Function that is continuous everywhere but differentiable nowhere
mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere
Weierstrass function (nowhere-differentiable function)
Weierstrass_function_(nowhere-differentiable_function)
Mathematical function
Mittag-Leffler function, introduced by Gösta Mittag-Leffler in 1903, can be defined by the Maclaurin series E α ( z ) = ∑ k = 0 ∞ z k Γ ( α k + 1 ) , {\displaystyle
Mittag-Leffler_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
Fourier transform of the probability density function
probability density function, then the characteristic function is the Fourier transform (with sign reversal) of the probability density function. Thus it provides
Characteristic function (probability theory)
Characteristic_function_(probability_theory)
Special functions of several complex variables
a generating function where the coefficient of q k {\displaystyle q^{k}} represents how many ways there are to write k {\displaystyle k} as a perfect
Theta_function
Approximation of a function by a polynomial
a k-times differentiable function around a given point by a polynomial of degree k, called the k-th-order Taylor polynomial. For a smooth function, the
Taylor's_theorem
Meromorphic function on the complex plane
{\displaystyle K} is the conductor of L ( K , s ) {\displaystyle L(K,s)} : q ( K ) = | d K | . {\displaystyle q(K)=|d_{K}|.} Thus, the complete L-function of K {\displaystyle
L-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
Types of special mathematical functions
d t = ∫ 0 x ∑ k = 0 ∞ ( − 1 ) k t s + k − 1 k ! d t = ∑ k = 0 ∞ ( − 1 ) k x s + k k ! ( s + k ) = x s ∑ k = 0 ∞ ( − x ) k k ! ( s + k ) {\displaystyle
Incomplete_gamma_function
Transcendental single-variable function
Clausen functions: S z ( θ ) = ∑ k = 1 ∞ sin k θ k z {\displaystyle \operatorname {S} _{z}(\theta )=\sum _{k=1}^{\infty }{\frac {\sin k\theta }{k^{z}}}}
Clausen_function
Generalization of the Riemann zeta function for algebraic number fields
Dedekind zeta function of an algebraic number field K, usually denoted ζ K ( s ) {\displaystyle \zeta _{K}(s)} , is an analytic function that represents
Dedekind_zeta_function
Set of all things that may be the input of a mathematical function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname
Domain_of_a_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
Mathematical function
k ( x ) {\displaystyle a_{k}(x)} are polynomials (not all zero). Basic examples of algebraic functions are polynomial functions, rational functions,
Algebraic_function
Special mathematical function
polylogarithm function is defined by a power series in z generalizing the Mercator series, which is also a Dirichlet series in s: Li s ( z ) = ∑ k = 1 ∞ z k k s
Polylogarithm
Mathematical function
second Chebyshev function ψ(x) is defined similarly, with the sum extending over all prime powers not exceeding x ψ ( x ) = ∑ k ∈ N ∑ p k ≤ x log p = ∑
Chebyshev_function
Result of repeatedly applying a mathematical function
into the function as input, and this process is repeated. For example, on the image on the right: L=F(K), M=F∘F(K)=F2(K). Iterated functions are studied
Iterated_function
Arithmetic function related to the divisors of an integer
theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number
Divisor_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
Fundamental theorem in condensed matter physics
Mathematically, they are written Bloch function ψ ( r ) = e i k ⋅ r u ( r ) {\displaystyle \psi (\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u(\mathbf {r}
Bloch's_theorem
Function returning minus 1, zero or plus 1
In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that has the value −1, +1 or 0 according to whether
Sign_function
Linear map or polynomial function of degree one
a k x k , {\displaystyle f(x_{1},\ldots ,x_{k})=b+a_{1}x_{1}+\cdots +a_{k}x_{k},} and the graph is a hyperplane of dimension k. A constant function is
Linear_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)
Function whose domain is the positive integers
k−1pik−1=∏i=1ω(n)(1+pik+pi2k+⋯+piaik). Setting k = 0 in the second product gives τ(n)=d(n)=(1+a1)(1+a2)⋯(1+aω(n)). φ(n), the Euler totient function,
Arithmetic_function
Economic formula of productivity
Wicksteed. The most common version of the function is given by: Y ( L , K ) = A L α K β {\displaystyle Y(L,K)=AL^{\alpha }K^{\beta }} where: Y is the total production
Cobb–Douglas production function
Cobb–Douglas_production_function
Logarithm to the base of the mathematical constant e
property still works for the complex exponential function, ez = ez+2kiπ, for all complex z and integers k. So the logarithm cannot be defined for the whole
Natural_logarithm
Polynomial function of degree two
quadratic function and the solutions of the corresponding quadratic equation. f ( x ) = a ( x − h ) 2 + k {\displaystyle f(x)=a(x-h)^{2}+k} is called
Quadratic_function
Number-theoretical function
squares function is an arithmetic function that gives the number of representations for a given positive integer n {\displaystyle n} as the sum of k {\displaystyle
Sum_of_squares_function
analogue. Digamma function, Polygamma function Incomplete beta function Incomplete gamma function K-function Multivariate gamma function: A generalization
List of mathematical functions
List_of_mathematical_functions
Arithmetical function
Jordan's totient function, denoted as J k ( n ) {\displaystyle J_{k}(n)} , where k {\displaystyle k} is a positive integer, is a function of a positive integer
Jordan's_totient_function
Type of mathematical function
{\displaystyle \{\varphi _{k}\}_{k}} which forms a basis for some function space of interest, hence the name. Sums of radial basis functions are typically used
Radial_basis_function
Function in thermodynamics and statistical physics
partition function describes the statistical properties of a system in thermodynamic equilibrium.[citation needed] Partition functions are functions of the
Partition function (statistical mechanics)
Partition_function_(statistical_mechanics)
Periodic distribution ("function") of "point-mass" Dirac delta sampling
as sha function, impulse train or sampling function) is a periodic generalized function with the formula Ш T ( t ) := ∑ k = − ∞ ∞ δ ( t − k T ) {\displaystyle
Dirac_comb
Function used in computer cryptography
science Do one-way functions exist? More unsolved problems in computer science In computer science, a one-way function is a function that is easy to compute
One-way_function
Polynomial function with logarithm terms
function in n is a polynomial in the logarithm of n, a k ( log n ) k + a k − 1 ( log n ) k − 1 + ⋯ + a 1 ( log n ) + a 0 . {\displaystyle a_{k}(\log
Polylogarithmic_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
Inverse functions of sin, cos, tan, etc.
trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the trigonometric functions, under
Inverse trigonometric functions
Inverse_trigonometric_functions
Function associated with an algebraic variety
function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as Z ( V , s ) = exp ( ∑ k = 1 ∞ N k k (
Local_zeta_function
Mathematical function, inverse of an exponential function
k=1∞(−1)k+1(z−1)kk. Equating the function ln(z) to this infinite sum (series) is shorthand for saying that the function can be approximated
Logarithm
Attribute of a mathematical function
residues can be calculated for any function f : C ∖ { a k } k → C {\displaystyle f:\mathbb {C} \smallsetminus \{a_{k}\}_{k}\rightarrow \mathbb {C} } that
Residue_(complex_analysis)
Type of activation function
linear unit) activation function is an activation function defined as the non-negative part of its argument, i.e., the ramp function: ReLU(x)=x+=max(0,x)=x+|x|2={xif x>0
Rectified_linear_unit
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
Function that is holomorphic on the whole complex plane
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane
Entire_function
Generalization of the inverse function theorem
k + 1 {\displaystyle C^{k+1}} function f, the linearization D P f : C k + 1 → C k {\displaystyle DP_{f}:C^{k+1}\to C^{k}} has a right inverse S : C k
Nash–Moser_theorem
Function representing the number of primes less than or equal to a given number
divisible by no pi is ⌊x⌋−∑i⌊xpi⌋+∑i<j⌊xpipj⌋−∑i<j<k⌊xpipjpk⌋+⋯ (where ⌊x⌋ denotes the floor function). This number is therefore equal to π(x)−π(x)+1 when
Prime-counting_function
Number of partitions of an integer
given by p k ( n ) = p k ( n − k ) + p k − 1 ( n ) {\displaystyle p_{k}(n)=p_{k}(n-k)+p_{k-1}(n)} and its generating function is ∑ n = 0 ∞ p k ( n ) q n
Partition function (number theory)
Partition_function_(number_theory)
Power series derived from a discrete probability distribution
probability mass function of X {\displaystyle X} is recovered by taking derivatives of G {\displaystyle G} , p ( k ) = Pr ( X = k ) = G ( k ) ( 0 ) k ! . {\displaystyle
Probability generating function
Probability_generating_function
Quickly growing function
holds: ∀m>1,∀n>1:Am(n)<FGHm(n) For fixed k, the function FGHk(n) can be computed by a LOOP program of nesting depth k: # INPUT (n) LOOP n: # nesting depth:
Ackermann_function
k ∈ N : ϕ ( x , k + 1 ) ≤ ϕ ( x , k ) {\displaystyle \forall k\in \mathbb {N} :\phi (x,k+1)\leq \phi (x,k)} Completely analogous a partial function f
Semicomputable_function
Concept in theoretical computer science
(April 1983). "The determination of the value of Rado's noncomputable function Σ(k) for four-state Turing machines". Mathematics of Computation. 40 (162):
Busy_beaver
Mathematical function associated to algebraic varieties
the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is a meromorphic function on the complex plane
Hasse–Weil_zeta_function
Function made from a set
(after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector
Minkowski_functional
Type of energy
In solid-state physics, the work function is the minimum thermodynamic work (i.e., energy) needed to remove an electron from a solid to a point in the
Work_function
Arithmetic function
summatory function Φ ( n ) {\displaystyle \Phi (n)} is a summatory function of Euler's totient function defined by Φ ( n ) := ∑ k = 1 n φ ( k ) , n ∈ N
Totient_summatory_function
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K FUNCTION
K FUNCTION
K FUNCTION
K FUNCTION
K FUNCTION
K FUNCTION
K FUNCTION
K FUNCTION
K FUNCTION
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