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Extension of the factorial function
( z ) = e − 1 2 + 0 − z + 1 z − 1 2 + 2 − z + 2 z − 2 2 + 4 − z + 3 z − 3 2 + 6 − z + 4 z − 4 2 + 8 − z + 5 z − 5 2 + 10 − z + ⋱ + e − 1 z + 0 − z +
Gamma_function
Mathematical function
the Riemann–Siegel Z function, the Riemann–Siegel zeta function, the Hardy function, the Hardy Z function and the Hardy zeta function. It can be defined
Z_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
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
Complex-differentiable (mathematical) function
regular functions. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point z 0 {\displaystyle
Holomorphic_function
Sigmoid shape special function
error function (also called the Gauss error function), often denoted by e r f {\displaystyle \mathbf {erf} } , is the function erf ( z ) = 2 π ∫ 0 z e −
Error_function
Mathematical function
∂ z 1 B ( z 1 , z 2 ) = B ( z 1 , z 2 ) ( Γ ′ ( z 1 ) Γ ( z 1 ) − Γ ′ ( z 1 + z 2 ) Γ ( z 1 + z 2 ) ) = B ( z 1 , z 2 ) ( ψ ( z 1 ) − ψ ( z 1 + z 2 )
Beta_function
Family of solutions to related differential equations
.. ( 1 z d d z ) m ( z n + 1 f n ( z ) ) = z n − m + 1 f n − m ( z ) , ( 1 z d d z ) m ( z − n f n ( z ) ) = ( − 1 ) m z − n − m f n + m ( z ) . {\displaystyle
Bessel_function
Mathematical function
digamma function is defined as the logarithmic derivative of the gamma function: ψ ( z ) = d d z ln Γ ( z ) = Γ ′ ( z ) Γ ( z ) . {\displaystyle \psi (z)={\frac
Digamma_function
Concept in mathematics
where 1 F 1 ( a ; b ; z ) = M ( a ; b ; z ) {\displaystyle \;_{1}F_{1}(a;b;z)=M(a;b;z)} is the confluent hypergeometric function. Other pairs of independent
Parabolic_cylinder_function
Association of one output to each input
function is illustrated by the multiplicative inverse of the Riemann zeta function: the determination of the domain of definition of the function z ↦
Function_(mathematics)
Formal power series
a ( z ) ⋅ S ( z ) + b ( z ) ⋅ z S ′ ( z ) + c ( z ) ⋅ z 2 S ″ ( z ) + d ( z ) ⋅ z 3 S ‴ ( z ) , {\displaystyle a(z)\cdot S(z)+b(z)\cdot zS'(z)+c(z)\cdot
Generating_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
Mathematical function, denoted exp(x) or e^x
quickly: e z = 1 + 2 z 2 − z + z 2 6 + z 2 10 + z 2 14 + ⋱ {\displaystyle e^{z}=1+{\cfrac {2z}{2-z+{\cfrac {z^{2}}{6+{\cfrac {z^{2}}{10+{\cfrac {z^{2}}{14+\ddots
Exponential_function
Meromorphic function
logarithm of the gamma function: ψ ( m ) ( z ) := d m d z m ψ ( z ) = d m + 1 d z m + 1 ln Γ ( z ) . {\displaystyle \psi ^{(m)}(z):={\frac {\mathrm {d}
Polygamma_function
Transforming a function in such a way that it only takes a single argument
the prototypical example, one begins with a function f : ( X × Y ) → Z {\displaystyle f:(X\times Y)\to Z} that takes two arguments, one from X {\displaystyle
Currying
Mathematical description of quantum state
| r , s z ⟩ = | r ⟩ | s z ⟩ {\displaystyle |\mathbf {r} ,s_{z}\rangle =|\mathbf {r} \rangle |s_{z}\rangle } . The position-space wave function of a single
Wave_function
Two-dimensional group theory table
and z functions in “linear functions, roatations”. So, Γtrans = 1B1u+1B2u+1B3u Rotational motion has Rx, Ry and Rz functions in “linear functions, roatations”
Character_table
Mathematical function, inverse of an exponential function
number z that satisfies 0 < z ≤ 2, the following formula holds: ln(z)=(z−1)11−(z−1)22+(z−1)33−(z−1)44+⋯=∑k=1∞(−1)k+1(z−1)kk. Equating the function ln(z) to
Logarithm
Special mathematical function
complex plane Li –3(z) Li –2(z) Li –1(z) Li0(z) Li1(z) Li2(z) Li3(z) The polylogarithm function is defined by a power series in z generalizing the Mercator
Polylogarithm
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
Class of mathematical function
functions f ( z ) = e z z and f ( z ) = sin z ( z − 1 ) 2 {\displaystyle f(z)={\frac {e^{z}}{z}}\quad {\text{and}}\quad f(z)={\frac {\sin {z}}{(z-1)^{2}}}}
Meromorphic_function
Mathematical function
the trigamma function, denoted ψ1(z) or ψ(1)(z), is the second of the polygamma functions, and is defined by ψ 1 ( z ) = d 2 d z 2 ln Γ ( z ) {\displaystyle
Trigamma_function
Mathematical function
log-gamma function log Γ ( z ) = − γ z − log z + ∑ n = 1 ∞ ( z n − log ( 1 + z n ) ) , {\displaystyle \log \Gamma \left(z\right)=-\gamma z-\log z+\sum
Riemann–Siegel_theta_function
Inverse functions of sin, cos, tan, etc.
z: ∫ arcsin ( z ) d z = z arcsin ( z ) + 1 − z 2 + C ∫ arccos ( z ) d z = z arccos ( z ) − 1 − z 2 + C ∫ arctan ( z ) d z = z arctan ( z )
Inverse trigonometric functions
Inverse_trigonometric_functions
Types of special mathematical functions
z ) = z s Γ ( s ) γ ∗ ( s , z ) , {\displaystyle \gamma (s,z)=z^{s}\,\Gamma (s)\,\gamma ^{*}(s,z),} extends the real lower incomplete gamma function as
Incomplete_gamma_function
Mathematical functions
z ) X ⁗ ( z ) = 4 X ′ ( z ) X ‴ ( z ) − 3 X ″ ( z ) 2 + 2 X ( z ) 2 , z ∈ C . {\displaystyle X(z)X''''(z)=4X'(z)X'''(z)-3X''(z)^{2}+2X(z)^{2},\quad z\in
Lemniscate_elliptic_functions
Smooth approximation of one-hot arg max
(unit) softmax function σ:ℝK→(0,1)K, where K>1, takes a tuple 𝐳=(z1,…,zK)∈ℝK and computes each component of vector σ(𝐳)∈(0,1)K with σ(𝐳)i=ezi∑j=1Kezj
Softmax_function
Type of function in mathematics
analytic at 0 {\displaystyle 0} . The corresponding function z ↦ | z | {\displaystyle z\mapsto |z|} on the complex numbers is not complex analytic on
Analytic_function
Extension of superfactorials to the complex numbers
In mathematics, the Barnes G-function G ( z ) {\displaystyle G(z)} is a function that is an extension of superfactorials to the complex numbers. It is
Barnes_G-function
Index of lists with the same name
Hardy zeta function, alternative names for the Z function Ruelle zeta function Selberg zeta function of a Riemann surface Shimizu L-function Shintani zeta
List_of_zeta_functions
Function that is discontinuous at rationals and continuous at irrationals
Thomae's function is a real-valued function of a real variable that can be defined as: f ( x ) = { 1 q if x = p q ( x is rational), with p ∈ Z and q
Thomae's_function
Representation of a mathematical function
In mathematics, the graph of a function f {\displaystyle f} is the set of ordered pairs ( x , y ) {\displaystyle (x,y)} , where f ( x ) = y . {\displaystyle
Graph_of_a_function
Function studied by Ramanujan
the Ramanujan tau function, studied by Srinivasa Ramanujan, is the function τ : N → Z {\displaystyle \tau :\mathbb {N} \to \mathbb {Z} } defined by ∑ n
Ramanujan_tau_function
In functional programming
function f : ( X × Y × Z ) → N {\displaystyle f\colon (X\times Y\times Z)\to N} , we might fix (or 'bind') the first argument, producing a function of
Partial_application
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
Special functions of several complex variables
function of z. Accordingly, the theta function is 1-periodic in z: ϑ ( z + 1 ; τ ) = ϑ ( z ; τ ) . {\displaystyle \vartheta (z+1;\tau )=\vartheta (z;\tau
Theta_function
Logarithm of a complex number
(z-z_{0})/(-z_{0})} .[citation needed] 1 z = 1 z 0 ⋅ 1 1 − z − z 0 − z 0 = ∑ n = 0 ∞ 1 z 0 ( z − z 0 − z 0 ) n = ∑ n = 0 ∞ ( − 1 ) n z 0 n + 1 ( z − z 0 ) n
Complex_logarithm
Special case of the polylogarithm
Spence's function), denoted as Li2(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the
Dilogarithm
Mathematical functions
mathematics, the inverse hyperbolic functions are inverses of the hyperbolic functions, analogous to the inverse circular functions. There are six in common use:
Inverse_hyperbolic_functions
Functions in mathematics
functions of three variables are given in the table below with r 2 = x 2 + y 2 + z 2 {\displaystyle r^{2}=x^{2}+y^{2}+z^{2}} : Harmonic functions that
Harmonic_function
Mathematical function
mathematics, the Mittag-Leffler functions are a family of special functions. They are complex-valued functions of a complex argument z, and moreover depend on
Mittag-Leffler_function
Description of continuous random distribution
Proof: Let Z {\displaystyle Z} be a collapsed random variable with probability density function p Z ( z ) = δ ( z ) {\displaystyle p_{Z}(z)=\delta (z)} (i.e
Probability_density_function
Complex complementary error function
Faddeeva function or Kramp function is a scaled complex complementary error function, w ( z ) := e − z 2 erfc ( − i z ) = erfcx ( − i z ) = e − z 2 ( 1
Faddeeva_function
Probability that random variable X is less than or equal to x
F Z ( z ) = F ℜ ( Z ) , ℑ ( Z ) ( ℜ ( z ) , ℑ ( z ) ) = P ( ℜ ( Z ) ≤ ℜ ( z ) , ℑ ( Z ) ≤ ℑ ( z ) ) . {\displaystyle F_{Z}(z)=F_{\Re {(Z)},\Im {(Z)}}(\Re
Cumulative distribution function
Cumulative_distribution_function
Function that takes two inputs
unique z∈Z such that (x,y,z) belongs to R. f(x,y) is then defined to be this z. Alternatively, a binary function may be interpreted as simply a function from
Binary_function
Solution of a confluent hypergeometric equation
, 2 , z ) = ( e z − 1 ) / z , M ( 1 , 3 , z ) = 2 ! ( e z − 1 − z ) / z 2 {\displaystyle M(1,2,z)=(e^{z}-1)/z,\ \ M(1,3,z)=2!(e^{z}-1-z)/z^{2}} etc
Confluent hypergeometric function
Confluent_hypergeometric_function
Branch of mathematics studying functions of a complex variable
f} at z 0 {\displaystyle z_{0}} is defined to be f ′ ( z 0 ) = lim z → z 0 f ( z ) − f ( z 0 ) z − z 0 . {\displaystyle f'(z_{0})=\lim _{z\to z_{0}}{\frac
Complex_analysis
Mathematical function whose derivative exists
the complex function f(z)=z+z‾2 corresponds to the 2-variable real function f(x,y)=x, which is differentiable at every point. This function is not complex-differentiable
Differentiable_function
Mathematical function
z ) ( ( 1 − m ) z − E ( z ) + m cd ( z ) sn ( z ) ) 2 m ( 1 − m ) , d d m cn ( z ) = sn ( z ) dn ( z ) ( ( m − 1 ) z + E ( z ) − m sn ( z
Jacobi_elliptic_functions
Function returning minus 1, zero or plus 1
function can be generalized to complex numbers as: sgn z = z | z | {\displaystyle \operatorname {sgn} z={\frac {z}{|z|}}} for any complex number z {\displaystyle
Sign_function
Number with a real and an imaginary part
example, the function f(z)=z‾ is differentiable as a function ℝ2→ℝ2, but is not complex differentiable. A real differentiable function is complex differentiable
Complex_number
Evaluation of a function on its argument
X(Y)=z⟺(∃D,R(X∈RD∧((Y,z)∈X)))∨(∀D,R(X∉RD∨(Y,z)∉X)∧z=∅), where RD denotes set exponentiation: the set of all functions from D to R. In prose: X(Y)=z if there
Function_application
Special function in the physical sciences
below, the Airy functions can be extended to the complex plane, giving entire functions. The asymptotic behaviour of the Airy functions as |z| goes to infinity
Airy_function
Conjecture on zeros of the zeta function
[x,y,z] = [Re(ζ(r + it)), Im(ζ(r + it)), t] with 0.1≤r≤0.9 and 1≤t≤51. The real part (red) and imaginary part (blue) of the Riemann zeta function ζ(s)
Riemann_hypothesis
Operation on mathematical functions
two functions, f {\displaystyle f} and g {\displaystyle g} , and returns a new function f ∘ g {\displaystyle f\circ g} . When the composite function f ∘
Function_composition
Special mathematical functions defined on the surface of a sphere
usual p functions ( ℓ = 1 {\displaystyle \ell =1} ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. The complex
Spherical_harmonics
Fundamental trigonometric functions
functional equation for the Riemann zeta-function, ζ(s)=2(2π)s−1Γ(1−s)sin(π2s)ζ(1−s). As a holomorphic function, sin z is a 2D solution of Laplace's equation:
Sine_and_cosine
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
Complex exponential in terms of sine and cosine
the function d f d z = f {\displaystyle {\frac {df}{dz}}=f} and f ( 0 ) = 1. {\displaystyle f(0)=1.} For complex z e z = 1 + z 1 ! + z 2 2 ! + z 3 3
Euler's_formula
Concept in mathematics
gamma function. There are multiple equivalent definitions of the K-function. The direct definition: K ( z ) = ( 2 π ) − z − 1 2 exp [ ( z 2 ) + ∫ 0 z −
K-function
Special mathematical function
about a similar function in 1887. The Lerch transcendent, is given by: Φ ( z , s , α ) = ∑ n = 0 ∞ z n ( n + α ) s {\displaystyle \Phi (z,s,\alpha )=\sum
Lerch_transcendent
Generalized mathematical function
analytic function f ( z ) {\displaystyle f(z)} in some neighbourhood of a point z = a {\displaystyle z=a} . This is the case for functions defined by
Multivalued_function
Class of functions behaving "like" periodic functions
\omega } if f ( z + ω ) = g ( z , f ( z ) ) {\displaystyle f(z+\omega )=g(z,f(z))} , where g {\displaystyle g} is a "simpler" function than f {\displaystyle
Quasiperiodic_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
Characteristic property of holomorphic functions
| f ( z ) − f ( z 0 ) − f ′ ( z 0 ) ( z − z 0 ) | / | z − z 0 | → 0 {\displaystyle |f(z)-f(z_{0})-f'(z_{0})(z-z_{0})|/|z-z_{0}|\to 0} as z → z 0 {\displaystyle
Cauchy–Riemann_equations
Analytic function on the upper half-plane with a certain behavior under the modular group
Unlike an ordinary periodic function, its symmetries include transformations such as replacing a complex number z by −1/z, and the transformation law
Modular_form
Function family in complex analysis
z} . A definition of antiholomorphic function follows: "[a] function f ( z ) = u + i v {\displaystyle f(z)=u+iv} of one or more complex variables z =
Antiholomorphic_function
Linear transform from the time domain to the frequency domain
X ( z ) z = z 2 z ( z 2 − 1.5 z + 0.5 ) = z z 2 − 1.5 z + 0.5 {\displaystyle {\frac {X(z)}{z}}={\frac {z^{2}}{z(z^{2}-1.5\,z+0.5)}}={\frac {z}{z^{2}-1
Z-transform
Set of functions between two fixed sets
(Y,Z)),\,{\widetilde {f}}(x)(y)=f(x,y)} is bijective, where Fun ( X , Y ) {\displaystyle \operatorname {Fun} (X,Y)} denotes the set of functions from
Function_space
Function specifying the behavior of a component in an electronic or control system
function can be written as: H ( z ) = Y ( z ) X ( z ) = Z { y [ n ] } Z { x [ n ] } . {\displaystyle H(z)={\frac {Y(z)}{X(z)}}={\frac {{\mathcal {Z
Transfer_function
Power series derived from a discrete probability distribution
function of X is defined as G ( z ) = E ( z X ) = ∑ x = 0 ∞ p ( x ) z x , {\displaystyle G(z)=\operatorname {E} (z^{X})=\sum _{x=0}^{\infty }p(x)z^{x}
Probability generating function
Probability_generating_function
Ratio of polynomial functions
analogy to a proper fraction in ℚ. In complex analysis, a rational function f(z)=P(z)Q(z) is the ratio of two polynomials with complex coefficients, where
Rational_function
Function in thermodynamics and statistical physics
classical and discrete, the canonical partition function is defined as Z = ∑ i e − β E i , {\displaystyle Z=\sum _{i}e^{-\beta E_{i}},} where i {\displaystyle
Partition function (statistical mechanics)
Partition_function_(statistical_mechanics)
Mathematical function
\sum _{k\in \mathbb {Z} }\exp \left(-\pi \cdot (kc)^{2}\right).} The integral of an arbitrary Gaussian function is ∫ − ∞ ∞ a exp ( − ( x − b
Gaussian_function
Solutions to Laplace's equation
cylindrical coordinates, ρ (radial coordinate), φ (polar angle), and z (height). Each function Vn(k) is the product of three terms, each depending on one coordinate
Cylindrical_harmonics
In mathematics, a solution to a modified form of the confluent hypergeometric equation
functions Mκ,μ(z), Wκ,μ(z), defined in terms of Kummer's confluent hypergeometric functions M and U by M κ , μ ( z ) = exp ( − z / 2 ) z μ + 1 2 M ( μ
Whittaker_function
Function that is holomorphic on the whole complex plane
entire function that is not a polynomial. Every entire function f ( z ) {\displaystyle f(z)} can be represented as a single power series: f ( z ) = ∑
Entire_function
Mathematical function
Often the functions to be minimized are not f i {\displaystyle f_{i}} but | f i − z i ∗ | {\displaystyle |f_{i}-z_{i}^{*}|} for some scalars z i ∗ {\displaystyle
Chebyshev_function
Angle of complex number about real axis
π]. In this article the multi-valued function will be denoted arg(z) and its principal value will be denoted Arg(z), but in some sources the capitalization
Argument_(complex_analysis)
Function in probability theory
separation. For a random field or stochastic process Z(x) on a domain D, a covariance function C(x, y) gives the covariance of the values of the random
Covariance_function
Indicator function of positive numbers
principal value of the complex logarithm of z. Other definitions that are undefined at H(0) include: A piecewise function: H ( x ) := { 1 , x > 0 0 , x < 0 {\displaystyle
Heaviside_step_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
Method of solution to differential equations
Heaviside step function, J ν ( z ) {\textstyle J_{\nu }(z)} is a Bessel function, I ν ( z ) {\textstyle I_{\nu }(z)} is a modified Bessel function of the first
Green's_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
Derivative of a function with multiple variables
of change of the function in the x-direction. Sometimes, for z=f(x,y,…), the partial derivative of z with respect to x is denoted as ∂z∂x. Since a partial
Partial_derivative
Mathematical function
reciprocal gamma function is the function f ( z ) = 1 Γ ( z ) , {\displaystyle f(z)={\frac {1}{\Gamma (z)}},} where Γ(z) denotes the gamma function. Since the
Reciprocal_gamma_function
Second-order partial differential equation
( x , y , z ) {\displaystyle f(x,y,z)} is a twice-differentiable real-valued function. The Laplace operator therefore maps a scalar function to another
Laplace's_equation
these functions satisfy the identity cm 3 z + sm 3 z = 1 {\displaystyle \operatorname {cm} ^{3}z+\operatorname {sm} ^{3}z=1} , as real functions they
Dixon_elliptic_functions
Arithmetic operation
(-\ln {z})}={\frac {\mathrm {W} (-\ln {z})}{-\ln {z}}}~,} where W represents Lambert's W function. This formula follows from the assumption that z z ⋅ ⋅
Tetration
Table of probabilities related to the normal distribution
called the unit normal table or Z table, is a mathematical table for the values of Φ, the cumulative distribution function of the normal distribution. It
Standard_normal_table
mathematics, the Anger function, introduced by C. T. Anger (1855), is a function defined as J ν ( z ) = 1 π ∫ 0 π cos ( ν θ − z sin θ ) d θ {\displaystyle
Anger_function
Class of periodic mathematical functions
z + ω 1 ) = f ( z ) and f ( z + ω 2 ) = f ( z ) , ∀ z ∈ C . {\displaystyle f(z+\omega _{1})=f(z){\text{ and }}f(z+\omega _{2})=f(z),\quad \forall z\in
Elliptic_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
Extension of the domain of an analytic function (mathematics)
U, and F is an analytic function defined on V such that F ( z ) = f ( z ) ∀ z ∈ U , {\displaystyle F(z)=f(z)\qquad \forall z\in U,} then F is called an
Analytic_continuation
Probability distribution
{\pi }}}\int _{z(-\infty )}^{z(x_{0})}w(z)\,dz\right).} Substituting the definition of the Faddeeva function (scaled complex error function) yields for the
Voigt_profile
One-dimensional complex manifold
functions such as z {\displaystyle {\sqrt {z}}} or log ( z ) {\displaystyle \log(z)} , e.g. the subset of pairs ( z , w ) ∈ C 2 {\displaystyle (z,w)\in
Riemann_surface
Inverse of the gamma function
Let Γ n − 1 ( z ) {\displaystyle \Gamma _{n}^{-1}(z)} be the n {\displaystyle n} -th branch of the gamma function, with Γ 0 − 1 ( z ) {\displaystyle
Inverse_gamma_function
Mathematical function
Wright omega function or Wright function, denoted ω, is defined in terms of the Lambert W function as: ω ( z ) = W ⌈ I m ( z ) − π 2 π ⌉ ( e z ) . {\displaystyle
Wright_omega_function
Distance from zero to a number
of a complex number z is the square root of z⋅z‾, which is therefore called the absolute square or squared modulus of z: |z|=z⋅z‾. This generalizes the
Absolute_value
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