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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
Extension of the factorial function
Daniel Bernoulli, the gamma function Γ ( z ) {\displaystyle \Gamma (z)} is defined for all complex numbers z {\displaystyle z} except non-positive integers
Gamma_function
Analytic function in mathematics
The Riemann zeta function or Euler–Riemann zeta function, denoted by the lowercase Greek letter ζ (zeta), is a mathematical function of a complex variable
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
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
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
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
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
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
The beta function is symmetric, meaning that B ( z 1 , z 2 ) = B ( z 2 , z 1 ) {\displaystyle \mathrm {B} (z_{1},z_{2})=\mathrm {B} (z_{2},z_{1})} for
Beta_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
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
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
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
Function that is holomorphic on the whole complex plane
functions such as the error function. If an entire function f ( z ) {\displaystyle f(z)} has a root at w {\displaystyle w} , then f ( z ) / ( z −
Entire_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)
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
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
Smooth approximation of one-hot arg max
softmax function σ : R K → ( 0 , 1 ) K {\displaystyle \sigma :\mathbb {R} ^{K}\to (0,1)^{K}} , where K > 1 {\displaystyle K>1} , takes a tuple z = ( z 1
Softmax_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
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
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
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
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
Type of mathematical function
multiple values, such as the elementary function z {\displaystyle {\sqrt {z}}} or log z {\displaystyle \log z} ) for every complex argument, except at
Elementary_function
Mathematical function, inverse of an exponential function
tangent function: ln ( z ) = 2 ⋅ artanh z − 1 z + 1 = 2 ( z − 1 z + 1 + 1 3 ( z − 1 z + 1 ) 3 + 1 5 ( z − 1 z + 1 ) 5 + ⋯ ) , {\displaystyle \ln(z)=2\cdot
Logarithm
Conjecture on zeros of the zeta function
function is real and non-zero. Using the expression for the zeta function on the critical line, ζ(1/2 + it) = Z(t)e−iθ(t), where Hardy's function, Z,
Riemann_hypothesis
Fundamental trigonometric functions
holomorphic function, sin z is a 2D solution of Laplace's equation: Δ u ( x 1 , x 2 ) = 0. {\displaystyle \Delta u(x_{1},x_{2})=0.} The complex sine function is
Sine_and_cosine
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
Operation on formal power series
function (OGF) of the sequence, denoted F ( z ) {\displaystyle F(z)} , and the exponential generating function (EGF) of the sequence, denoted F ^ ( z
Generating function transformation
Generating_function_transformation
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
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
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
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
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
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
Function that takes two inputs
binary function if and only if for any x ∈ X {\displaystyle x\in X} and y ∈ Y {\displaystyle y\in Y} , there exists a unique z ∈ Z {\displaystyle z\in Z} such
Binary_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
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
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
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
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
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
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
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
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
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
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
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
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
Evaluation of a function on its argument
is a function, and ( Y , z ) ∈ X ∅ otherwise {\displaystyle X(Y)=\left\{{\begin{array}{lll}z&{\text{if }}X{\text{ is a function, and }}(Y,z)\in X\\\varnothing
Function_application
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
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
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
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 whose derivative exists
using z = x + i y {\textstyle z=x+iy} , f ( z ) = z + z ¯ 2 {\textstyle f(z)={\frac {z+{\overline {z}}}{2}}} becomes the 2-variable real function f ( x
Differentiable_function
Statement in complex analysis; formerly the Bieberbach conjecture
z ) = z + ⋯ {\displaystyle f(z)=z+\cdots } is a schlicht function then φ ( z ) = z ( f ( z 2 ) / z 2 ) 1 / 2 {\displaystyle \varphi (z)=z(f(z^{2})/z^{2})^{1/2}}
De_Branges's_theorem
Number with a real and an imaginary part
{\displaystyle z_{0}} if the limit lim z → z 0 f ( z ) − f ( z 0 ) z − z 0 {\displaystyle \lim _{z\to z_{0}}{f(z)-f(z_{0}) \over z-z_{0}}} exists (in
Complex_number
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
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
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
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
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
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
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
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
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
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
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
Arithmetic operation
numbers z with the definition: ∞ z = ⋅ ⋅ z z z = e − W ( − ln z ) = W ( − ln z ) − ln z , {\displaystyle {}^{\infty }z=\cdot ^{\cdot ^{z^{z^{z}}}}=e^{-\mathrm
Tetration
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
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
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
Mathematical function relating circular and hyperbolic functions
those secant functions: d d z gd z = sech z , d d z gd − 1 z = sec z . {\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} z}}\operatorname
Gudermannian_function
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
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
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
equation: z 2 d 2 y d z 2 + z d y d z + ( z 2 − ν 2 ) y = z μ + 1 . {\displaystyle z^{2}{\frac {d^{2}y}{dz^{2}}}+z{\frac {dy}{dz}}+(z^{2}-\nu ^{2})y=z^{\mu
Lommel_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
Probability distribution
density function (or density): φ ( z ) = e − z 2 / 2 2 π . {\displaystyle \varphi (z)={\frac {e^{-z^{2}/2}}{\sqrt {2\pi }}}\,.} The variable z {\displaystyle
Normal_distribution
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
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
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
electrical network synthesis. They are complex functions, Z(s), of a complex variable, s. A rational function is defined to have the PR property if it has
Positive-real_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
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
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
Function uniquely mapping two numbers into a single number
ElegantUnpair [ z ] := { { z − ⌊ z ⌋ 2 , ⌊ z ⌋ } if z − ⌊ z ⌋ 2 < ⌊ z ⌋ , { ⌊ z ⌋ , z − ⌊ z ⌋ 2 − ⌊ z ⌋ } if z − ⌊ z ⌋ 2 ≥ ⌊ z ⌋ . {\displaystyle
Pairing_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
Mathematical Function
function (named after Adrien-Marie Legendre) is a special function whose Taylor series is also a Dirichlet series, given by χ ν ( z ) = ∑ k = 0 ∞ z 2
Legendre_chi_function
Function with two complex number "periods"
v} are periods of a function f {\displaystyle f} means that f ( z + u ) = f ( z + v ) = f ( z ) {\displaystyle f(z+u)=f(z+v)=f(z)\,} for all values of
Doubly_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
Indicator function of positive numbers
The Heaviside step function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a step function named after Oliver Heaviside
Heaviside_step_function
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
Special function defined by an integral
exponential times the function U ( 1 , 1 , z ) {\displaystyle U(1,1,z)} : E 1 ( z ) = e − z U ( 1 , 1 , z ) {\displaystyle E_{1}(z)=e^{-z}U(1,1,z)} The exponential
Exponential_integral
Mathematics function in complex analysis
z ¯ = 1 / z {\displaystyle {\overline {z}}=1/z} . Thus, the Schwarz function of the unit circle is S ( z ) = 1 / z {\displaystyle S(z)=1/z} . A more complicated
Schwarz_function
mathematics, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as Z ( V , s ) = exp (
Local_zeta_function
gamma function, then the Bessel–Clifford function is defined by the series C n ( z ) = ∑ k = 0 ∞ π ( k + n ) z k k ! {\displaystyle {\mathcal {C}}_{n}(z)=\sum
Bessel–Clifford_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
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