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...

Daniel Bernoulli, the gamma function Γ ( z ) {\displaystyle \Gamma (z)} is defined for all complex numbers z {\displaystyle z} except non-positive integers...

functions have the following property: if z and w are any complex numbers, then w e w = z {\displaystyle we^{w}=z} holds if and only if w = W k ( z )...

The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter ζ (zeta), is a mathematical function of a complex variable defined...

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...

erf (−z) = −erf z means that the error function is an odd function. This directly results from the fact that the integrand e−t2 is an even function (the...

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...

.. ( 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...

functional of V − z , {\displaystyle V-z,} which is a continuous sublinear function on X {\displaystyle X} since V − z {\displaystyle V-z} is convex, absorbing...

| 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...

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...

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...

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...

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...

entire 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 − w...

hypergeometric function 2F1(a,b;c;z) is a special function represented by the hypergeometric series, that includes many other special functions as specific...

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...

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, which is also a Dirichlet...

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}...

mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of...

x , z ) ∈ R ⟹ y = z {\displaystyle (x,y)\in R\land (x,z)\in R\implies y=z\qquad } Partial functions are defined similarly to ordinary functions, with...

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...

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...

, 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...

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...

G-function G(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and...

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 =...

probability distribution over predicted output classes. The softmax function takes as input a vector z of K real numbers, and normalizes it into a probability distribution...

Thomae's function is a real-valued function of a real variable that can be defined as:: 531  f ( x ) = { 1 q if  x = p q ( x  is rational), with  p ∈ Z  and ...

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...