Special functions

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Gamma function画像

• Gamma function
• Gamma function (graph)
• The Inverse of the Gamma Function (Graph)
• Imperfect Gamma Function
• Log-gamma function
• Polygamma function

Beta function画像

• Beta function
• Improper beta function
• Incomplete beta function (graph)
• Improper beta function (inverse function)

Zeta function画像

• Zeta function
• The Zeta Function (Graph)
• Dirichlet's theta function
• Dirichlet's theta function (graph)

Bessel functions画像

• Bessel functions
• Bessel functions (zeroes)
• Class 1 Bessel Functions (Graph)
• Bessel Functions of the Second Kind (Graph)
• Modified Bessel functions
• Type 1 Modified Bessel Functions (Graph)
• Type 2 Modified Bessel Functions (Graph)
• Hankel function
• Hankel function (graph)

Spherical Bessel functions画像

• Spherical Bessel functions
• Spherical Bessel functions (zeroes)
• Type 1 spherical Bessel functions (graph)
• Type 2 spherical Bessel functions (graph)
• Modified Bessel functions
• Type 1 Modified Spherical Bessel Functions (Graph)
• Type 2 Modified Spherical Bessel Functions (Graph)
• Spherical Hankel function
• Spherical Hankel function (graph)

Airy function画像

• Airy functions Ai, Bi
• Airy functions Ai, Bi (zero point)
• Airy Functions Ai, Bi (Graph)
• The first derivatives of the Airy functions, Ai' and Bi'
• First derivatives Ai' and Bi' of the Airy function (at the origin)
• First derivatives Ai' and Bi' of the Airy function (graph)

Kelvin function画像

• Type 1 Kelvin function
• Type 1 Kelvin Function (Graph)
• First-order derivative of the first-kind Kelvin function (graph)
• Type II Kelvin function
• Type 2 Kelvin Function (Graph)
• First Derivative of the Second-Kind Kelvin Function (Graph)
• Type 3 Kelvin function
• Type 3 Kelvin Function (Graph)
• First Derivative of the Type-3 Kelvin Function (Graph)

Struve function画像

• Struve function
• Struve function (graph)
• Modified Struve function
• Modified Struve Functions (Graph)

Error function画像

• Error function
• Error Function (Graph)
• Error function (inverse function)

Legendre function画像

• Legendre function
• Legendre's function

Bernoulli number画像

• Bernoulli number
• Bernoulli numbers (graph)
• Euler number

Sterling number画像

• Type I Stirling number
• Type 1 Stirling numbers (table)
• Type-2 Stirling number
• Type II Stirling numbers (table)

Fibonacci numbers画像

• Fibonacci numbers
• Fibonacci numbers (graph)
• Fibonacci Function (Graph)

Double factorial画像

• Double factorial
• Double Factorial (Graph)

Pockhammer function画像

• Pockhammer function
• Pockhammer function (graph)

Brillouin function画像

• Brillouin function
• Langevin function
• Brillouin function (graph)
• Langevin function (graph)

Activation function画像

• Sigmoid function
• The First Derivative of the Sigmoid Function
• The Second Derivative of the Sigmoid Function
• Sigmoid Function (Graph)
• Soft sign function
• First derivative of a soft sign function
• Soft Sign Function (Graph)
• Soft Plus Function
• First derivative of the soft-plus function
• Soft Plus Function (Graph)
• Softmax function
• ReLU
• ReLU (graph)
• Leaky ReLU
• Leaky ReLU (graph)
• tanh(x) function
• The first derivative of the tanh(x) function
• The second derivative of the tanh(x) function
• The tanh(x) Function (Graph)

Integral function画像

• Exponential integral En(x)
• Exponential Integral En(x) (Graph)
• Exponential integral Ei(x)
• Exponential Integral Ei(x) (Graph)
• Logarithmic integral li(x)
• Logarithmic integral li(x) (graph)
• Sine integral Si(x)
• Cosine integral Ci(x)
• Trigonometric Integrals Si(x), Ci(x) (Graph)
• Hyperbolic sine integral Shi(x)
• Hyperbolic cosine integral Chi(x)
• Integrals of hyperbolic functions Shi(x) and Chi(x) (graph)

Fresnel integral画像

• Fresnel sine integral S(x)
• Fresnel cosine integral C(x)
• Fresnel integrals S(x), C(x) (graph)

Oval integral画像

• Type 1 complete elliptic integral K(k)
• Type II complete elliptic integral E(k)
• Total elliptic integrals K(k), E(k) (graph)
• Type 3 complete elliptic integral Π(n,k)
• Type 1 Improper Elliptic Integral F(x,k)
• Type 1 Improper Elliptic Integral F(φ,k)
• Type II Improper Elliptic Integral E(x,k)
• Type 2 Improper Elliptic Integral E(φ,k)
• Type 3 Improper Elliptic Integral Π(x,n,k)
• Type 3 Improper Elliptic Integral Π(φ,n,k)

Elliptic functions画像

• SN function sn(u,k)
• CN function cn(u,k)
• DN function dn(u,k)
• Elliptic Functions sn, cn, dn (Graph)
• Amplitude function am(u,k)
• Amplitude Function am(u,k) (Graph)
• SD function sd(u,k)
• CD function cd(u,k)
• ND function nd(u,k)
• Inverse SN function arcsn(x, k)
• Inverse CN function arccn(x,k)
• Inverse DN function arcdn(x, k)

Orthogonal polynomials画像

• Hermite polynomials (graph)
• Legendre polynomials (graph)
• Legendre polynomials (graph)
• Type 1 Chebyshev polynomials (graph)
• Type II Chebyshev polynomials (graph)
• Lager Polynomials (Graph)
• Lager polynomials (graph)
• Geigenbauer polynomials (graph)
• Jacobian polynomial (graph)
• Spherical harmonic functions
• Spherical Harmonic Functions (Graph)

Hypergeometric function画像

• Gaussian hypergeometric function
• Type 1 Convergent Hypergeometric Function
• Type 2 Convergent Hypergeometric Function
• Type 1 Whittaker function
• Type 1 Whittaker function (graph)
• Type 2 Hoyt-Tucker Function
• Type 2 Whittaker function (graph)