Special Mathematical Functions The torch.special module provides special mathematical functions including gamma functions, error functions, Bessel functions, and orthogonal polynomials.
Gamma Functions gammaln torch.special.gammaln( input , * , out = None )
Computes the natural logarithm of the absolute value of the gamma function.
Natural log of the absolute value of Gamma function: ln(|Γ(input)|)
digamma torch.special.digamma( input , * , out = None )
Computes the logarithmic derivative of the gamma function (also called psi function).
Digamma function: ψ(x) = d/dx ln(Γ(x)) = Γ'(x)/Γ(x)
polygamma torch.special.polygamma(n, input , * , out = None )
Computes the nth derivative of the digamma function.
The order of the polygamma function (n ≥ 0).
multigammaln torch.special.multigammaln( input , p, * , out = None )
Computes the multivariate log-gamma function with dimension p.
The number of dimensions.
gammainc / gammaincc torch.special.gammainc( input , other, * , out = None ) torch.special.gammaincc( input , other, * , out = None )
Computes the regularized lower/upper incomplete gamma function.
The first non-negative input tensor.
The second non-negative input tensor.
Error Functions erf torch.special.erf( input , * , out = None )
Computes the error function.
Error function: erf(x) = (2/√π) ∫₀ˣ e^(-t²) dt
erfc torch.special.erfc( input , * , out = None )
Computes the complementary error function: erfc(x) = 1 - erf(x).
erfcx torch.special.erfcx( input , * , out = None )
Computes the scaled complementary error function: erfcx(x) = e^(x²) * erfc(x).
erfinv torch.special.erfinv( input , * , out = None )
Computes the inverse error function.
The input tensor (values must be in (-1, 1)).
ndtr torch.special.ndtr( input , * , out = None )
Computes the area under the standard Gaussian probability density function, integrated from minus infinity to input.
Standard normal CDF: Φ(x) = (1/√(2π)) ∫₋∞ˣ e^(-t²/2) dt
ndtri torch.special.ndtri( input , * , out = None )
Computes the inverse of ndtr (quantile function for standard normal distribution).
log_ndtr torch.special.log_ndtr( input , * , out = None )
Computes the log of the standard normal CDF.
Bessel Functions bessel_j0 / bessel_j1 torch.special.bessel_j0( input , * , out = None ) torch.special.bessel_j1( input , * , out = None )
Bessel function of the first kind of order 0 and 1.
bessel_y0 / bessel_y1 torch.special.bessel_y0( input , * , out = None ) torch.special.bessel_y1( input , * , out = None )
Bessel function of the second kind of order 0 and 1.
modified_bessel_i0 / modified_bessel_i1 torch.special.modified_bessel_i0( input , * , out = None ) torch.special.modified_bessel_i1( input , * , out = None )
Modified Bessel function of the first kind of order 0 and 1.
modified_bessel_k0 / modified_bessel_k1 torch.special.modified_bessel_k0( input , * , out = None ) torch.special.modified_bessel_k1( input , * , out = None )
Modified Bessel function of the second kind of order 0 and 1.
i0 / i0e / i1 / i1e torch.special.i0( input , * , out = None ) torch.special.i0e( input , * , out = None ) # exponentially scaled torch.special.i1( input , * , out = None ) torch.special.i1e( input , * , out = None ) # exponentially scaled
Modified Bessel functions (aliases and scaled versions).
spherical_bessel_j0 torch.special.spherical_bessel_j0( input , * , out = None )
Spherical Bessel function of the first kind of order 0.
Orthogonal Polynomials chebyshev_polynomial_t / u / v / w torch.special.chebyshev_polynomial_t( input , n, * , out = None ) torch.special.chebyshev_polynomial_u( input , n, * , out = None ) torch.special.chebyshev_polynomial_v( input , n, * , out = None ) torch.special.chebyshev_polynomial_w( input , n, * , out = None )
Chebyshev polynomials of the first, second, third, and fourth kinds.
Degree of the polynomial.
hermite_polynomial_h / he torch.special.hermite_polynomial_h( input , n, * , out = None ) # Physicist's torch.special.hermite_polynomial_he( input , n, * , out = None ) # Probabilist's
Hermite polynomials.
laguerre_polynomial_l torch.special.laguerre_polynomial_l( input , n, * , out = None )
Laguerre polynomial.
legendre_polynomial_p torch.special.legendre_polynomial_p( input , n, * , out = None )
Legendre polynomial.
Other Special Functions expit torch.special.expit( input , * , out = None )
Computes the expit (logistic sigmoid) function: expit(x) = 1/(1 + e^(-x)).
logit torch.special.logit( input , eps = None , * , out = None )
Computes the logit function (inverse of expit).
Epsilon for input clamping. If provided, input is clamped to [eps, 1-eps].
xlog1py / xlogy torch.special.xlog1py( input , other, * , out = None ) torch.special.xlogy( input , other, * , out = None )
Computes input * log1p(other) and input * log(other) with special handling for zero inputs.
entr torch.special.entr( input , * , out = None )
Computes the entropy: entr(x) = -x * ln(x) for x > 0, 0 for x = 0, and -∞ for x < 0.
exp2 / expm1 / log1p torch.special.exp2( input , * , out = None ) # 2^x torch.special.expm1( input , * , out = None ) # e^x - 1 torch.special.log1p( input , * , out = None ) # log(1 + x)
Exponential and logarithm variants.
sinc torch.special.sinc( input , * , out = None )
Computes the normalized sinc function: sinc(x) = sin(πx)/(πx) for x ≠ 0, 1 for x = 0.
zeta torch.special.zeta( input , other, * , out = None )
Computes the Hurwitz zeta function: ζ(x, q) = Σ(k=0 to ∞) 1/(k+q)^x.
airy_ai torch.special.airy_ai( input , * , out = None )
Airy function Ai(x).
Example Usage Gamma Functions
Error Functions
Bessel Functions
Orthogonal Polynomials
Sigmoid and Logit
Special Logarithms
import torch import torch.special as special # Gamma function logarithm x = torch.tensor([ 1.0 , 2.0 , 3.0 , 4.0 ]) gammaln_x = special.gammaln(x) print ( f "ln(Γ(x)): { gammaln_x } " ) # Note: Γ(n) = (n-1)! for positive integers # So Γ(4) = 3! = 6, ln(6) ≈ 1.79 # Digamma function digamma_x = special.digamma(x) print ( f "ψ(x): { digamma_x } " ) # Polygamma (second derivative) polygamma_x = special.polygamma( 2 , x) print ( f "ψ''(x): { polygamma_x } " ) # Incomplete gamma functions a = torch.tensor([ 2.0 , 3.0 ]) x = torch.tensor([ 1.0 , 2.0 ]) lower = special.gammainc(a, x) upper = special.gammaincc(a, x) print ( f "Lower + Upper = { lower + upper } " ) # Should be all 1's
Use Cases
Statistical Distributions
ndtr, ndtri: Normal distribution CDF and quantileserf, erfc: Related to normal distributiongammaln: Log-likelihood in gamma distributionsdigamma, polygamma: Maximum likelihood estimation
Bessel functions: Solving wave equations, filters
sinc: Sinc interpolation, digital signal processingOrthogonal polynomials: Spectral methods, approximation theory
expit, logit: Logistic regression, binary classificationsoftmax, log_softmax: Multi-class classificationxlogy, entr: Information theory, KL divergencegammaln: Variational inference, Bayesian methods