gsl

Octave bindings to the GNU Scientific Library

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Special functions

gsl_sf_airy_Ai
Computes the Airy function Ai(x) with an accuracy specified by mode.
gsl_sf_airy_Ai_deriv
Computes the Airy function derivative Ai'(x) with an accuracy specified by mode.
gsl_sf_airy_Ai_deriv_scaled
Computes the derivative of the scaled Airy function S_A(x) Ai(x).
gsl_sf_airy_Ai_scaled
Computes a scaled version of the Airy function S_A(x) Ai(x).
gsl_sf_airy_Bi
Computes the Airy function Bi(x) with an accuracy specified by mode.
gsl_sf_airy_Bi_deriv
Computes the Airy function derivative Bi'(x) with an accuracy specified by mode.
gsl_sf_airy_Bi_deriv_scaled
Computes the derivative of the scaled Airy function S_B(x) Bi(x).
gsl_sf_airy_Bi_scaled
Computes a scaled version of the Airy function S_B(x) Bi(x).
gsl_sf_airy_zero_Ai
Computes the location of the s-th zero of the Airy function Ai(x).
gsl_sf_airy_zero_Ai_deriv
Computes the location of the s-th zero of the Airy function derivative Ai(x).
gsl_sf_airy_zero_Bi
Computes the location of the s-th zero of the Airy function Bi(x).
gsl_sf_airy_zero_Bi_deriv
Computes the location of the s-th zero of the Airy function derivative Bi(x).
gsl_sf_atanint
Computes the Arctangent integral AtanInt(x) = \int_0^x dt \arctan(t)/t.
gsl_sf_bessel_il_scaled
Computes the scaled regular modified spherical Bessel function of order l, \exp(-|x|) i_l(x)
gsl_sf_bessel_il_scaled_array
This routine computes the values of the scaled regular modified spherical Bessel functions \exp(-|x|) i_l(x) for l from 0 to lmax inclusive for lmax >= 0.
gsl_sf_bessel_In
Computes the regular modified cylindrical Bessel function of order n, I_n(x).
gsl_sf_bessel_In_array
his routine computes the values of the regular modified cylindrical Bessel functions I_n(x) for n from nmin to nmax inclusive.
gsl_sf_bessel_In_scaled
Computes the scaled regular modified cylindrical Bessel function of order n, \exp(-|x|) I_n(x)
gsl_sf_bessel_In_scaled_array
This routine computes the values of the scaled regular cylindrical Bessel functions \exp(-|x|) I_n(x) for n from nmin to nmax inclusive.
gsl_sf_bessel_Inu
Computes the regular modified Bessel function of fractional order nu, I_\nu(x) for x>0, \nu>0.
gsl_sf_bessel_Inu_scaled
Computes the scaled regular modified Bessel function of fractional order nu, \exp(-|x|)I_\nu(x) for x>0, \nu>0.
gsl_sf_bessel_jl
Computes the regular spherical Bessel function of order l, j_l(x), for l >= 0 and x >= 0.
gsl_sf_bessel_jl_array
Computes the values of the regular spherical Bessel functions j_l(x) for l from 0 to lmax inclusive for lmax >= 0 and x >= 0.
gsl_sf_bessel_jl_steed_array
This routine uses Steed’s method to compute the values of the regular spherical Bessel functions j_l(x) for l from 0 to lmax inclusive for lmax >= 0 and x >= 0.
gsl_sf_bessel_Jn
Computes the regular cylindrical Bessel function of order n, J_n(x).
gsl_sf_bessel_Jn_array
This routine computes the values of the regular cylindrical Bessel functions J_n(x) for n from nmin to nmax inclusive.
gsl_sf_bessel_Jnu
Computes the regular cylindrical Bessel function of fractional order nu, J_\nu(x).
gsl_sf_bessel_kl_scaled
Computes the scaled irregular modified spherical Bessel function of order l, \exp(x) k_l(x), for x>0.
gsl_sf_bessel_kl_scaled_array
This routine computes the values of the scaled irregular modified spherical Bessel functions \exp(x) k_l(x) for l from 0 to lmax inclusive for lmax >= 0 and x>0.
gsl_sf_bessel_Kn
Computes the irregular modified cylindrical Bessel function of order n, K_n(x), for x > 0.
gsl_sf_bessel_Kn_array
This routine computes the values of the irregular modified cylindrical Bessel functions K_n(x) for n from nmin to nmax inclusive.
gsl_sf_bessel_Kn_scaled
ERR contains an estimate of the absolute error in the value Z.
gsl_sf_bessel_Kn_scaled_array
This routine computes the values of the scaled irregular cylindrical Bessel functions \exp(x) K_n(x) for n from nmin to nmax inclusive.
gsl_sf_bessel_Knu
Computes the irregular modified Bessel function of fractional order nu, K_\nu(x) for x>0, \nu>0.
gsl_sf_bessel_Knu_scaled
Computes the scaled irregular modified Bessel function of fractional order nu, \exp(+|x|) K_\nu(x) for x>0, \nu>0.
gsl_sf_bessel_lnKnu
Computes the logarithm of the irregular modified Bessel function of fractional order nu, \ln(K_\nu(x)) for x>0, \nu>0.
gsl_sf_bessel_yl
Computes the irregular spherical Bessel function of order l, y_l(x), for l >= 0.
gsl_sf_bessel_yl_array
This routine computes the values of the irregular spherical Bessel functions y_l(x) for l from 0 to lmax inclusive for lmax >= 0.
gsl_sf_bessel_Yn
Computes the irregular cylindrical Bessel function of order n, Y_n(x), for x>0.
gsl_sf_bessel_Yn_array
This routine computes the values of the irregular cylindrical Bessel functions Y_n(x) for n from nmin to nmax inclusive.
gsl_sf_bessel_Ynu
Computes the irregular cylindrical Bessel function of fractional order nu, Y_\nu(x).
gsl_sf_bessel_zero_J0
Computes the location of the s-th positive zero of the Bessel function J_0(x).
gsl_sf_bessel_zero_J1
Computes the location of the s-th positive zero of the Bessel function J_1(x).
gsl_sf_bessel_zero_Jnu
Computes the location of the n-th positive zero of the Bessel function J_x().
gsl_sf_beta
Computes the Beta Function, B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b) for a > 0, b > 0.
gsl_sf_beta_inc
Computes the normalized incomplete Beta function
gsl_sf_Chi
Computes the integral
gsl_sf_choose
Computes the combinatorial factor n choose m = n!/(m!(n-m)!).
gsl_sf_Ci
Computes the Cosine integral Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0.
gsl_sf_clausen
The Clausen function is defined by the following integral,
gsl_sf_conicalP_0
Computes the conical function P^0_{-1/2 + i \lambda}(x) for x > -1.
gsl_sf_conicalP_1
Computes the conical function P^1_{-1/2 + i \lambda}(x) for x > -1.
gsl_sf_conicalP_cyl_reg
Computes the Regular Cylindrical Conical Function P^{-m}_{-1/2 + i \lambda}(x), for x > -1, m >= -1.
gsl_sf_conicalP_half
Computes the irregular Spherical Conical Function P^{1/2}_{-1/2 + i \lambda}(x) for x > -1.
gsl_sf_conicalP_mhalf
Computes the regular Spherical Conical Function P^{-1/2}_{-1/2 + i \lambda}(x) for x > -1.
gsl_sf_conicalP_sph_reg
Computes the Regular Spherical Conical Function P^{-1/2-l}_{-1/2 + i \lambda}(x), for x > -1, l >= -1.
gsl_sf_coupling_3j
computes the Wigner 3-j coefficient,
gsl_sf_coupling_6j
computes the Wigner 6-j coefficient,
gsl_sf_coupling_9j
computes the Wigner 9-j coefficient,
gsl_sf_dawson
The Dawson integral is defined by \exp(-x^2) \int_0^x dt \exp(t^2).
gsl_sf_debye_1
The Debye functions are defined by the integral
gsl_sf_debye_2
The Debye functions are defined by the integral
gsl_sf_debye_3
The Debye functions are defined by the integral
gsl_sf_debye_4
The Debye functions are defined by the integral
gsl_sf_debye_5
The Debye functions are defined by the integral
gsl_sf_debye_6
The Debye functions are defined by the integral
gsl_sf_dilog
Computes the dilogarithm for a real argument.
gsl_sf_doublefact
Compute the double factorial n!! = n(n-2)(n-4)\dots.
gsl_sf_ellint_D
This function computes the incomplete elliptic integral D(\phi,k) which is defined through the Carlson form RD(x,y,z) by the following relation,
gsl_sf_ellint_E
This routine computes the elliptic integral E(\phi,k) to the accuracy specified by the mode variable mode.
gsl_sf_ellint_Ecomp
Computes the complete elliptic integral E(k) to the accuracy specified by the mode variable mode.
gsl_sf_ellint_F
This routine computes the elliptic integral F(\phi,k) to the accuracy specified by the mode variable mode.
gsl_sf_ellint_Kcomp
Computes the complete elliptic integral K(k) pi --- 2 /
gsl_sf_ellint_P
This routine computes the incomplete elliptic integral \Pi(\phi,k,n) to the accuracy specified by the mode variable mode.
gsl_sf_ellint_Pcomp
Computes the complete elliptic integral \Pi(k,n) to the accuracy specified by the mode variable mode.
gsl_sf_ellint_RC
This routine computes the incomplete elliptic integral RC(x,y) to the accuracy specified by the mode variable mode.
gsl_sf_ellint_RD
This routine computes the incomplete elliptic integral RD(x,y,z) to the accuracy specified by the mode variable mode.
gsl_sf_ellint_RF
This routine computes the incomplete elliptic integral RF(x,y,z) to the accuracy specified by the mode variable mode.
gsl_sf_ellint_RJ
This routine computes the incomplete elliptic integral RJ(x,y,z,p) to the accuracy specified by the mode variable mode.
gsl_sf_erf
Computes the error function erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).
gsl_sf_erfc
Computes the complementary error function erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2).
gsl_sf_erf_Q
Computes the upper tail of the Gaussian probability function Q(x) = (1/(2\pi)) \int_x^\infty dt \exp(-t^2/2).
gsl_sf_erf_Z
Computes the Gaussian probability function Z(x) = (1/(2\pi)) \exp(-x^2/2).
gsl_sf_eta
Computes the eta function \eta(s) for arbitrary s.
gsl_sf_eta_int
Computes the eta function \eta(n) for integer n.
gsl_sf_expint_3
Computes the exponential integral Ei_3(x) = \int_0^x dt \exp(-t^3) for x >= 0.
gsl_sf_expint_E1
Computes the exponential integral E_1(x),
gsl_sf_expint_E2
Computes the second-order exponential integral E_2(x),
gsl_sf_expint_Ei
Computes the exponential integral E_i(x),
gsl_sf_expm1
Computes the quantity \exp(x)-1 using an algorithm that is accurate for small x.
gsl_sf_exp_mult
These routines exponentiate x and multiply by the factor y to return the product y \exp(x).
gsl_sf_exprel
Computes the quantity (\exp(x)-1)/x using an algorithm that is accurate for small x.
gsl_sf_exprel_2
Computes the quantity 2(\exp(x)-1-x)/x^2 using an algorithm that is accurate for small x.
gsl_sf_exprel_n
Computes the N-relative exponential, which is the n-th generalization of the functions gsl_sf_exprel and gsl_sf_exprel2.
gsl_sf_fact
Computes the factorial n!.
gsl_sf_fermi_dirac_half
Computes the complete Fermi-Dirac integral F_{1/2}(x).
gsl_sf_fermi_dirac_inc_0
Computes the incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x).
gsl_sf_fermi_dirac_int
Computes the complete Fermi-Dirac integral with an integer index of j, F_j(x) = (1/\Gamma(j+1)) \int_0^\infty dt (t^j /(\exp(t-x)+1)).
gsl_sf_fermi_dirac_mhalf
Computes the complete Fermi-Dirac integral F_{-1/2}(x).
gsl_sf_gamma
Computes the Gamma function \Gamma(x), subject to x not being a negative integer.
gsl_sf_gamma_inc
These functions compute the incomplete Gamma Function the normalization factor included in the previously defined functions: \Gamma(a,x) = \int_x\infty dt t^{a-1} \exp(-t) for a real and x >= 0.
gsl_sf_gamma_inc_P
Computes the complementary normalized incomplete Gamma Function P(a,x) = 1/\Gamma(a) \int_0^x dt t^{a-1} \exp(-t) for a > 0, x >= 0.
gsl_sf_gamma_inc_Q
Computes the normalized incomplete Gamma Function Q(a,x) = 1/\Gamma(a) \int_x\infty dt t^{a-1} \exp(-t) for a > 0, x >= 0.
gsl_sf_gammainv
Computes the reciprocal of the gamma function, 1/\Gamma(x) using the real Lanczos method.
gsl_sf_gammastar
Computes the regulated Gamma Function \Gamma^*(x) for x > 0.
gsl_sf_gegenpoly_array
This function computes an array of Gegenbauer polynomials C^{(\lambda)}_n(x) for n = 0, 1, 2, \dots, nmax, subject to \lambda > -1/2, nmax >= 0.
gsl_sf_gegenpoly_n
These functions evaluate the Gegenbauer polynomial C^{(\lambda)}_n(x) for n, lambda, x subject to \lambda > -1/2, n >= 0.
gsl_sf_hazard
The hazard function for the normal distrbution, also known as the inverse Mill\'s ratio, is defined as h(x) = Z(x)/Q(x) = \sqrt{2/\pi \exp(-x^2 / 2) / \erfc(x/\sqrt 2)}.
gsl_sf_hydrogenicR
This routine computes the n-th normalized hydrogenic bound state radial wavefunction,
gsl_sf_hyperg_0F1
Computes the hypergeometric function 0F1(c,x).
gsl_sf_hyperg_1F1
Primary Confluent Hypergoemetric U function A&E 13.1.3 All inputs are double as is the output.
gsl_sf_hyperg_1F1_int
Primary Confluent Hypergoemetric U function A&E 13.1.3 m and n are integers.
gsl_sf_hyperg_2F0
Computes the hypergeometric function 2F0(a,b,x).
gsl_sf_hyperg_2F1
Computes the Gauss hypergeometric function 2F1(a,b,c,x) = F(a,b,c,x) for |x| < 1.
gsl_sf_hyperg_U
Secondary Confluent Hypergoemetric U function A&E 13.1.3 All inputs are double as is the output.
gsl_sf_hyperg_U_int
Secondary Confluent Hypergoemetric U function A&E 13.1.3 m and n are integers.
gsl_sf_hzeta
Computes the Hurwitz zeta function \zeta(s,q) for s > 1, q > 0.
gsl_sf_laguerre_n
Computes the generalized Laguerre polynomial L^a_n(x) for a > -1 and n >= 0.
gsl_sf_lambert_W0
These compute the principal branch of the Lambert W function, W_0(x).
gsl_sf_lambert_Wm1
These compute the secondary real-valued branch of the Lambert W function, W_{-1}(x).
gsl_sf_legendre_array
Calculate all normalized associated Legendre polynomials for 0 <= l <= lmax and 0 <= m <= l for |x| <= 1.
gsl_sf_legendre_deriv_array
Calculate all normalized associated Legendre polynomials and their first derivatives for 0 <= l <= lmax and 0 <= m <= l for |x| <= 1.
gsl_sf_legendre_Pl
These functions evaluate the Legendre polynomial P_l(x) for a specific value of l, x subject to l >= 0, |x| <= 1
gsl_sf_legendre_Pl_array
These functions compute arrays of Legendre polynomials P_l(x) and derivatives dP_l(x)/dx, for l = 0, \dots, lmax, |x| <= 1.
gsl_sf_legendre_Plm
Computes the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1.
gsl_sf_legendre_Plm_array
Compute arrays of Legendre polynomials P_l^m(x) for m >= 0, l = |m|, ..., lmax, |x| <= 1.
gsl_sf_legendre_Plm_deriv_array
Compute arrays of Legendre polynomials P_l^m(x) and derivatives dP_l^m(x)/dx for m >= 0, l = |m|, ..., lmax, |x| <= 1.
gsl_sf_legendre_Ql
Computes the Legendre function Q_l(x) for x > -1, x != 1 and l >= 0.
gsl_sf_legendre_sphPlm
Computes the normalized associated Legendre polynomial $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ suitable for use in spherical harmonics.
gsl_sf_legendre_sphPlm_array
Computes an array of normalized associated Legendre functions sqrt((2l+1)/(4*pi)) * sqrt((l-m)!/(l+m)!) Plm (x) for m >= 0, l = |m|, ..., lmax, |x| <= 1.0
gsl_sf_legendre_sphPlm_deriv_array
Computes an array of normalized associated Legendre functions sqrt((2l+1)/(4*pi)) * sqrt((l-m)!/(l+m)!) Plm (x) for m >= 0, l = |m|, ..., lmax, |x| <= 1.0 and their derivatives.
gsl_sf_lnbeta
Computes the logarithm of the Beta Function, \log(B(a,b)) for a > 0, b > 0.
gsl_sf_lnchoose
Computes the logarithm of n choose m.
gsl_sf_lncosh
Computes \log(\cosh(x)) for any x.
gsl_sf_lndoublefact
Computes the logarithm of the double factorial of n, \log(n!!).
gsl_sf_lnfact
Computes the logarithm of the factorial of n, \log(n!).
gsl_sf_lngamma
Computes the logarithm of the Gamma function, \log(\Gamma(x)), subject to x not a being negative integer.
gsl_sf_lnpoch
Computes the logarithm of the Pochhammer symbol, \log((a)_x) = \log(\Gamma(a + x)/\Gamma(a)) for a > 0, a+x > 0.
gsl_sf_lnsinh
Computes \log(\sinh(x)) for x > 0.
gsl_sf_log_1plusx
Computes \log(1 + x) for x > -1 using an algorithm that is accurate for small x.
gsl_sf_log_1plusx_mx
Computes \log(1 + x) - x for x > -1 using an algorithm that is accurate for small x.
gsl_sf_log_erfc
Computes the logarithm of the complementary error function \log(\erfc(x)).
gsl_sf_mathieu_a
Computes the characteristic values a_n(q) of the Mathieu function ce_n(q,x).
gsl_sf_mathieu_b
Computes the characteristic values b_n(q) of the Mathieu function se_n(q,x).
gsl_sf_mathieu_ce
This routine computes the angular Mathieu function ce_n(q,x).
gsl_sf_mathieu_Mc
This routine computes the radial j-th kind Mathieu function Mc_n^{(j)}(q,x) of order n.
gsl_sf_mathieu_Ms
This routine computes the radial j-th kind Mathieu function Ms_n^{(j)}(q,x) of order n.
gsl_sf_mathieu_se
This routine computes the angular Mathieu function se_n(q,x).
gsl_sf_poch
Computes the Pochhammer symbol
gsl_sf_pochrel
Computes the relative Pochhammer symbol ((a,x) - 1)/x where (a,x) = (a)_x := \Gamma(a + x)/\Gamma(a).
gsl_sf_psi
Computes the digamma function \psi(x) for general x, x \ne 0.
gsl_sf_psi_1_int
Computes the Trigamma function \psi(n) for positive integer n.
gsl_sf_psi_1piy
Computes the real part of the digamma function on the line 1+i y, Re[\psi(1 + i y)].
gsl_sf_psi_n
Computes the polygamma function \psi^{(m)}(x) for m >= 0, x > 0.
gsl_sf_Shi
Computes the integral Shi(x) = \int_0^x dt \sinh(t)/t.
gsl_sf_Si
Computes the Sine integral Si(x) = \int_0^x dt \sin(t)/t.
gsl_sf_sinc
Computes \sinc(x) = \sin(\pi x) / (\pi x) for any value of x.
gsl_sf_synchrotron_1
Computes the first synchrotron function x \int_x^\infty dt K_{5/3}(t) for x >= 0.
gsl_sf_synchrotron_2
Computes the second synchrotron function x K_{2/3}(x) for x >= 0.
gsl_sf_taylorcoeff
Computes the Taylor coefficient x^n / n! for x >= 0, n >= 0.
gsl_sf_transport_2
Computes the transport function J(2,x).
gsl_sf_transport_3
Computes the transport function J(3,x).
gsl_sf_transport_4
Computes the transport function J(4,x).
gsl_sf_transport_5
Computes the transport function J(5,x).
gsl_sf_zeta
Computes the Riemann zeta function \zeta(s) for arbitrary s, s \ne 1.
gsl_sf_zeta_int
Computes the Riemann zeta function \zeta(n) for integer n, n \ne 1.
gsl_sf_zetam1
Computes \zeta(s) - 1 for arbitrary s, s \ne 1, where \zeta denotes the Riemann zeta function.
gsl_sf_zetam1_int
Computes \zeta(s) - 1 for integer n, n \ne 1, where \zeta denotes the Riemann zeta function.
gsl_zt_zeta
Computes the Riemann zeta function \zeta(s) for arbitrary s, s \ne 1.

Support

gsl_sf
gsl_sf is an oct-file containing Octave bindings to the special functions of the GNU Scientific Library (GSL).

Package: gsl