GSL special function
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GSL special function

For more information about the functions see the documentation of GSL.

FunctionDescription
gsl_log1p(x)log(1+x)
gsl_expm1(x)exp(x)-1
gsl_hypot(x,y)sqrt{x^2 + y^2}
gsl_acosh(x)arccosh(x)
gsl_asinh(x)arcsinh(x)
gsl_atanh(x)arctanh(x)
airy_Ai(x)Airy function Ai(x)
airy_Bi(x)Airy function Bi(x)
airy_Ais(x)scaled version of the Airy function S_A(x) Ai(x)
airy_Bis(x)scaled version of the Airy function S_B(x) Bi(x)
airy_Aid(x)Airy function derivative Ai'(x)
airy_Bid(x)Airy function derivative Bi'(x)
airy_Aids(x)derivative of the scaled Airy function S_A(x) Ai(x)
airy_Bids(x)derivative of the scaled Airy function S_B(x) Bi(x)
airy_0_Ai(s)s-th zero of the Airy function Ai(x)
airy_0_Bi(s)s-th zero of the Airy function Bi(x)
airy_0_Aid(s)s-th zero of the Airy function derivative Ai'(x)
airy_0_Bid(s)s-th zero of the Airy function derivative Bi'(x)
bessel_JJ0(x)regular cylindrical Bessel function of zeroth order, J_0(x)
bessel_JJ1(x)regular cylindrical Bessel function of first order, J_1(x)
bessel_Jn(n,x)regular cylindrical Bessel function of order n, J_n(x)
bessel_YY0(x)irregular cylindrical Bessel function of zeroth order, Y_0(x)
bessel_YY1(x)irregular cylindrical Bessel function of first order, Y_1(x)
bessel_Yn(n,x)irregular cylindrical Bessel function of order n, Y_n(x)
bessel_I0(x)regular modified cylindrical Bessel function of zeroth order, I_0(x)
bessel_I1(x)regular modified cylindrical Bessel function of first order, I_1(x)
bessel_In(n,x)regular modified cylindrical Bessel function of order n, I_n(x)
bessel_II0s(x)scaled regular modified cylindrical Bessel function of zeroth order, exp (-|x|) I_0(x)
bessel_II1s(x)scaled regular modified cylindrical Bessel function of first order, exp(-|x|) I_1(x)
bessel_Ins(n,x)scaled regular modified cylindrical Bessel function of order n, exp(-|x|) I_n(x)
bessel_K0(x)irregular modified cylindrical Bessel function of zeroth order, K_0(x)
bessel_K1(x)irregular modified cylindrical Bessel function of first order, K_1(x)
bessel_Kn(n,x)irregular modified cylindrical Bessel function of order n, K_n(x)
bessel_KK0s(x)scaled irregular modified cylindrical Bessel function of zeroth order, exp (x) K_0(x)
bessel_KK1s(x)scaled irregular modified cylindrical Bessel function of first order, exp(x) K_1(x)
bessel_Kns(n,x)scaled irregular modified cylindrical Bessel function of order n, exp(x) K_n(x)
bessel_j0(x)regular spherical Bessel function of zeroth order, j_0(x)
bessel_j1(x)regular spherical Bessel function of first order, j_1(x)
bessel_j2(x)regular spherical Bessel function of second order, j_2(x)
bessel_jl(l,x)regular spherical Bessel function of order l, j_l(x)
bessel_y0(x)irregular spherical Bessel function of zeroth order, y_0(x)
bessel_y1(x)irregular spherical Bessel function of first order, y_1(x)
bessel_y2(x)irregular spherical Bessel function of second order, y_2(x)
bessel_yl(l,x)irregular spherical Bessel function of order l, y_l(x)
bessel_i0s(x)scaled regular modified spherical Bessel function of zeroth order, exp(-|x|) i_0(x)
bessel_i1s(x)scaled regular modified spherical Bessel function of first order, exp(-|x|) i_1(x)
bessel_i2s(x)scaled regular modified spherical Bessel function of second order, exp(-|x|) i_2(x)
bessel_ils(l,x)scaled regular modified spherical Bessel function of order l, exp(-|x|) i_l(x)
bessel_k0s(x)scaled irregular modified spherical Bessel function of zeroth order, exp(x) k_0(x)
bessel_k1s(x)scaled irregular modified spherical Bessel function of first order, exp(x) k_1(x)
bessel_k2s(x)scaled irregular modified spherical Bessel function of second order, exp(x) k_2(x)
bessel_kls(l,x)scaled irregular modified spherical Bessel function of order l, exp(x) k_l(x)
bessel_Jnu(nu,x)regular cylindrical Bessel function of fractional order nu, J_\nu(x)
bessel_Ynu(nu,x)irregular cylindrical Bessel function of fractional order nu, Y_\nu(x)
bessel_Inu(nu,x)regular modified Bessel function of fractional order nu, I_\nu(x)
bessel_Inus(nu,x)scaled regular modified Bessel function of fractional order nu, exp(-|x|) I_\nu(x)
bessel_Knu(nu,x)irregular modified Bessel function of fractional order nu, K_\nu(x)
bessel_lnKnu(nu,x)logarithm of the irregular modified Bessel function of fractional order nu,ln(K_\nu(x))
bessel_Knus(nu,x)scaled irregular modified Bessel function of fractional order nu, exp(|x|) K_\nu(x)
bessel_0_J0(s)s-th positive zero of the Bessel function J_0(x)
bessel_0_J1(s)s-th positive zero of the Bessel function J_1(x)
bessel_0_Jnu(nu,s)s-th positive zero of the Bessel function J_nu(x)
clausen(x)Clausen integral Cl_2(x)
hydrogenicR_1(Z,R)lowest-order normalized hydrogenic bound state radial wavefunction R_1 := 2Z \sqrt{Z} \exp(-Z r)
hydrogenicR(n,l,Z,R)n-th normalized hydrogenic bound state radial wavefunction
dawson(x)Dawson's integral
debye_1(x)first-order Debye function D_1(x) = (1/x) \int_0^x dt (t/(e^t - 1))
debye_2(x)second-order Debye function D_2(x) = (2/x^2) \int_0^x dt (t^2/(e^t - 1))
debye_3(x)third-order Debye function D_3(x) = (3/x^3) \int_0^x dt (t^3/(e^t - 1))
debye_4(x)fourth-order Debye function D_4(x) = (4/x^4) \int_0^x dt (t^4/(e^t - 1))
dilog(x)dilogarithm
ellint_Kc(k)complete elliptic integral K(k)
ellint_Ec(k)complete elliptic integral E(k)
ellint_F(phi,k)incomplete elliptic integral F(phi,k)
ellint_E(phi,k)incomplete elliptic integral E(phi,k)
ellint_P(phi,k,n)incomplete elliptic integral P(phi,k,n)
ellint_D(phi,k,n)incomplete elliptic integral D(phi,k,n)
ellint_RC(x,y)incomplete elliptic integral RC(x,y)
ellint_RD(x,y,z)incomplete elliptic integral RD(x,y,z)
ellint_RF(x,y,z)incomplete elliptic integral RF(x,y,z)
ellint_RJ(x,y,z)incomplete elliptic integral RJ(x,y,z,p)
gsl_erf(x)error function erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2)
gsl_erfc(x)complementary error function erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2)
log_erfc(x)logarithm of the complementary error function \log(\erfc(x))
erf_Z(x)Gaussian probability function Z(x) = (1/(2\pi)) \exp(-x^2/2)
erf_Q(x)upper tail of the Gaussian probability function Q(x) = (1/(2\pi)) \int_x^\infty dt \exp(-t^2/2)
gsl_exp(x)exponential function
exprel(x)(exp(x)-1)/x using an algorithm that is accurate for small x
exprel_2(x)2(exp(x)-1-x)/x^2 using an algorithm that is accurate for small x
exprel_n(n,x)n-relative exponential, which is the n-th generalization of the functions `gsl_sf_exprel'
exp_int_E1(x)exponential integral E_1(x), E_1(x) := Re \int_1^\infty dt \exp(-xt)/t
exp_int_E2(x)second-order exponential integral E_2(x), E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2
exp_int_Ei(x)exponential integral E_i(x), Ei(x) := PV(\int_{-x}^\infty dt \exp(-t)/t)
shi(x)Shi(x) = \int_0^x dt sinh(t)/t
chi(x)integral Chi(x) := Re[ gamma_E + log(x) + \int_0^x dt (cosh[t]-1)/t]
expint_3(x)exponential integral Ei_3(x) = \int_0^x dt exp(-t^3) for x >= 0
si(x)Sine integral Si(x) = \int_0^x dt sin(t)/t
ci(x)Cosine integral Ci(x) = -\int_x^\infty dt cos(t)/t for x > 0
atanint(x)Arctangent integral AtanInt(x) = \int_0^x dt arctan(t)/t
fermi_dirac_m1(x)complete Fermi-Dirac integral with an index of -1, F_{-1}(x) = e^x / (1 + e^x)
fermi_dirac_0(x)complete Fermi-Dirac integral with an index of 0, F_0(x) = \ln(1 + e^x)
fermi_dirac_1(x)complete Fermi-Dirac integral with an index of 1, F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1))
fermi_dirac_2(x)complete Fermi-Dirac integral with an index of 2, F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1))
fermi_dirac_int(j,x)complete Fermi-Dirac integral with an index of j, F_j(x) = (1/Gamma(j+1)) \int_0^\infty dt (t^j /(exp(t-x)+1))
fermi_dirac_mhalf(x)complete Fermi-Dirac integral F_{-1/2}(x)
fermi_dirac_half(x)complete Fermi-Dirac integral F_{1/2}(x)
fermi_dirac_3half(x)complete Fermi-Dirac integral F_{3/2}(x)
fermi_dirac_inc_0(x,b)incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x)
gamma(x)Gamma function
lngamma(x)logarithm of the Gamma function
gammastar(x)regulated Gamma Function \Gamma^*(x) for x > 0
gammainv(x)reciprocal of the gamma function, 1/Gamma(x) using the real Lanczos method.
taylorcoeff(n,x)Taylor coefficient x^n / n! for x >= 0
fact(n)factorial n!
doublefact(n)double factorial n!! = n(n-2)(n-4)...
lnfact(n)logarithm of the factorial of n, log(n!)
lndoublefact(n)logarithm of the double factorial log(n!!)
choose(n,m)combinatorial factor `n choose m' = n!/(m!(n-m)!)
lnchoose(n,m)logarithm of `n choose m'
poch(a,x)Pochhammer symbol (a)_x := \Gamma(a + x)/\Gamma(x)
lnpoch(a,x)logarithm of the Pochhammer symbol (a)_x := \Gamma(a + x)/\Gamma(x)
pochrel(a,x)relative Pochhammer symbol ((a,x) - 1)/x where (a,x) = (a)_x := \Gamma(a + x)/\Gamma(a)
gamma_inc_Q(a,x)normalized incomplete Gamma Function P(a,x) = 1/Gamma(a) \int_x\infty dt t^{a-1} exp(-t) for a > 0, x >= 0
gamma_inc_P(a,x)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_beta(a,b)Beta Function, B(a,b) = Gamma(a) Gamma(b)/Gamma(a+b) for a > 0, b > 0
lnbeta(a,b)logarithm of the Beta Function, log(B(a,b)) for a > 0, b > 0
betainc(a,b,x)normalize incomplete Beta function B_x(a,b)/B(a,b) for a > 0, b > 0
gegenpoly_1(lambda,x)Gegenbauer polynomial C^{lambda}_1(x)
gegenpoly_2(lambda,x)Gegenbauer polynomial C^{lambda}_2(x)
gegenpoly_3(lambda,x)Gegenbauer polynomial C^{lambda}_3(x)
gegenpoly_n(n,lambda,x)Gegenbauer polynomial C^{lambda}_n(x)
hyperg_0F1(c,x)hypergeometric function 0F1(c,x)
hyperg_1F1i(m,n,x)confluent hypergeometric function 1F1(m,n,x) = M(m,n,x) for integer parameters m, n
hyperg_1F1(a,b,x)confluent hypergeometric function 1F1(m,n,x) = M(m,n,x) for general parameters a,b
hyperg_Ui(m,n,x)confluent hypergeometric function U(m,n,x) for integer parameters m,n
hyperg_U(a,b,x)confluent hypergeometric function U(a,b,x)
hyperg_2F1(a,b,c,x)Gauss hypergeometric function 2F1(a,b,c,x)
hyperg_2F1c(ar,ai,c,x)Gauss hypergeometric function 2F1(a_R + i a_I, a_R - i a_I, c, x) with complex parameters
hyperg_2F1r(ar,ai,c,x)renormalized Gauss hypergeometric function 2F1(a,b,c,x) / Gamma(c)
hyperg_2F1cr(ar,ai,c,x)renormalized Gauss hypergeometric function 2F1(a_R + i a_I, a_R - i a_I, c, x) / Gamma(c)
hyperg_2F0(a,b,x)hypergeometric function 2F0(a,b,x)
laguerre_1(a,x)generalized Laguerre polynomials L^a_1(x)
laguerre_2(a,x)generalized Laguerre polynomials L^a_2(x)
laguerre_3(a,x)generalized Laguerre polynomials L^a_3(x)
lambert_W0(x)principal branch of the Lambert W function, W_0(x)
lambert_Wm1(x)secondary real-valued branch of the Lambert W function, W_{-1}(x)
legendre_P1(x)Legendre polynomials P_1(x)
legendre_P2(x)Legendre polynomials P_2(x)
legendre_P3(x)Legendre polynomials P_3(x)
legendre_Pl(l,x)Legendre polynomials P_l(x)
legendre_Q0(x)Legendre polynomials Q_0(x)
legendre_Q1(x)Legendre polynomials Q_1(x)
legendre_Ql(l,x)Legendre polynomials Q_l(x)
legendre_Plm(l,m,x)associated Legendre polynomial P_l^m(x)
legendre_sphPlm(l,m,x)normalized associated Legendre polynomial $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ suitable for use in spherical harmonics
conicalP_half(lambda,x)irregular Spherical Conical Function P^{1/2}_{-1/2 + i \lambda}(x) for x > -1
conicalP_mhalf(lambda,x)regular Spherical Conical Function P^{-1/2}_{-1/2 + i \lambda}(x) for x > -1
conicalP_0(lambda,x)conical function P^0_{-1/2 + i \lambda}(x) for x > -1
conicalP_1(lambda,x)conical function P^1_{-1/2 + i \lambda}(x) for x > -1
conicalP_sphreg(l,lambda,x)Regular Spherical Conical Function P^{-1/2-l}_{-1/2 + i \lambda}(x) for x > -1, l >= -1
conicalP_cylreg(l,lambda,x)Regular Cylindrical Conical Function P^{-m}_{-1/2 + i \lambda}(x) for x > -1, m >= -1
legendre_H3d_0(lambda,eta)zeroth radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, L^{H3d}_0(lambda,eta) := sin(lambda eta)/(lambda sinh(eta)) for eta >= 0
legendre_H3d_1(lambda,eta)zeroth radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, L^{H3d}_1(lambda,eta) := 1/sqrt{lambda^2 + 1} sin(lambda eta)/(lambda sinh(eta)) (coth(eta) - lambda cot(lambda eta)) for eta >= 0
legendre_H3d(l,lambda,eta)L'th radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space eta >= 0, l >= 0
gsl_log(x)logarithm of X
loga(x)logarithm of the magnitude of X, log(|x|)
logp(x)log(1 + x) for x > -1 using an algorithm that is accurate for small x
logm(x)log(1 + x) - x for x > -1 using an algorithm that is accurate for small x
gsl_pow(x,n)power x^n for integer n
psii(n)digamma function psi(n) for positive integer n
psi(x)digamma function psi(n) for general x
psiy(y)real part of the digamma function on the line 1+i y, Re[psi(1 + i y)]
ps1i(n)Trigamma function psi'(n) for positive integer n
ps_n(m,x)polygamma function psi^{(m)}(x) for m >= 0, x > 0
synchrotron_1(x)first synchrotron function x \int_x^\infty dt K_{5/3}(t) for x >= 0
synchrotron_2(x)second synchrotron function x K_{2/3}(x) for x >= 0
transport_2(x)transport function J(2,x)
transport_3(x)transport function J(3,x)
transport_4(x)transport function J(4,x)
transport_5(x)transport function J(5,x)
hypot(x,y)hypotenuse function \sqrt{x^2 + y^2}
sinc(x)sinc(x) = sin(pi x) / (pi x)
lnsinh(x)log(sinh(x)) for x > 0
lncosh(x)log(cosh(x))
zetai(n)Riemann zeta function zeta(n) for integer N
gsl_zeta(s)Riemann zeta function zeta(s) for arbitrary s
hzeta(s,q)Hurwitz zeta function zeta(s,q) for s > 1, q > 0
etai(n)eta function eta(n) for integer n
eta(s)eta function eta(s) for arbitrary s
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