| 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_J0(x) | regular cylindrical Bessel function of zeroth order, J_0(x) |
| bessel_J1(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_Y0(x) | irregular cylindrical Bessel function of zeroth order, Y_0(x) |
| bessel_Y1(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_I0s(x) | scaled regular modified cylindrical Bessel function of zeroth order, exp (-|x|) I_0(x) |
| bessel_I1s(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_K0s(x) | scaled irregular modified cylindrical Bessel function of zeroth order, exp (x) K_0(x) |
| bessel_K1s(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 n!! = n(n-2)(n-4)... |
| 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 |