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view main/gsl/doc/gsl.texi @ 11838:b9f4ede9342a octave-forge
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author | carandraug |
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date | Wed, 19 Jun 2013 02:15:35 +0000 |
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\input texinfo @ifnottex @node Top @top Octave gsl @end ifnottex gsl_sf -*- texinfo -*- @deftypefn {Loadable Function} {} gsl_sf () Octave bindings to the GNU Scientific Library. All GSL functions can be called with by the GSL names within octave. @end deftypefn clausen -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} clausen (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} clausen (@dots{}) The Clausen function is defined by the following integral, Cl_2(x) = - \\int_0^x dt \\log(2 \\sin(t/2)) It is related to the dilogarithm by Cl_2(\\theta) = \\Im Li_2(\\exp(i \\theta)). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn dawson -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} dawson (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} dawson (@dots{}) The Dawson integral is defined by \\exp(-x^2) \\int_0^x dt \\exp(t^2). A table of Dawson integral can be found in Abramowitz & Stegun, Table 7.5. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn debye_1 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} debye_1 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} debye_1 (@dots{}) The Debye functions are defined by the integral D_n(x) = n/x^n \\int_0^x dt (t^n/(e^t - 1)). For further information see Abramowitz & Stegun, Section 27.1. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn debye_2 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} debye_2 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} debye_2 (@dots{}) The Debye functions are defined by the integral D_n(x) = n/x^n \\int_0^x dt (t^n/(e^t - 1)). For further information see Abramowitz & Stegun, Section 27.1. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn debye_3 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} debye_3 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} debye_3 (@dots{}) The Debye functions are defined by the integral D_n(x) = n/x^n \\int_0^x dt (t^n/(e^t - 1)). For further information see Abramowitz & Stegun, Section 27.1. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn debye_4 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} debye_4 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} debye_4 (@dots{}) The Debye functions are defined by the integral D_n(x) = n/x^n \\int_0^x dt (t^n/(e^t - 1)). For further information see Abramowitz & Stegun, Section 27.1. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn erf_gsl -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} erf_gsl (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} erf_gsl (@dots{}) These routines compute the error function erf(x) = (2/\\sqrt(\\pi)) \\int_0^x dt \\exp(-t^2). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn erfc_gsl -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} erfc_gsl (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} erfc_gsl (@dots{}) These routines compute the complementary error function erfc(x) = 1 - erf(x) = (2/\\sqrt(\\pi)) \\int_x^\\infty \\exp(-t^2). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn log_erfc -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} log_erfc (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} log_erfc (@dots{}) These routines compute the logarithm of the complementary error function \\log(\\erfc(x)). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn erf_Z -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} erf_Z (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} erf_Z (@dots{}) These routines compute the Gaussian probability function Z(x) = (1/(2\\pi)) \\exp(-x^2/2). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn erf_Q -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} erf_Q (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} erf_Q (@dots{}) These routines compute the upper tail of the Gaussian probability function Q(x) = (1/(2\\pi)) \\int_x^\\infty dt \\exp(-t^2/2). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn hazard -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} hazard (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} hazard (@dots{}) 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)@}. It decreases rapidly as x approaches -\\infty and asymptotes to h(x) \\sim x as x approaches +\\infty. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn expm1 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} expm1 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} expm1 (@dots{}) These routines compute the quantity \\exp(x)-1 using an algorithm that is accurate for small x. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn exprel -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} exprel (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} exprel (@dots{}) These routines compute the quantity (\\exp(x)-1)/x using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion (\\exp(x)-1)/x = 1 + x/2 + x^2/(2*3) + x^3/(2*3*4) + \\dots. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn exprel_2 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} exprel_2 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} exprel_2 (@dots{}) These routines compute the quantity 2(\\exp(x)-1-x)/x^2 using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion 2(\\exp(x)-1-x)/x^2 = 1 + x/3 + x^2/(3*4) + x^3/(3*4*5) + \\dots. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn expint_E1 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} expint_E1 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} expint_E1 (@dots{}) These routines compute the exponential integral E_1(x), E_1(x) := Re \\int_1^\\infty dt \\exp(-xt)/t. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn expint_E2 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} expint_E2 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} expint_E2 (@dots{}) These routines compute the second-order exponential integral E_2(x), E_2(x) := \\Re \\int_1^\\infty dt \\exp(-xt)/t^2. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn expint_Ei -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} expint_Ei (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} expint_Ei (@dots{}) These routines compute the exponential integral E_i(x), Ei(x) := - PV(\\int_@{-x@}^\\infty dt \\exp(-t)/t) where PV denotes the principal value of the integral. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn Shi -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} Shi (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} Shi (@dots{}) These routines compute the integral Shi(x) = \\int_0^x dt \\sinh(t)/t. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn Chi -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} Chi (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} Chi (@dots{}) These routines compute the integral Chi(x) := Re[ \\gamma_E + \\log(x) + \\int_0^x dt (\\cosh[t]-1)/t] , where \\gamma_E is the Euler constant. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn expint_3 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} expint_3 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} expint_3 (@dots{}) These routines compute the exponential integral Ei_3(x) = \\int_0^x dt \\exp(-t^3) for x >= 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn Si -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} Si (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} Si (@dots{}) These routines compute the Sine integral Si(x) = \\int_0^x dt \\sin(t)/t. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn Ci -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} Ci (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} Ci (@dots{}) These routines compute the Cosine integral Ci(x) = -\\int_x^\\infty dt \\cos(t)/t for x > 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn atanint -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} atanint (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} atanint (@dots{}) These routines compute the Arctangent integral AtanInt(x) = \\int_0^x dt \\arctan(t)/t. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn fermi_dirac_mhalf -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} fermi_dirac_mhalf (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} fermi_dirac_mhalf (@dots{}) These routines compute the complete Fermi-Dirac integral F_@{-1/2@}(x). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn fermi_dirac_half -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} fermi_dirac_half (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} fermi_dirac_half (@dots{}) These routines compute the complete Fermi-Dirac integral F_@{1/2@}(x). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn fermi_dirac_3half -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} fermi_dirac_3half (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} fermi_dirac_3half (@dots{}) These routines compute the complete Fermi-Dirac integral F_@{3/2@}(x). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn gamma_gsl -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} gamma_gsl (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} gamma_gsl (@dots{}) These routines compute the Gamma function \\Gamma(x), subject to x not being a negative integer. The function is computed using the real Lanczos method. The maximum value of x such that \\Gamma(x) is not considered an overflow is given by the macro GSL_SF_GAMMA_XMAX and is 171.0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn lngamma_gsl -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} lngamma_gsl (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} lngamma_gsl (@dots{}) These routines compute the logarithm of the Gamma function, \\log(\\Gamma(x)), subject to x not a being negative integer. For x<0 the real part of \\log(\\Gamma(x)) is returned, which is equivalent to \\log(|\\Gamma(x)|). The function is computed using the real Lanczos method. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn gammastar -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} gammastar (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} gammastar (@dots{}) These routines compute the regulated Gamma Function \\Gamma^*(x) for x > 0. The regulated gamma function is given by, \\Gamma^*(x) = \\Gamma(x)/(\\sqrt@{2\\pi@} x^@{(x-1/2)@} \\exp(-x)) = (1 + (1/12x) + ...) for x \\to \\infty and is a useful suggestion of Temme. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn gammainv_gsl -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} gammainv_gsl (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} gammainv_gsl (@dots{}) These routines compute the reciprocal of the gamma function, 1/\\Gamma(x) using the real Lanczos method. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn lambert_W0 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} lambert_W0 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} lambert_W0 (@dots{}) These compute the principal branch of the Lambert W function, W_0(x). Lambert\\'s W functions, W(x), are defined to be solutions of the equation W(x) \\exp(W(x)) = x. This function has multiple branches for x < 0; however, it has only two real-valued branches. We define W_0(x) to be the principal branch, where W > -1 for x < 0, and W_@{-1@}(x) to be the other real branch, where W < -1 for x < 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn lambert_Wm1 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} lambert_Wm1 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} lambert_Wm1 (@dots{}) These compute the secondary real-valued branch of the Lambert W function, W_@{-1@}(x). Lambert\\'s W functions, W(x), are defined to be solutions of the equation W(x) \\exp(W(x)) = x. This function has multiple branches for x < 0; however, it has only two real-valued branches. We define W_0(x) to be the principal branch, where W > -1 for x < 0, and W_@{-1@}(x) to be the other real branch, where W < -1 for x < 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn log_1plusx -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} log_1plusx (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} log_1plusx (@dots{}) These routines compute \\log(1 + x) for x > -1 using an algorithm that is accurate for small x. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn log_1plusx_mx -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} log_1plusx_mx (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} log_1plusx_mx (@dots{}) These routines compute \\log(1 + x) - x for x > -1 using an algorithm that is accurate for small x. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn psi -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} psi (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} psi (@dots{}) These routines compute the digamma function \\psi(x) for general x, x \ e 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn psi_1piy -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} psi_1piy (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} psi_1piy (@dots{}) These routines compute the real part of the digamma function on the line 1+i y, Re[\\psi(1 + i y)]. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn synchrotron_1 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} synchrotron_1 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} synchrotron_1 (@dots{}) These routines compute the first synchrotron function x \\int_x^\\infty dt K_@{5/3@}(t) for x >= 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn synchrotron_2 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} synchrotron_2 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} synchrotron_2 (@dots{}) These routines compute the second synchrotron function x K_@{2/3@}(x) for x >= 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn transport_2 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} transport_2 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} transport_2 (@dots{}) These routines compute the transport function J(2,x). The transport functions J(n,x) are defined by the integral representations J(n,x) := \\int_0^x dt t^n e^t /(e^t - 1)^2. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn transport_3 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} transport_3 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} transport_3 (@dots{}) These routines compute the transport function J(3,x). The transport functions J(n,x) are defined by the integral representations J(n,x) := \\int_0^x dt t^n e^t /(e^t - 1)^2. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn transport_4 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} transport_4 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} transport_4 (@dots{}) These routines compute the transport function J(4,x). The transport functions J(n,x) are defined by the integral representations J(n,x) := \\int_0^x dt t^n e^t /(e^t - 1)^2. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn transport_5 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} transport_5 (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} transport_5 (@dots{}) These routines compute the transport function J(5,x). The transport functions J(n,x) are defined by the integral representations J(n,x) := \\int_0^x dt t^n e^t /(e^t - 1)^2. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn sinc_gsl -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} sinc_gsl (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} sinc_gsl (@dots{}) These routines compute \\sinc(x) = \\sin(\\pi x) / (\\pi x) for any value of x. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn lnsinh -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} lnsinh (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} lnsinh (@dots{}) These routines compute \\log(\\sinh(x)) for x > 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn lncosh -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} lncosh (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} lncosh (@dots{}) These routines compute \\log(\\cosh(x)) for any x. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn zeta -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} zeta (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} zeta (@dots{}) These routines compute the Riemann zeta function \\zeta(s) for arbitrary s, s \ e 1. The Riemann zeta function is defined by the infinite sum \\zeta(s) = \\sum_@{k=1@}^\\infty k^@{-s@}. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn eta -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} eta (@var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} eta (@dots{}) These routines compute the eta function \\eta(s) for arbitrary s. The eta function is defined by \\eta(s) = (1-2^@{1-s@}) \\zeta(s). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_Jn -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} bessel_Jn (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} bessel_Jn (@dots{}) These routines compute the regular cylindrical Bessel function of order n, J_n(x). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_Yn -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} bessel_Yn (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} bessel_Yn (@dots{}) These routines compute the irregular cylindrical Bessel function of order n, Y_n(x), for x>0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_In -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} bessel_In (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} bessel_In (@dots{}) These routines compute the regular modified cylindrical Bessel function of order n, I_n(x). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_In_scaled -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} bessel_In_scaled (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} bessel_In_scaled (@dots{}) These routines compute the scaled regular modified cylindrical Bessel function of order n, \\exp(-|x|) I_n(x) @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_Kn -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} bessel_Kn (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} bessel_Kn (@dots{}) These routines compute the irregular modified cylindrical Bessel function of order n, K_n(x), for x > 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_Kn_scaled -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} bessel_Kn_scaled (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} bessel_Kn_scaled (@dots{}) @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_jl -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} bessel_jl (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} bessel_jl (@dots{}) These routines compute the regular spherical Bessel function of order l, j_l(x), for l >= 0 and x >= 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_yl -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} bessel_yl (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} bessel_yl (@dots{}) These routines compute the irregular spherical Bessel function of order l, y_l(x), for l >= 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_il_scaled -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} bessel_il_scaled (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} bessel_il_scaled (@dots{}) These routines compute the scaled regular modified spherical Bessel function of order l, \\exp(-|x|) i_l(x) @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_kl_scaled -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} bessel_kl_scaled (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} bessel_kl_scaled (@dots{}) These routines compute the scaled irregular modified spherical Bessel function of order l, \\exp(x) k_l(x), for x>0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn exprel_n -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} exprel_n (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} exprel_n (@dots{}) These routines compute the N-relative exponential, which is the n-th generalization of the functions gsl_sf_exprel and gsl_sf_exprel2. The N-relative exponential is given by, exprel_N(x) = N!/x^N (\\exp(x) - \\sum_@{k=0@}^@{N-1@} x^k/k!) = 1 + x/(N+1) + x^2/((N+1)(N+2)) + ... = 1F1 (1,1+N,x) @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn fermi_dirac_int -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} fermi_dirac_int (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} fermi_dirac_int (@dots{}) These routines compute 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)). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn taylorcoeff -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} taylorcoeff (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} taylorcoeff (@dots{}) These routines compute the Taylor coefficient x^n / n! for x >= 0, n >= 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn legendre_Pl -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} legendre_Pl (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} legendre_Pl (@dots{}) These functions evaluate the Legendre polynomial P_l(x) for a specific value of l, x subject to l >= 0, |x| <= 1 @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn legendre_Ql -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} legendre_Ql (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} legendre_Ql (@dots{}) These routines compute the Legendre function Q_l(x) for x > -1, x != 1 and l >= 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn psi_n -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} psi_n (@var{n}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} psi_n (@dots{}) These routines compute the polygamma function \\psi^@{(m)@}(x) for m >= 0, x > 0. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_Jnu -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} bessel_Jnu (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} bessel_Jnu (@dots{}) These routines compute the regular cylindrical Bessel function of fractional order nu, J_\ u(x). @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_Ynu -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} bessel_Ynu (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} bessel_Ynu (@dots{}) These routines compute the irregular cylindrical Bessel function of fractional order nu, Y_\ u(x). @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_Inu -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} bessel_Inu (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} bessel_Inu (@dots{}) These routines compute the regular modified Bessel function of fractional order nu, I_\ u(x) for x>0, \ u>0. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_Inu_scaled -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} bessel_Inu_scaled (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} bessel_Inu_scaled (@dots{}) These routines compute the scaled regular modified Bessel function of fractional order nu, \\exp(-|x|)I_\ u(x) for x>0, \ u>0. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_Knu -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} bessel_Knu (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} bessel_Knu (@dots{}) These routines compute the irregular modified Bessel function of fractional order nu, K_\ u(x) for x>0, \ u>0. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_lnKnu -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} bessel_lnKnu (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} bessel_lnKnu (@dots{}) These routines compute the logarithm of the irregular modified Bessel function of fractional order nu, \\ln(K_\ u(x)) for x>0, \ u>0. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_Knu_scaled -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} bessel_Knu_scaled (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} bessel_Knu_scaled (@dots{}) These routines compute the scaled irregular modified Bessel function of fractional order nu, \\exp(+|x|) K_\ u(x) for x>0, \ u>0. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn exp_mult -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} exp_mult (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} exp_mult (@dots{}) These routines exponentiate x and multiply by the factor y to return the product y \\exp(x). @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn fermi_dirac_inc_0 -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} fermi_dirac_inc_0 (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} fermi_dirac_inc_0 (@dots{}) These routines compute the incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \\ln(1 + e^@{b-x@}) - (b-x). @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn poch -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} poch (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} poch (@dots{}) These routines compute the Pochhammer symbol (a)_x := \\Gamma(a + x)/\\Gamma(a), subject to a and a+x not being negative integers. The Pochhammer symbol is also known as the Apell symbol. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn lnpoch -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} lnpoch (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} lnpoch (@dots{}) These routines compute the logarithm of the Pochhammer symbol, \\log((a)_x) = \\log(\\Gamma(a + x)/\\Gamma(a)) for a > 0, a+x > 0. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn pochrel -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} pochrel (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} pochrel (@dots{}) These routines compute the relative Pochhammer symbol ((a,x) - 1)/x where (a,x) = (a)_x := \\Gamma(a + x)/\\Gamma(a). @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn gamma_inc_Q -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} gamma_inc_Q (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} gamma_inc_Q (@dots{}) These routines compute 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. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn gamma_inc_P -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} gamma_inc_P (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} gamma_inc_P (@dots{}) These routines compute 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. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn gamma_inc -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} gamma_inc (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} gamma_inc (@dots{}) 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. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn beta_gsl -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} beta_gsl (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} beta_gsl (@dots{}) These routines compute the Beta Function, B(a,b) = \\Gamma(a)\\Gamma(b)/\\Gamma(a+b) for a > 0, b > 0. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn lnbeta -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} lnbeta (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} lnbeta (@dots{}) These routines compute the logarithm of the Beta Function, \\log(B(a,b)) for a > 0, b > 0. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn hyperg_0F1 -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} hyperg_0F1 (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} hyperg_0F1 (@dots{}) These routines compute the hypergeometric function 0F1(c,x). @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn conicalP_half -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} conicalP_half (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} conicalP_half (@dots{}) These routines compute the irregular Spherical Conical Function P^@{1/2@}_@{-1/2 + i \\lambda@}(x) for x > -1. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn conicalP_mhalf -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} conicalP_mhalf (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} conicalP_mhalf (@dots{}) These routines compute the regular Spherical Conical Function P^@{-1/2@}_@{-1/2 + i \\lambda@}(x) for x > -1. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn conicalP_0 -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} conicalP_0 (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} conicalP_0 (@dots{}) These routines compute the conical function P^0_@{-1/2 + i \\lambda@}(x) for x > -1. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn conicalP_1 -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} conicalP_1 (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} conicalP_1 (@dots{}) These routines compute the conical function P^1_@{-1/2 + i \\lambda@}(x) for x > -1. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn hzeta -*- texinfo -*- @deftypefn {Loadable Function} {@var{z} =} hzeta (@var{x}, @var{y}) @deftypefnx {Loadable Function} {[@var{z}, @var{err}] =} hzeta (@dots{}) These routines compute the Hurwitz zeta function \\zeta(s,q) for s > 1, q > 0. @var{err} contains an estimate of the absolute error in the value @var{z}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn airy_Ai -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} airy_Ai (@var{x}, @var{mode}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} airy_Ai (@dots{}) These routines compute the Airy function Ai(x) with an accuracy specified by mode. The second argument @var{mode} must be an integer corresponding to @table @asis @item 0 = GSL_PREC_DOUBLE Double-precision, a relative accuracy of approximately @code{2 * 10^-16}. @item 1 = GSL_PREC_SINGLE Single-precision, a relative accuracy of approximately @code{10^-7}. @item 2 = GSL_PREC_APPROX Approximate values, a relative accuracy of approximately @code{5 * 10^-4}. @end table @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn airy_Bi -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} airy_Bi (@var{x}, @var{mode}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} airy_Bi (@dots{}) These routines compute the Airy function Bi(x) with an accuracy specified by mode. The second argument @var{mode} must be an integer corresponding to @table @asis @item 0 = GSL_PREC_DOUBLE Double-precision, a relative accuracy of approximately @code{2 * 10^-16}. @item 1 = GSL_PREC_SINGLE Single-precision, a relative accuracy of approximately @code{10^-7}. @item 2 = GSL_PREC_APPROX Approximate values, a relative accuracy of approximately @code{5 * 10^-4}. @end table @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn airy_Ai_scaled -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} airy_Ai_scaled (@var{x}, @var{mode}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} airy_Ai_scaled (@dots{}) These routines compute a scaled version of the Airy function S_A(x) Ai(x). For x>0 the scaling factor S_A(x) is \\exp(+(2/3) x^(3/2)), and is 1 for x<0. The second argument @var{mode} must be an integer corresponding to @table @asis @item 0 = GSL_PREC_DOUBLE Double-precision, a relative accuracy of approximately @code{2 * 10^-16}. @item 1 = GSL_PREC_SINGLE Single-precision, a relative accuracy of approximately @code{10^-7}. @item 2 = GSL_PREC_APPROX Approximate values, a relative accuracy of approximately @code{5 * 10^-4}. @end table @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn airy_Bi_scaled -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} airy_Bi_scaled (@var{x}, @var{mode}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} airy_Bi_scaled (@dots{}) These routines compute a scaled version of the Airy function S_B(x) Bi(x). For x>0 the scaling factor S_B(x) is exp(-(2/3) x^(3/2)), and is 1 for x<0. The second argument @var{mode} must be an integer corresponding to @table @asis @item 0 = GSL_PREC_DOUBLE Double-precision, a relative accuracy of approximately @code{2 * 10^-16}. @item 1 = GSL_PREC_SINGLE Single-precision, a relative accuracy of approximately @code{10^-7}. @item 2 = GSL_PREC_APPROX Approximate values, a relative accuracy of approximately @code{5 * 10^-4}. @end table @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn airy_Ai_deriv -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} airy_Ai_deriv (@var{x}, @var{mode}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} airy_Ai_deriv (@dots{}) These routines compute the Airy function derivative Ai'(x) with an accuracy specified by mode. The second argument @var{mode} must be an integer corresponding to @table @asis @item 0 = GSL_PREC_DOUBLE Double-precision, a relative accuracy of approximately @code{2 * 10^-16}. @item 1 = GSL_PREC_SINGLE Single-precision, a relative accuracy of approximately @code{10^-7}. @item 2 = GSL_PREC_APPROX Approximate values, a relative accuracy of approximately @code{5 * 10^-4}. @end table @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn airy_Bi_deriv -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} airy_Bi_deriv (@var{x}, @var{mode}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} airy_Bi_deriv (@dots{}) These routines compute the Airy function derivative Bi'(x) with an accuracy specified by mode. The second argument @var{mode} must be an integer corresponding to @table @asis @item 0 = GSL_PREC_DOUBLE Double-precision, a relative accuracy of approximately @code{2 * 10^-16}. @item 1 = GSL_PREC_SINGLE Single-precision, a relative accuracy of approximately @code{10^-7}. @item 2 = GSL_PREC_APPROX Approximate values, a relative accuracy of approximately @code{5 * 10^-4}. @end table @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn airy_Ai_deriv_scaled -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} airy_Ai_deriv_scaled (@var{x}, @var{mode}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} airy_Ai_deriv_scaled (@dots{}) These routines compute the derivative of the scaled Airy function S_A(x) Ai(x). The second argument @var{mode} must be an integer corresponding to @table @asis @item 0 = GSL_PREC_DOUBLE Double-precision, a relative accuracy of approximately @code{2 * 10^-16}. @item 1 = GSL_PREC_SINGLE Single-precision, a relative accuracy of approximately @code{10^-7}. @item 2 = GSL_PREC_APPROX Approximate values, a relative accuracy of approximately @code{5 * 10^-4}. @end table @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn airy_Bi_deriv_scaled -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} airy_Bi_deriv_scaled (@var{x}, @var{mode}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} airy_Bi_deriv_scaled (@dots{}) These routines compute the derivative of the scaled Airy function S_B(x) Bi(x). The second argument @var{mode} must be an integer corresponding to @table @asis @item 0 = GSL_PREC_DOUBLE Double-precision, a relative accuracy of approximately @code{2 * 10^-16}. @item 1 = GSL_PREC_SINGLE Single-precision, a relative accuracy of approximately @code{10^-7}. @item 2 = GSL_PREC_APPROX Approximate values, a relative accuracy of approximately @code{5 * 10^-4}. @end table @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn ellint_Kcomp -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} ellint_Kcomp (@var{x}, @var{mode}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} ellint_Kcomp (@dots{}) These routines compute the complete elliptic integral K(k) @tex \beforedisplay $$ \eqalign{ K(k) &= \int_0^{\pi/2} {dt \over \sqrt{(1 - k^2 \sin^2(t))}} \cr } $$ \afterdisplay @end tex The notation used here is based on Carlson, @cite{Numerische Mathematik} 33 (1979) and differs slightly from that used by Abramowitz & Stegun, where the functions are given in terms of the parameter @math{m = k^2}. The second argument @var{mode} must be an integer corresponding to @table @asis @item 0 = GSL_PREC_DOUBLE Double-precision, a relative accuracy of approximately @code{2 * 10^-16}. @item 1 = GSL_PREC_SINGLE Single-precision, a relative accuracy of approximately @code{10^-7}. @item 2 = GSL_PREC_APPROX Approximate values, a relative accuracy of approximately @code{5 * 10^-4}. @end table @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn ellint_Ecomp -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} ellint_Ecomp (@var{x}, @var{mode}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} ellint_Ecomp (@dots{}) These routines compute the complete elliptic integral E(k) to the accuracy specified by the mode variable mode. @tex \beforedisplay $$ \eqalign{ E(k) &= \int_0^{\pi/2} \sqrt{(1 - k^2 \sin^2(t))} dt \cr } $$ \afterdisplay @end tex The notation used here is based on Carlson, @cite{Numerische Mathematik} 33 (1979) and differs slightly from that used by Abramowitz & Stegun, where the functions are given in terms of the parameter @math{m = k^2}. The second argument @var{mode} must be an integer corresponding to @table @asis @item 0 = GSL_PREC_DOUBLE Double-precision, a relative accuracy of approximately @code{2 * 10^-16}. @item 1 = GSL_PREC_SINGLE Single-precision, a relative accuracy of approximately @code{10^-7}. @item 2 = GSL_PREC_APPROX Approximate values, a relative accuracy of approximately @code{5 * 10^-4}. @end table @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn airy_zero_Ai -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} airy_zero_Ai (@var{n}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} airy_zero_Ai (@dots{}) These routines compute the location of the s-th zero of the Airy function Ai(x). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn airy_zero_Bi -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} airy_zero_Bi (@var{n}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} airy_zero_Bi (@dots{}) These routines compute the location of the s-th zero of the Airy function Bi(x). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn airy_zero_Ai_deriv -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} airy_zero_Ai_deriv (@var{n}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} airy_zero_Ai_deriv (@dots{}) These routines compute the location of the s-th zero of the Airy function derivative Ai(x). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn airy_zero_Bi_deriv -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} airy_zero_Bi_deriv (@var{n}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} airy_zero_Bi_deriv (@dots{}) These routines compute the location of the s-th zero of the Airy function derivative Bi(x). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_zero_J0 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} bessel_zero_J0 (@var{n}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} bessel_zero_J0 (@dots{}) These routines compute the location of the s-th positive zero of the Bessel function J_0(x). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn bessel_zero_J1 -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} bessel_zero_J1 (@var{n}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} bessel_zero_J1 (@dots{}) These routines compute the location of the s-th positive zero of the Bessel function J_1(x). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn psi_1_int -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} psi_1_int (@var{n}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} psi_1_int (@dots{}) These routines compute the Trigamma function \\psi(n) for positive integer n. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn zeta_int -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} zeta_int (@var{n}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} zeta_int (@dots{}) These routines compute the Riemann zeta function \\zeta(n) for integer n, n \ e 1. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn eta_int -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} eta_int (@var{n}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} eta_int (@dots{}) These routines compute the eta function \\eta(n) for integer n. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn legendre_Plm -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} legendre_Plm (@var{n}, @var{m}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} legendre_Plm (@dots{}) These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1. @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn legendre_sphPlm -*- texinfo -*- @deftypefn {Loadable Function} {@var{y} =} legendre_sphPlm (@var{n}, @var{m}, @var{x}) @deftypefnx {Loadable Function} {[@var{y}, @var{err}] =} legendre_sphPlm (@dots{}) These routines compute 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. The parameters must satisfy m >= 0, l >= m, |x| <= 1. Theses routines avoid the overflows that occur for the standard normalization of P_l^m(x). @var{err} contains an estimate of the absolute error in the value @var{y}. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn hyperg_U -*- texinfo -*- @deftypefn {Loadable Function} {@var{out} =} hyperg_U (@var{x0}, @var{x1}, @var{x2}) @deftypefnx {Loadable Function} {[@var{out}, @var{err}] =} hyperg_U (@dots{}) Secondary Confluent Hypergoemetric U function A&E 13.1.3 All input are double as is the output. @var{err} contains an estimate of the absolute error in the value @var{out}.a. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn hyperg_1F1 -*- texinfo -*- @deftypefn {Loadable Function} {@var{out} =} hyperg_1F1 (@var{x0}, @var{x1}, @var{x2}) @deftypefnx {Loadable Function} {[@var{out}, @var{err}] =} hyperg_1F1 (@dots{}) Primary Confluent Hypergoemetric U function A&E 13.1.3 All inputs are double as is the output. @var{err} contains an estimate of the absolute error in the value @var{out}.a. This function is from the GNU Scientific Library, see @url{http://www.gnu.org/software/gsl/} for documentation. @end deftypefn @bye