# leto/math--gsl

Clean up SF pod

 @@ -596,7 +596,7 @@ Math::GSL::SF - Special Functions =head1 SYNOPSIS - use Math::GSL::SF qw /:all/; + use Math::GSL::SF qw/:all/; =head1 DESCRIPTION @@ -618,7 +618,7 @@ Here is a list of all included functions: =item C -- These routines compute the Airy function Ai($x) with an accuracy specified by$mode. $mode should be$GSL_PREC_DOUBLE, $GSL_PREC_SINGLE or$GSL_PREC_APPROX. $result is a gsl_sf_result structure. + These routines compute the Airy function Ai($x) with an accuracy specified by $mode.$mode should be $GSL_PREC_DOUBLE,$GSL_PREC_SINGLE or $GSL_PREC_APPROX.$result is a gsl_sf_result structure. =back @@ -628,7 +628,7 @@ Here is a list of all included functions: =item C -- These routines compute the Airy function Bi($x) with an accuracy specified by$mode. $mode should be$GSL_PREC_DOUBLE, $GSL_PREC_SINGLE or$GSL_PREC_APPROX. $result is a gsl_sf_result structure. + These routines compute the Airy function Bi($x) with an accuracy specified by $mode.$mode should be $GSL_PREC_DOUBLE,$GSL_PREC_SINGLE or $GSL_PREC_APPROX.$result is a gsl_sf_result structure. =back @@ -638,7 +638,7 @@ Here is a list of all included functions: =item C -- 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. $result is a gsl_sf_result structure. + These routines compute a scaled version of the Airy function S_A($x) Ai($x). For$x>0 the scaling factor S_A($x) is \exp(+(2/3)$x**(3/2)), and is 1 for $x<0.$result is a gsl_sf_result structure. =back @@ -648,7 +648,7 @@ Here is a list of all included functions: =item C -- 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. $result is a gsl_sf_result structure. + These routines compute a scaled version of the Airy function S_B($x) Bi($x). For$x>0 the scaling factor S_B($x) is exp(-(2/3)$x**(3/2)), and is 1 for $x<0.$result is a gsl_sf_result structure. =back @@ -658,7 +658,7 @@ Here is a list of all included functions: =item C -- These routines compute the Airy function derivative Ai'($x) with an accuracy specified by$mode. $result is a gsl_sf_result structure. + These routines compute the Airy function derivative Ai'($x) with an accuracy specified by $mode.$result is a gsl_sf_result structure. =back @@ -668,7 +668,7 @@ Here is a list of all included functions: =item C --These routines compute the Airy function derivative Bi'($x) with an accuracy specified by$mode. $result is a gsl_sf_result structure. +These routines compute the Airy function derivative Bi'($x) with an accuracy specified by $mode.$result is a gsl_sf_result structure. =back @@ -678,7 +678,7 @@ Here is a list of all included functions: =item C --These routines compute the scaled Airy function derivative 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. $result is a gsl_sf_result structure. +These routines compute the scaled Airy function derivative S_A(x) Ai'(x). For x>0 the scaling factor S_A(x) is \exp(+(2/3) x^(3/2)), and is 1 for x<0.$result is a gsl_sf_result structure. =back @@ -688,7 +688,7 @@ Here is a list of all included functions: =item C --These routines compute the scaled Airy function derivative 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. $result is a gsl_sf_result structure. +These routines compute the scaled Airy function derivative S_B(x) Bi'(x). For x>0 the scaling factor S_B(x) is exp(-(2/3) x^(3/2)), and is 1 for x<0.$result is a gsl_sf_result structure. =back @@ -698,7 +698,7 @@ Here is a list of all included functions: =item C --These routines compute the location of the s-th zero of the Airy function Ai($x).$result is a gsl_sf_result structure. +These routines compute the location of the s-th zero of the Airy function Ai($x).$result is a gsl_sf_result structure. =back @@ -708,7 +708,7 @@ Here is a list of all included functions: =item C --These routines compute the location of the s-th zero of the Airy function Bi($x).$result is a gsl_sf_result structure. +These routines compute the location of the s-th zero of the Airy function Bi($x).$result is a gsl_sf_result structure. =back @@ -718,7 +718,7 @@ Here is a list of all included functions: =item C --These routines compute the location of the s-th zero of the Airy function derivative Ai'(x). $result is a gsl_sf_result structure. +These routines compute the location of the s-th zero of the Airy function derivative Ai'(x).$result is a gsl_sf_result structure. =back @@ -728,7 +728,7 @@ Here is a list of all included functions: =item C -- These routines compute the location of the s-th zero of the Airy function derivative Bi'(x). $result is a gsl_sf_result structure. + These routines compute the location of the s-th zero of the Airy function derivative Bi'(x).$result is a gsl_sf_result structure. =back @@ -738,7 +738,7 @@ Here is a list of all included functions: =item C --These routines compute the regular cylindrical Bessel function of zeroth order, J_0($x).$result is a gsl_sf_result structure. +These routines compute the regular cylindrical Bessel function of zeroth order, J_0($x).$result is a gsl_sf_result structure. =back @@ -748,7 +748,7 @@ Here is a list of all included functions: =item C -- These routines compute the regular cylindrical Bessel function of first order, J_1($x).$result is a gsl_sf_result structure. + These routines compute the regular cylindrical Bessel function of first order, J_1($x).$result is a gsl_sf_result structure. =back @@ -758,13 +758,18 @@ Here is a list of all included functions: =item C --These routines compute the regular cylindrical Bessel function of order n, J_n($x). +These routines compute the regular cylindrical Bessel function of order n, J_n($x). =back =over -=item C - This routine computes the values of the regular cylindrical Bessel functions J_n($x) for n from$nmin to $nmax inclusive, storing the results in the array$result_array. The values are computed using recurrence relations for efficiency, and therefore may differ slightly from the exact values. +=item C + +This routine computes the values of the regular cylindrical Bessel functions +J_n($x) for n from$nmin to $nmax inclusive, storing the results in the array +$result_array. The values are computed using recurrence relations for +efficiency, and therefore may differ slightly from the exact values. =back @@ -774,7 +779,7 @@ Here is a list of all included functions: =item C -- These routines compute the irregular spherical Bessel function of zeroth order, y_0(x) = -\cos(x)/x. + These routines compute the irregular spherical Bessel function of zeroth order, y_0(x) = -\cos(x)/x. =back @@ -784,7 +789,7 @@ Here is a list of all included functions: =item C --These routines compute the irregular spherical Bessel function of first order, y_1(x) = -(\cos(x)/x + \sin(x))/x. +These routines compute the irregular spherical Bessel function of first order, y_1(x) = -(\cos(x)/x + \sin(x))/x. =back @@ -794,15 +799,15 @@ Here is a list of all included functions: =item C --These routines compute the irregular cylindrical Bessel function of order $n, Y_n(x), for x>0. +These routines compute the irregular cylindrical Bessel function of order$n, Y_n(x), for x>0. =back =over =item C -- + =back @@ -812,7 +817,7 @@ Here is a list of all included functions: =item C --These routines compute the regular modified cylindrical Bessel function of zeroth order, I_0(x). +These routines compute the regular modified cylindrical Bessel function of zeroth order, I_0(x). =back @@ -822,7 +827,7 @@ Here is a list of all included functions: =item C --These routines compute the regular modified cylindrical Bessel function of first order, I_1(x). +These routines compute the regular modified cylindrical Bessel function of first order, I_1(x). =back @@ -832,15 +837,15 @@ Here is a list of all included functions: =item C --These routines compute the regular modified cylindrical Bessel function of order $n, I_n(x). +These routines compute the regular modified cylindrical Bessel function of order$n, I_n(x). =back =over =item C -- + =back @@ -850,7 +855,7 @@ Here is a list of all included functions: =item C --These routines compute the scaled regular modified cylindrical Bessel function of zeroth order \exp(-|x|) I_0(x). +These routines compute the scaled regular modified cylindrical Bessel function of zeroth order \exp(-|x|) I_0(x). =back @@ -860,7 +865,7 @@ Here is a list of all included functions: =item C --These routines compute the scaled regular modified cylindrical Bessel function of first order \exp(-|x|) I_1(x). +These routines compute the scaled regular modified cylindrical Bessel function of first order \exp(-|x|) I_1(x). =back @@ -870,15 +875,15 @@ Here is a list of all included functions: =item C --These routines compute the scaled regular modified cylindrical Bessel function of order $n, \exp(-|x|) I_n(x) +These routines compute the scaled regular modified cylindrical Bessel function of order$n, \exp(-|x|) I_n(x) =back =over =item C -- + =back @@ -888,7 +893,7 @@ Here is a list of all included functions: =item C --These routines compute the irregular modified cylindrical Bessel function of zeroth order, K_0(x), for x > 0. +These routines compute the irregular modified cylindrical Bessel function of zeroth order, K_0(x), for x > 0. =back @@ -898,7 +903,7 @@ Here is a list of all included functions: =item C --These routines compute the irregular modified cylindrical Bessel function of first order, K_1(x), for x > 0. +These routines compute the irregular modified cylindrical Bessel function of first order, K_1(x), for x > 0. =back @@ -908,15 +913,15 @@ Here is a list of all included functions: =item C --These routines compute the irregular modified cylindrical Bessel function of order $n, K_n(x), for x > 0. +These routines compute the irregular modified cylindrical Bessel function of order$n, K_n(x), for x > 0. =back =over =item C -- + =back @@ -926,7 +931,7 @@ Here is a list of all included functions: =item C --These routines compute the scaled irregular modified cylindrical Bessel function of zeroth order \exp(x) K_0(x) for x>0. +These routines compute the scaled irregular modified cylindrical Bessel function of zeroth order \exp(x) K_0(x) for x>0. =back @@ -936,7 +941,7 @@ Here is a list of all included functions: =item C -- + =back @@ -946,15 +951,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -964,7 +969,7 @@ Here is a list of all included functions: =item C -- + =back @@ -974,7 +979,7 @@ Here is a list of all included functions: =item C -- + =back @@ -984,7 +989,7 @@ Here is a list of all included functions: =item C -- + =back @@ -994,23 +999,23 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back =over =item C -- + =back @@ -1020,7 +1025,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1030,7 +1035,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1040,7 +1045,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1050,15 +1055,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -1068,7 +1073,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1078,7 +1083,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1088,7 +1093,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1098,15 +1103,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -1116,7 +1121,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1126,7 +1131,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1136,7 +1141,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1146,15 +1151,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -1164,15 +1169,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -1182,7 +1187,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1192,7 +1197,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1202,7 +1207,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1212,7 +1217,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1222,7 +1227,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1232,7 +1237,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1242,7 +1247,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1252,7 +1257,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1262,7 +1267,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1272,7 +1277,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1282,7 +1287,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1292,7 +1297,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1320,7 +1325,7 @@ Here is a list of all included functions: =item C -- These routines compute the Wigner 3-j coefficient, + These routines compute the Wigner 3-j coefficient, (ja jb jc ma mb mc) where the arguments are given in half-integer units, ja = $two_ja/2, ma =$two_ma/2, etc. @@ -1333,7 +1338,7 @@ Here is a list of all included functions: =item C -- These routines compute the Wigner 6-j coefficient, + These routines compute the Wigner 6-j coefficient, {ja jb jc jd je jf} where the arguments are given in half-integer units, ja = $two_ja/2, ma =$two_ma/2, etc. @@ -1346,7 +1351,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1356,7 +1361,7 @@ Here is a list of all included functions: =item C --These routines compute the Wigner 9-j coefficient, +These routines compute the Wigner 9-j coefficient, {ja jb jc jd je jf @@ -1371,7 +1376,7 @@ Here is a list of all included functions: =item C --These routines compute the value of Dawson's integral for $x. +These routines compute the value of Dawson's integral for$x. =back @@ -1381,7 +1386,7 @@ Here is a list of all included functions: =item C --These routines compute the first-order Debye function D_1(x) = (1/x) \int_0^x dt (t/(e^t - 1)). +These routines compute the first-order Debye function D_1(x) = (1/x) \int_0^x dt (t/(e^t - 1)). =back @@ -1391,7 +1396,7 @@ Here is a list of all included functions: =item C --These routines compute the second-order Debye function D_2(x) = (2/x^2) \int_0^x dt (t^2/(e^t - 1)). +These routines compute the second-order Debye function D_2(x) = (2/x^2) \int_0^x dt (t^2/(e^t - 1)). =back @@ -1401,7 +1406,7 @@ Here is a list of all included functions: =item C --These routines compute the third-order Debye function D_3(x) = (3/x^3) \int_0^x dt (t^3/(e^t - 1)). +These routines compute the third-order Debye function D_3(x) = (3/x^3) \int_0^x dt (t^3/(e^t - 1)). =back @@ -1411,7 +1416,7 @@ Here is a list of all included functions: =item C --These routines compute the fourth-order Debye function D_4(x) = (4/x^4) \int_0^x dt (t^4/(e^t - 1)). +These routines compute the fourth-order Debye function D_4(x) = (4/x^4) \int_0^x dt (t^4/(e^t - 1)). =back @@ -1421,7 +1426,7 @@ Here is a list of all included functions: =item C --These routines compute the fifth-order Debye function D_5(x) = (5/x^5) \int_0^x dt (t^5/(e^t - 1)). +These routines compute the fifth-order Debye function D_5(x) = (5/x^5) \int_0^x dt (t^5/(e^t - 1)). =back @@ -1431,7 +1436,7 @@ Here is a list of all included functions: =item C --These routines compute the sixth-order Debye function D_6(x) = (6/x^6) \int_0^x dt (t^6/(e^t - 1)). +These routines compute the sixth-order Debye function D_6(x) = (6/x^6) \int_0^x dt (t^6/(e^t - 1)). =back @@ -1441,7 +1446,7 @@ Here is a list of all included functions: =item C -- These routines compute the dilogarithm for a real argument. In Lewin's notation this is Li_2(x), the real part of the dilogarithm of a real x. It is defined by the integral representation Li_2(x) = - \Re \int_0^x ds \log(1-s) / s. Note that \Im(Li_2(x)) = 0 for x <= 1, and -\pi\log(x) for x > 1. Note that Abramowitz & Stegun refer to the Spence integral S(x)=Li_2(1-x) as the dilogarithm rather than Li_2(x). + These routines compute the dilogarithm for a real argument. In Lewin's notation this is Li_2(x), the real part of the dilogarithm of a real x. It is defined by the integral representation Li_2(x) = - \Re \int_0^x ds \log(1-s) / s. Note that \Im(Li_2(x)) = 0 for x <= 1, and -\pi\log(x) for x > 1. Note that Abramowitz & Stegun refer to the Spence integral S(x)=Li_2(1-x) as the dilogarithm rather than Li_2(x). =back @@ -1463,7 +1468,7 @@ Here is a list of all included functions: =item C - This function multiplies $x and$y with associated absolute errors $dx and$dy. The product xy +/- xy \sqrt((dx/x)^2 +(dy/y)^2) is stored in $result. -- + =back @@ -1474,7 +1479,7 @@ Here is a list of all included functions: =item C --These routines compute the complete elliptic integral K($k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2. +These routines compute the complete elliptic integral K($k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2. =back @@ -1484,7 +1489,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1494,7 +1499,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1504,7 +1509,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1514,7 +1519,7 @@ Here is a list of all included functions: =item C --These routines compute the incomplete elliptic integral F($phi,$k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2. +These routines compute the incomplete elliptic integral F($phi,$k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2. =back @@ -1524,7 +1529,7 @@ Here is a list of all included functions: =item C --These routines compute the incomplete elliptic integral E($phi,$k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2. +These routines compute the incomplete elliptic integral E($phi,$k) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameter m = k^2. =back @@ -1534,7 +1539,7 @@ Here is a list of all included functions: =item C --These routines compute the incomplete elliptic integral \Pi(\phi,k,n) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n. +These routines compute the incomplete elliptic integral \Pi(\phi,k,n) to the accuracy specified by the mode variable mode. Note that Abramowitz & Stegun define this function in terms of the parameters m = k^2 and \sin^2(\alpha) = k^2, with the change of sign n \to -n. =back @@ -1544,7 +1549,7 @@ Here is a list of all included functions: =item C --These functions compute the incomplete elliptic integral D(\phi,k) which is defined through the Carlson form RD(x,y,z) by the following relation, D(\phi,k,n) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1). The argument$n is not used and will be removed in a future release. +These functions compute the incomplete elliptic integral D(\phi,k) which is defined through the Carlson form RD(x,y,z) by the following relation, D(\phi,k,n) = (1/3)(\sin(\phi))^3 RD (1-\sin^2(\phi), 1-k^2 \sin^2(\phi), 1). The argument $n is not used and will be removed in a future release. =back @@ -1554,7 +1559,7 @@ Here is a list of all included functions: =item C -- These routines compute the incomplete elliptic integral RC($x,$y) to the accuracy specified by the mode variable$mode. + These routines compute the incomplete elliptic integral RC($x,$y) to the accuracy specified by the mode variable $mode. =back @@ -1564,7 +1569,7 @@ Here is a list of all included functions: =item C -- These routines compute the incomplete elliptic integral RD($x,$y,$z) to the accuracy specified by the mode variable $mode. + These routines compute the incomplete elliptic integral RD($x,$y,$z) to the accuracy specified by the mode variable $mode. =back @@ -1574,7 +1579,7 @@ Here is a list of all included functions: =item C -- These routines compute the incomplete elliptic integral RF($x,$y,$z) to the accuracy specified by the mode variable $mode. + These routines compute the incomplete elliptic integral RF($x,$y,$z) to the accuracy specified by the mode variable $mode. =back @@ -1584,7 +1589,7 @@ Here is a list of all included functions: =item C -- These routines compute the incomplete elliptic integral RJ($x,$y,$z,$p) to the accuracy specified by the mode variable$mode. + These routines compute the incomplete elliptic integral RJ($x,$y,$z,$p) to the accuracy specified by the mode variable $mode. =back @@ -1596,7 +1601,7 @@ Here is a list of all included functions: =item C --These routines compute the complementary error function erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2). +These routines compute the complementary error function erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2). =back @@ -1606,7 +1611,7 @@ Here is a list of all included functions: =item C --These routines compute the logarithm of the complementary error function \log(\erfc(x)). +These routines compute the logarithm of the complementary error function \log(\erfc(x)). =back @@ -1616,7 +1621,7 @@ Here is a list of all included functions: =item C --These routines compute the error function erf(x), where erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2). +These routines compute the error function erf(x), where erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2). =back @@ -1626,7 +1631,7 @@ Here is a list of all included functions: =item C --These routines compute the Gaussian probability density function Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2). +These routines compute the Gaussian probability density function Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2). =back @@ -1636,7 +1641,7 @@ Here is a list of all included functions: =item C -- These routines compute the upper tail of the Gaussian probability function Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2). The hazard function for the normal distribution, 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. + These routines compute the upper tail of the Gaussian probability function Q(x) = (1/\sqrt{2\pi}) \int_x^\infty dt \exp(-t^2/2). The hazard function for the normal distribution, 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. =back @@ -1646,7 +1651,7 @@ Here is a list of all included functions: =item C -- These routines compute the hazard function for the normal distribution. + These routines compute the hazard function for the normal distribution. =back @@ -1656,7 +1661,7 @@ Here is a list of all included functions: =item C -- These routines provide an exponential function \exp(x) using GSL semantics and error checking. + These routines provide an exponential function \exp(x) using GSL semantics and error checking. =back @@ -1672,7 +1677,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1688,7 +1693,7 @@ Here is a list of all included functions: =item C --These routines compute the quantity \exp(x)-1 using an algorithm that is accurate for small x. +These routines compute the quantity \exp(x)-1 using an algorithm that is accurate for small x. =back @@ -1698,7 +1703,7 @@ Here is a list of all included functions: =item C --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. +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. =back @@ -1708,7 +1713,7 @@ Here is a list of all included functions: =item C --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. +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. =back @@ -1718,7 +1723,7 @@ Here is a list of all included functions: =item C --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, +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) @@ -1743,7 +1748,7 @@ Here is a list of all included functions: =item C --These routines compute the exponential integral E_1(x), E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t. +These routines compute the exponential integral E_1(x), E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t. =back @@ -1753,7 +1758,7 @@ Here is a list of all included functions: =item C --These routines compute the second-order exponential integral E_2(x), +These routines compute the second-order exponential integral E_2(x), E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2. @@ -1765,7 +1770,7 @@ Here is a list of all included functions: =item C --These routines compute the exponential integral E_n(x) of order n, +These routines compute the exponential integral E_n(x) of order n, E_n(x) := \Re \int_1^\infty dt \exp(-xt)/t^n. @@ -1777,7 +1782,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1787,7 +1792,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1797,7 +1802,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1807,7 +1812,7 @@ Here is a list of all included functions: =item C --These routines compute the exponential integral Ei(x), Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t) where PV denotes the principal value of the integral. +These routines compute the exponential integral Ei(x), Ei(x) := - PV(\int_{-x}^\infty dt \exp(-t)/t) where PV denotes the principal value of the integral. =back @@ -1817,7 +1822,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1827,7 +1832,7 @@ Here is a list of all included functions: =item C --These routines compute the integral Shi(x) = \int_0^x dt \sinh(t)/t. +These routines compute the integral Shi(x) = \int_0^x dt \sinh(t)/t. =back @@ -1837,7 +1842,7 @@ Here is a list of all included functions: =item C --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 (available as$M_EULER from the Math::GSL::Const module). +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 (available as $M_EULER from the Math::GSL::Const module). =back @@ -1847,7 +1852,7 @@ Here is a list of all included functions: =item C --These routines compute the third-order exponential integral Ei_3(x) = \int_0^xdt \exp(-t^3) for x >= 0. +These routines compute the third-order exponential integral Ei_3(x) = \int_0^xdt \exp(-t^3) for x >= 0. =back @@ -1857,7 +1862,7 @@ Here is a list of all included functions: =item C --These routines compute the Sine integral Si(x) = \int_0^x dt \sin(t)/t. +These routines compute the Sine integral Si(x) = \int_0^x dt \sin(t)/t. =back @@ -1867,7 +1872,7 @@ Here is a list of all included functions: =item C --These routines compute the Cosine integral Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0. +These routines compute the Cosine integral Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0. =back @@ -1877,7 +1882,7 @@ Here is a list of all included functions: =item C --These routines compute the complete Fermi-Dirac integral with an index of -1. This integral is given by F_{-1}(x) = e^x / (1 + e^x). +These routines compute the complete Fermi-Dirac integral with an index of -1. This integral is given by F_{-1}(x) = e^x / (1 + e^x). =back @@ -1887,7 +1892,7 @@ Here is a list of all included functions: =item C --These routines compute the complete Fermi-Dirac integral with an index of 0. This integral is given by F_0(x) = \ln(1 + e^x). +These routines compute the complete Fermi-Dirac integral with an index of 0. This integral is given by F_0(x) = \ln(1 + e^x). =back @@ -1897,7 +1902,7 @@ Here is a list of all included functions: =item C --These routines compute the complete Fermi-Dirac integral with an index of 1, F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1)). +These routines compute the complete Fermi-Dirac integral with an index of 1, F_1(x) = \int_0^\infty dt (t /(\exp(t-x)+1)). =back @@ -1907,7 +1912,7 @@ Here is a list of all included functions: =item C --These routines compute the complete Fermi-Dirac integral with an index of 2, F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1)). +These routines compute the complete Fermi-Dirac integral with an index of 2, F_2(x) = (1/2) \int_0^\infty dt (t^2 /(\exp(t-x)+1)). =back @@ -1917,7 +1922,7 @@ Here is a list of all included functions: =item C --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)). +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)). =back @@ -1927,7 +1932,7 @@ Here is a list of all included functions: =item C --These routines compute the complete Fermi-Dirac integral F_{-1/2}(x). +These routines compute the complete Fermi-Dirac integral F_{-1/2}(x). =back @@ -1937,7 +1942,7 @@ Here is a list of all included functions: =item C --These routines compute the complete Fermi-Dirac integral F_{1/2}(x). +These routines compute the complete Fermi-Dirac integral F_{1/2}(x). =back @@ -1947,7 +1952,7 @@ Here is a list of all included functions: =item C --These routines compute the complete Fermi-Dirac integral F_{3/2}(x). +These routines compute the complete Fermi-Dirac integral F_{3/2}(x). =back @@ -1957,7 +1962,7 @@ Here is a list of all included functions: =item C --These routines compute the incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x). +These routines compute the incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x). =back @@ -1967,7 +1972,7 @@ Here is a list of all included functions: =item C --These functions evaluate the Legendre polynomial P_l(x) for a specific value of l, x subject to l >= 0, |x| <= 1 +These functions evaluate the Legendre polynomial P_l(x) for a specific value of l, x subject to l >= 0, |x| <= 1 =back @@ -1977,7 +1982,7 @@ Here is a list of all included functions: =item C -- + =back @@ -1995,7 +2000,7 @@ Here is a list of all included functions: =item C --These functions evaluate the Legendre polynomials P_l(x) using explicit representations for l=1, 2, 3. +These functions evaluate the Legendre polynomials P_l(x) using explicit representations for l=1, 2, 3. =back @@ -2005,7 +2010,7 @@ Here is a list of all included functions: =item C --These routines compute the Legendre function Q_0(x) for x > -1, x != 1. +These routines compute the Legendre function Q_0(x) for x > -1, x != 1. =back @@ -2015,7 +2020,7 @@ Here is a list of all included functions: =item C --These routines compute the Legendre function Q_1(x) for x > -1, x != 1. +These routines compute the Legendre function Q_1(x) for x > -1, x != 1. =back @@ -2025,7 +2030,7 @@ Here is a list of all included functions: =item C --These routines compute the Legendre function Q_l(x) for x > -1, x != 1 and l >= 0. +These routines compute the Legendre function Q_l(x) for x > -1, x != 1 and l >= 0. =back @@ -2035,7 +2040,7 @@ Here is a list of all included functions: =item C --These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1. +These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1. =back @@ -2045,7 +2050,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2055,7 +2060,7 @@ Here is a list of all included functions: =item C --These routines compute the normalized associated Legendre polynomial$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$suitable for use in spherical harmonics. The parameters must satisfy m >= 0, l >= m, |x| <= 1. Theses routines avoid the overflows that occur for the standard normalization of P_l^m(x). +These routines compute the normalized associated Legendre polynomial$\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). =back @@ -2065,7 +2070,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2081,7 +2086,7 @@ Here is a list of all included functions: =item C --These routines compute the logarithm of the Gamma function, \log(\Gamma(x)), subject to x not being a negative integer or zero. 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. +These routines compute the logarithm of the Gamma function, \log(\Gamma(x)), subject to x not being a negative integer or zero. 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. =back @@ -2097,7 +2102,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2107,7 +2112,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2117,15 +2122,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -2135,7 +2140,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2145,7 +2150,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2155,7 +2160,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2165,7 +2170,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2175,7 +2180,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2185,7 +2190,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2195,7 +2200,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2205,7 +2210,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2215,7 +2220,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2225,7 +2230,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2235,15 +2240,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -2253,7 +2258,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2263,7 +2268,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2273,15 +2278,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -2291,7 +2296,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2301,7 +2306,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2319,7 +2324,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2329,7 +2334,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2341,7 +2346,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2351,7 +2356,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2361,7 +2366,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2371,15 +2376,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -2389,15 +2394,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -2407,7 +2412,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2417,7 +2422,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2427,7 +2432,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2437,7 +2442,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2447,7 +2452,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2465,7 +2470,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2475,7 +2480,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2485,7 +2490,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2495,7 +2500,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2505,7 +2510,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2515,7 +2520,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2525,7 +2530,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2535,7 +2540,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2545,7 +2550,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2555,7 +2560,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2565,7 +2570,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2575,7 +2580,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2585,15 +2590,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -2603,7 +2608,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2613,15 +2618,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -2631,7 +2636,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2641,7 +2646,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2651,7 +2656,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2661,7 +2666,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2671,23 +2676,23 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back =over =item C -- + =back @@ -2697,7 +2702,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2707,7 +2712,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2717,7 +2722,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2727,7 +2732,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2737,7 +2742,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2747,7 +2752,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2757,7 +2762,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2767,15 +2772,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -2785,7 +2790,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2795,7 +2800,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2805,15 +2810,15 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back @@ -2823,7 +2828,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2833,7 +2838,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2843,7 +2848,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2853,7 +2858,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2863,7 +2868,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2873,7 +2878,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2883,7 +2888,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2893,7 +2898,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2903,31 +2908,31 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back =over =item C -- + =back =over =item C -- + =back @@ -2937,7 +2942,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2947,7 +2952,7 @@ Here is a list of all included functions: =item C -- + =back @@ -2957,23 +2962,23 @@ Here is a list of all included functions: =item C -- + =back =over =item C -- + =back =over =item C -- +