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List of the special functions callable in SFL and Lua

 

Here is a template for the description of all the functions

 

FUNCTION CALL	Text that appears in the list box to select the functions
Brief description of the function

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The list

AiryAi(x)	Airy Ai
This routine compute the Airy function Ai(x)

AiryBi(x)	Airy Bi
This routine compute the Airy function Bi(x)

AiryAiScaled(x)	Airy Ai scaled
This routine 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

AiryBiScaled(x)	Airy Bi scaled
This routine 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.

AiryAiDeriv(x)	Airy Ai derivate
This routine compute the Airy function derivative Ai'(x)

AiryBiDeriv(x)	Airy Bi derivate
This routine compute the Airy function derivative Bi'(x)

AiryAiDerivScaled(x)	Airy Ai derivate scaled
This routine 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

AiryBiDerivScaled(x)	Airy Bi derivate scaled
This routine 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

AiryAiZero(s)	Airy Ai zero
This routine compute the location of the s-th zero of the Airy function Ai(x)

AiryBiZero(s)	Airy Bi zero
This routine compute the location of the s-th zero of the Airy function Bi(x)

AiryAiZeroDeriv(s)	Airy Ai zero derivate
This routine compute the location of the s-th zero of the Airy function derivative Ai'(x)

AiryBiZeroDeriv(s)	Airy Bi zero derivate
This routine compute the location of the s-th zero of the Airy function derivative Bi'(x).

BesselJ0(x)	Bessel regular cylindrical J0
These routines compute the regular cylindrical Bessel function of zeroth order, J_0(x)

BesselJ1(x)	Bessel regular cylindrical J1
These routines compute the regular cylindrical Bessel function of first order, J_1(x).

BesselJn(x,n)	Bessel regular cylindrical Jn
These routines compute the regular cylindrical Bessel function of order n, J_n(x).

BesselY0(x)	Bessel irregular cylindrical Y0
These routines compute the irregular cylindrical Bessel function of zeroth order, Y_0(x), for x>0.

BesselY1(x)	Bessel irregular cylindrical Y1
These routines compute the irregular cylindrical Bessel function of first order, Y_1(x), for x>0.

BesselYn(x,n)	Bessel irregular cylindrical Yn
These routines compute the irregular cylindrical Bessel function of order n, Y_n(x), for x>0.

BesselI0(x)	Bessel regular modified cylindrical I0
These routines compute the regular modified cylindrical Bessel function of zeroth order, I_0(x).

BesselI1(x)	Bessel regular modified cylindrical I1
These routines compute the regular modified cylindrical Bessel function of first order, I_1(x).

BesselIn(x,n)	Bessel regular modified cylindrical In
These routines compute the regular modified cylindrical Bessel function of order n, I_n(x).

BesselI0Scaled(x)	Bessel regular modified cylindrical I0 scaled
These routines compute the scaled regular modified cylindrical Bessel function of zeroth order \exp(-|x|) I_0(x).

BesselI1Scaled(x)	Bessel regular modified cylindrical I1 scaled
These routines compute the scaled regular modified cylindrical Bessel function of first order \exp(-|x|) I_1(x).

BesselInScaled(x,n)	Bessel regular modified cylindrical In scaled
These routines compute the scaled regular modified cylindrical Bessel function of order n, \exp(-|x|) I_n(x)

BesselK0(x)	Bessel irregular modified cylindrical K0
These routines compute the irregular modified cylindrical Bessel function of zeroth order, K_0(x), for x > 0.

BesselK1(x)	Bessel irregular modified cylindrical K1
These routines compute the irregular modified cylindrical Bessel function of first order, K_1(x), for x > 0.

BesselKn(x,n)	Bessel irregular modified cylindrical Kn
These routines compute the irregular modified cylindrical Bessel function of order n, K_n(x), for x > 0.

BesselK0Scaled(x)	Bessel irregular modified cylindrical K0 scaled
These routines compute the scaled irregular modified cylindrical Bessel function of zeroth order \exp(x) K_0(x) for x>0.

BesselK1Scaled(x)	Bessel irregular modified cylindrical K1 scaled
These routines compute the scaled irregular modified cylindrical Bessel function of first order \exp(x) K_1(x) for x>0.

BesselKnScaled(x,n)	Bessel irregular modified cylindrical Kn scaled
These routines compute the scaled irregular modified cylindrical Bessel function of order n, \exp(x) K_n(x), for x>0.

Besselj0(x)	Bessel regular spherical j0
These routines compute the regular spherical Bessel function of zeroth order, j_0(x) = \sin(x)/x.

Besselj1(x)	Bessel regular spherical j1
These routines compute the regular spherical Bessel function of first order, j_1(x) = (\sin(x)/x - \cos(x))/x.

Besselj2(x)	Bessel regular spherical j2
These routines compute the regular spherical Bessel function of second order, j_2(x) = ((3/x^2 - 1)\sin(x) - 3\cos(x)/x)/x.

Besseljn(x,n)	Bessel regular spherical jn
These routines compute the regular spherical Bessel function of order n, j_n(x), for n >= 0 and x >= 0.

Bessely0(x)	Bessel irregular spherical y0
These routines compute the irregular spherical Bessel function of zeroth order, y_0(x) = -\cos(x)/x.

Bessely1(x)	Bessel irregular spherical y1
These routines compute the irregular spherical Bessel function of first order, y_1(x) = -(\cos(x)/x + \sin(x))/x.

Bessely2(x)	Bessel irregular spherical y2
These routines compute the irregular spherical Bessel function of second order, y_2(x) = (-3/x^3 + 1/x)\cos(x) - (3/x^2)\sin(x).

Besselyn(x,n)	Bessel irregular spherical yn
These routines compute the irregular spherical Bessel function of order n, y_n(x), for n >= 0.

Besseli0Scaled(x)	Bessel regular modified spherical i0 scaled
These routines compute the scaled regular modified spherical Bessel function of zeroth order, \exp(-|x|) i_0(x).

Besseli1Scaled(x)	Bessel regular modified spherical i1 scaled
These routines compute the scaled regular modified spherical Bessel function of first order, \exp(-|x|) i_1(x).

Besseli2Scaled(x)	Bessel regular modified spherical i2 scaled
These routines compute the scaled regular modified spherical Bessel function of second order, \exp(-|x|) i_2(x)

BesselinScaled(x,n)	Bessel regular modified spherical in scaled
These routines compute the scaled regular modified spherical Bessel function of order n, \exp(-|x|) i_n(x)

Besselk0Scaled(x)	Bessel irregular modified spherical k0 scaled
These routines compute the scaled irregular modified spherical Bessel function of zeroth order, \exp(x) k_0(x), for x>0.

Besselk1Scaled(x)	Bessel irregular modified spherical k1 scaled
These routines compute the scaled irregular modified spherical Bessel function of first order, \exp(x) k_1(x), for x>0.

Besselk2Scaled(x)	Bessel irregular modified spherical k2 scaled
These routines compute the scaled irregular modified spherical Bessel function of second order, \exp(x) k_2(x), for x>0.

BesselknScaled(x,n)	Bessel irregular modified spherical kn scaled
These routines compute the scaled irregular modified spherical Bessel function of order n, \exp(x) k_n(x), for x>0.

BesselJnu(x,nu)	Bessel regular cylindrical of fractional order Jnu
These routines compute the regular cylindrical Bessel function of fractional order \nu, J_\nu(x).

BesselYnu(x,nu)	Bessel irregular cylindrical of fractional order Ynu
These routines compute the irregular cylindrical Bessel function of fractional order \nu, Y_\nu(x).

BesselInu(x,nu)	Bessel regular modified cylindrical of fractional order Inu
These routines compute the regular modified Bessel function of fractional order \nu, I_\nu(x) for x>0, \nu>0.

BesselKnu(x,nu)	Bessel irregular modified cylindrical of fractional order Knu
These routines compute the irregular modified Bessel function of fractional order \nu, K_\nu(x) for x>0, \nu>0.

BessellnKnu(x,nu)	Bessel ln of irregular modified cylindrical of fractional order Knu
These routines compute the logarithm of the irregular modified Bessel function of fractional order \nu, \ln(K_\nu(x)) for x>0, \nu>0.

BesselKnuScaled(x,nu)	Bessel irregular modified cylindrical of fractional order Knu scaled
These routines compute the scaled irregular modified Bessel function of fractional order \nu, \exp(+|x|) K_\nu(x) for x>0, \nu>0.

BesselJ0Zero(s)	Bessel regular cylindrical J0 zero
These routines compute the location of the s-th positive zero of the Bessel function J_0(x).

BesselJ1Zero(s)	Bessel regular cylindrical J1 zero
These routines compute the location of the s-th positive zero of the Bessel function J_1(x).

BesselJnuZero(s,nu)	Bessel regular cylindrical of fractional order Jnu zero
These routines compute the location of the s-th positive zero of the Bessel function J_\nu(x). The current implementation does not support negative values of nu.

Clausen(x)	Clausen
These routines compute the Clausen integral Cl_2(x).

Dawson(x)	Dawson
These routines compute the value of Dawson's integral for x.

Debye1(x)	Debye order 1
These routines compute the first-order Debye function D_1(x) = (1/x) \int_0^x dt (t/(e^t - 1)).

Debye2(x)	Debye order 2
These routines compute the second-order Debye function D_2(x) = (2/x^2) \int_0^x dt (t^2/(e^t - 1)).

Debye3(x)	Debye order 3
These routines compute the third-order Debye function D_3(x) = (3/x^3) \int_0^x dt (t^3/(e^t - 1)).

Debye4(x)	Debye order 4
These routines compute the fourth-order Debye function D_4(x) = (4/x^4) \int_0^x dt (t^4/(e^t - 1)).

Debye5(x)	Debye order 5
These routines compute the fifth-order Debye function D_5(x) = (5/x^5) \int_0^x dt (t^5/(e^t - 1)).

Debye6(x)	Debye order 6
These routines compute the sixth-order Debye function D_6(x) = (6/x^6) \int_0^x dt (t^6/(e^t - 1)).

Dilogarithm(x)	Dilogarithm
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.

EllintKcomp(k)	Elliptic integrals complete Legendre form K
These routines compute the complete elliptic integral K(k)

EllintEcomp(k)	Elliptic integrals complete Legendre form E
These routines compute the complete elliptic integral E(k)

EllintPcomp(k,n)	Elliptic integrals complete Legendre form P
These routines compute the complete elliptic integral \Pi(k,n)

EllintF(phi,k)	Elliptic integrals incomplete Legendre form F
These routines compute the incomplete elliptic integral F(\phi,k)

EllintE(phi,k)	Elliptic integrals incomplete Legendre form E
These routines compute the incomplete elliptic integral E(\phi,k)

EllintP(phi,k,n)	Elliptic integrals incomplete Legendre form P
These routines compute the incomplete elliptic integral \Pi(\phi,k,n)

EllintD(phi,k)	Elliptic integrals incomplete Legendre form D
These functions compute the incomplete elliptic integral D(\phi,k)

EllintRC(x,y)	Elliptic integrals incomplete Carlson form RC
These routines compute the incomplete elliptic integral RC(x,y)

EllintRD(x,y,z)	Elliptic integrals incomplete Carlson form RD
These routines compute the incomplete elliptic integral RD(x,y,z)

EllintRF(x,y,z)	Elliptic integrals incomplete Carlson form RF
These routines compute the incomplete elliptic integral RF(x,y,z)

EllintRJ(x,y,z,p)	Elliptic integrals incomplete Carlson form RJ
These routines compute the incomplete elliptic integral RJ(x,y,z,p)

EllfuncSn(u,m)	Elliptic functions of Jacobi sn
This function computes the Jacobian elliptic functions sn(u,m)

EllfuncCn(u,m)	Elliptic functions of Jacobi cn
This function computes the Jacobian elliptic functions cn(u,m)

EllfuncDn(u,m)	Elliptic functions of Jacobi dn
This function computes the Jacobian elliptic functions dn(u,m)

Erf(x)	Error function erf
These routines compute the error function erf(x), where erf(x) = (2/\sqrt(\pi)) \int_0^x dt \exp(-t^2).

Erfc(x)	Error function complementary erfc
These routines compute the complementary error function erfc(x) = 1 - erf(x) = (2/\sqrt(\pi)) \int_x^\infty \exp(-t^2).

LnErfc(x)	Error function log complementary log(erfc)
These routines compute the logarithm of the complementary error function \log(\erfc(x)).

ErfZ(x)	Probability, gaussian density function
These routines compute the Gaussian probability density function Z(x) = (1/\sqrt{2\pi}) \exp(-x^2/2).

ErfQ(x)	Probability, upper tail gaussian density function
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).

Hazard(x)	Probability, hazard function for the normal distribution
These routines compute the hazard function for the normal distribution.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)

ExpIntE1(x)	Exponential integral E1
These routines compute the exponential integral E_1(x), E_1(x) := \Re \int_1^\infty dt \exp(-xt)/t.

ExpIntE2(x)	Exponential integral E2
These routines compute the second-order exponential integral E_2(x), E_2(x) := \Re \int_1^\infty dt \exp(-xt)/t^2.

ExpIntEn(n,x)	Exponential integral En
These routines compute the exponential integral E_n(x) of order n, E_n(x) := \Re \int_1^\infty dt \exp(-xt)/t^n.

ExpIntEi(x)	Exponential integral Ei
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.

ExpIntEi3(x)	Exponential integral third order Ei_3
These routines compute the third-order exponential integral Ei_3(x) = \int_0^xdt \exp(-t^3) for x >= 0.

Shi(x)	Hyperbolic integral Shi
These routines compute the integral Shi(x) = \int_0^x dt \sinh(t)/t.

Chi(x)	Hyperbolic integral Chi
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

Si(x)	Sine integral Si
These routines compute the Sine integral Si(x) = \int_0^x dt \sin(t)/t.

Ci(x)	Cosine integral Ci
These routines compute the Cosine integral Ci(x) = -\int_x^\infty dt \cos(t)/t for x > 0.

AtanInt(x)	Arctangent integral AtanInt
These routines compute the Arctangent integral, which is defined as AtanInt(x) = \int_0^x dt \arctan(t)/t.

FermiDiracM1(x)	Complete Fermi-Dirac integral of index -1
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). The complete Fermi-Dirac integral F_j(x) is given by,F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))

FermiDirac0(x)	Complete Fermi-Dirac integral of index 0
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). The complete Fermi-Dirac integral F_j(x) is given by,F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))

FermiDirac1(x)	Complete Fermi-Dirac integral of index 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)). The complete Fermi-Dirac integral F_j(x) is given by,F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))

FermiDirac2(x)	Complete Fermi-Dirac integral of index 2
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)). The complete Fermi-Dirac integral F_j(x) is given by,F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))

FermiDiracj(j,x)	Complete Fermi-Dirac integral of index j
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)). The complete Fermi-Dirac integral F_j(x) is given by,F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))

FermiDiracMhalf(x)	Complete Fermi-Dirac integral of index -1/2
These routines compute the complete Fermi-Dirac integral F_{-1/2}(x). The complete Fermi-Dirac integral F_j(x) is given by,F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))

FermiDirachalf(x)	Complete Fermi-Dirac integral of index 1/2
These routines compute the complete Fermi-Dirac integral F_{1/2}(x). The complete Fermi-Dirac integral F_j(x) is given by,F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))

FermiDirac3half(x)	Complete Fermi-Dirac integral of index 3/2
These routines compute the complete Fermi-Dirac integral F_{3/2}(x). The complete Fermi-Dirac integral F_j(x) is given by,F_j(x) := (1/\Gamma(j+1)) \int_0^\infty dt (t^j / (\exp(t-x) + 1))

FermiDiracIncj(x,b)	Incomplete Fermi-Dirac integral of index j
These routines compute the incomplete Fermi-Dirac integral with an index of zero, F_0(x,b) = \ln(1 + e^{b-x}) - (b-x). The incomplete Fermi-Dirac integral F_j(x,b) is given by, F_j(x,b) := (1/\Gamma(j+1)) \int_b^\infty dt (t^j / (\Exp(t-x) + 1))

Gamma(x)	Gamma
These routines compute the Gamma function \Gamma(x), subject to x not being a negative integer or zero.The Gamma function is defined by the following integral, \Gamma(x) = \int_0^\infty dt t^{x-1} \exp(-t)

LnGamma(x)	Ln Gamma
These routines compute the logarithm of the Gamma function, \log(\Gamma(x)), subject to x not being a negative integer or zero.

GammaStar(x)	Gamma star
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

GammaInv(x)	Gamma inv
These routines compute the reciprocal of the gamma function, 1/\Gamma(x)

GammaInc(a,x)	Unnormalized incomplete Gamma Function
These functions compute the unnormalized incomplete Gamma Function \Gamma(a,x) = \int_x^\infty dt t^{a-1} \exp(-t) for a real and x >= 0.

GammaIncQ(a,x)	Normalized incomplete Gamma Function
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.

GammaIncP(a,x)	Complementary Normalized incomplete Gamma Function
These routines compute the complementary normalized incomplete Gamma Function P(a,x) = 1 - Q(a,x) = 1/\Gamma(a) \int_0^x dt t^{a-1} \exp(-t) for a > 0, x >= 0.

Fact(n)	Factorial
These routines compute the factorial n!. The factorial is related to the Gamma function by n! = \Gamma(n+1).

DoubleFact(n)	Double Factorial
These routines compute the double factorial n!! = n(n-2)(n-4) \dots

LnFact(n)	Ln Factorial
These routines compute the logarithm of the factorial of n, \log(n!).

LnDoubleFact(n)	Ln Double Factorial
These routines compute the logarithm of the double factorial of n, \log(n!!).

Choose(n,m)	Combination
These routines compute the combinatorial factor n choose m = n!/(m!(n-m)!)

LnChoose(n,m)	Ln Combination
These routines compute the logarithm of n choose m. This is equivalent to the sum \log(n!) - \log(m!) - \log((n-m)!).

TaylorCoef(n,x)	Taylor coefficient
These routines compute the Taylor coefficient x^n / n! for x >= 0, n >= 0.

Poch(a,x)	Pochhammer
These routines compute the Pochhammer symbol (a)_x = \Gamma(a + x)/\Gamma(a), subject to a and a+x not being negative integers or zero.

LnPoch(a,x)	Ln Pochhammer
These routines compute the logarithm of the Pochhammer symbol, \log((a)_x) = \log(\Gamma(a + x)/\Gamma(a)) for a > 0, a+x > 0.

PochRel(a,x)	Pochhammer relative
These routines compute the relative Pochhammer symbol ((a)_x - 1)/x where (a)_x = \Gamma(a + x)/\Gamma(a).

Beta(a,b)	Beta
These routines compute the Beta Function, B(a,b) = \Gamma(a)\Gamma(b)/\Gamma(a+b) subject to a and b not being negative integers.

LnBeta(a,b)	Ln Beta
These routines compute the logarithm of the Beta Function, \log(B(a,b)) subject to a and b not being negative integers.

BetaInc(a,b,x)	Incomplete Beta
These routines compute the normalized incomplete Beta function I_x(a,b)=B_x(a,b)/B(a,b) where B_x(a,b) = \int_0^x t^{a-1} (1-t)^{b-1} dt for 0 <= x <= 1.

GegenPoly1(lambda,x)	Gegenbauer Functions 1
These functions evaluate the Gegenbauer polynomials C^{(\lambda)}_n(x) for n =1

GegenPoly2(lambda,x)	Gegenbauer Functions 2
These functions evaluate the Gegenbauer polynomials C^{(\lambda)}_n(x) for n =2

GegenPoly3(lambda,x)	Gegenbauer Functions 3
These functions evaluate the Gegenbauer polynomials C^{(\lambda)}_n(x) for n =3

GegenPolyn(n,lambda,x)	Gegenbauer Functions n
These functions evaluate the Gegenbauer polynomial C^{(\lambda)}_n(x) for a specific value of n, lambda, x subject to \lambda > -1/2, n >= 0.

Hyperg0F1(c,x)	Hypergeometric 0F1
These routines compute the hypergeometric function 0F1(c,x).

Hyperg1F1int(m,n,x)	Hypergeometric 1F1 int
These routines compute the confluent hypergeometric function 1F1(m,n,x) = M(m,n,x) for integer parameters m, n.

Hyperg1F1(a,b,x)	Hypergeometric 1F1
These routines compute the confluent hypergeometric function 1F1(a,b,x) = M(a,b,x) for general parameters a, b.

HypergUint(m,n,x)	Hypergeometric U int
These routines compute the confluent hypergeometric function U(m,n,x) for integer parameters m, n.

HypergU(a,b,x)	Hypergeometric U
These routines compute the confluent hypergeometric function U(a,b,x).

Hyperg2F1(a,b,c,x)	Gauss Hypergeometric 2F1
These routines compute the Gauss hypergeometric function 2F1(a,b,c,x) = F(a,b,c,x) for |x| < 1.

Hyperg2F1conj(a_R,a_I,c,x)	Gauss Hypergeometric 2F1 conj
These routines compute the Gauss hypergeometric function 2F1(a_R + i a_I, a_R - i a_I, c, x) with complex parameters for |x| < 1.

Hyperg2F1renor(a,b,c,x)	Gauss Hypergeometric 2F1 renorm
These routines compute the renormalized Gauss hypergeometric function 2F1(a,b,c,x) / \Gamma(c) for |x| < 1.

Hyperg2F1conjrenor(a_R,a_I,c,x)	Gauss Hypergeometric 2F1 conj renorm
These routines compute the renormalized Gauss hypergeometric function 2F1(a_R + i a_I, a_R - i a_I, c, x) / \Gamma(c) for |x| < 1.

Hyperg2F0(a,b,x)	Hypergeometric 2F0
These routines compute the hypergeometric function 2F0(a,b,x). The series representation is a divergent hypergeometric series. However, for x < 0 we have 2F0(a,b,x) = (-1/x)^a U(a,1+a-b,-1/x)

Laguerre1(a,x)	Laguerre Functions 1
These routines evaluate the generalized Laguerre polynomials L^a_1(x)

Laguerre2(a,x)	Laguerre Functions 2
These routines evaluate the generalized Laguerre polynomials L^a_2(x)

Laguerre3(a,x)	Laguerre Functions 3
These routines evaluate the generalized Laguerre polynomials L^a_3(x)

Laguerren(n,a,x)	Laguerre Functions n
These routines evaluate the generalized Laguerre polynomials L^a_n(x) for a > -1, n >= 0.

LambertW0(x)	Lambert W0
These compute the principal branch of the Lambert W function, W_0(x).

LambertWm1(x)	Lambert Wm1
These compute the secondary real-valued branch of the Lambert W function, W_{-1}(x).

LegendreP1(x)	Legendre P1
These functions evaluate the Legendre polynomials P_n(x) using explicit representations for n=1

LegendreP2(x)	Legendre P2
These functions evaluate the Legendre polynomials P_n(x) using explicit representations for n=2

LegendreP3(x)	Legendre P3
These functions evaluate the Legendre polynomials P_n(x) using explicit representations for n=3

LegendrePn(n,x)	Legendre Pn
These functions evaluate the Legendre polynomial P_n(x) for a specific value of n, x subject to n >= 0, |x| <= 1

LegendreQ0(x)	Legendre Q0
These routines compute the Legendre function Q_0(x) for x > -1, x != 1.

LegendreQ1(x)	Legendre Q1
These routines compute the Legendre function Q_1(x) for x > -1, x != 1.

LegendreQn(n,x)	Legendre Qn
These routines compute the Legendre function Q_n(x) for x > -1, x != 1 and n >= 0.

LegendrePlm(l,m,x)	Associated Legendre
These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, |x| <= 1.

LegendreSphPlm(l,m,x)	Normalized Associated Legendre
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

DigammaInt(n)	Digamma for integer
These routines compute the digamma function \psi(n) for positive integer n. The digamma function is also called the Psi function.

Digamma(x)	Digamma
These routines compute the digamma function \psi(x) for general x, x \ne 0.

DigammaPiy(y)	Real part of the digamma function on the line 1+iy
These routines compute the real part of the digamma function on the line 1+i y, \Re[\psi(1 + i y)].

TrigammaInt(n)	Trigamma for integer
These routines compute the Trigamma function \psi'(n) for positive integer n.

Trigamma(x)	Trigamma
These routines compute the Trigamma function \psi'(x) for general x.

Polygamma(n,x)	Polygamma
These routines compute the polygamma function \psi^{(n)}(x) for n >= 0, x > 0.

Synchrotron1(x)	Synchrotron Function 1
These routines compute the first synchrotron function x \int_x^\infty dt K_{5/3}(t) for x >= 0.

Synchrotron2(x)	Synchrotron Function 2
These routines compute the second synchrotron function x K_{2/3}(x) for x >= 0.

Transport2(x)	Transport Function 2
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.

Transport3(x)	Transport Function 3
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.

Transport4(x)	Transport Function 4
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.

Transport5(x)	Transport Function 5
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.

ZetaInt(n)	Rieman Zeta Function int
These routines compute the Riemann zeta function \zeta(n) for integer n, n \ne 1.

Zeta(s)	Rieman Zeta Function
These routines compute the Riemann zeta function \zeta(s) for arbitrary s, s \ne 1.

Zetam1Int(n)	Rieman Zeta Function minus one int
These routines compute \zeta(n) - 1 for integer n, n \ne 1.

Zetam1(s)	Rieman Zeta Function minus one
These routines compute \zeta(s) - 1 for arbitrary s, s \ne 1.

HurwitzZeta(s,q)	Hurwitz Zeta Function
These routines compute the Hurwitz zeta function \zeta(s,q) for s > 1, q > 0.

EtaInt(n)	Eta Function int
These routines compute the eta function \eta(n) for integer n.

Eta(s)	Eta Function
These routines compute the eta function \eta(s) for arbitrary s.