Special functionsFor more information about the functions see the documentation of GSL.
| Function | Description |
|---|---|
| Ai(x) | Airy function Ai(x) |
| Bi(x) | Airy function Bi(x) |
| Ais(x) | scaled version of the Airy function SAi(x) |
| Bis(x) | scaled version of the Airy function SBi(x) |
| Aid(x) | Airy function derivative Ai'(x) |
| Bid(x) | Airy function derivative Bi'(x) |
| Aids(x) | derivative of the scaled Airy function SAi(x) |
| Bids(x) | derivative of the scaled Airy function SBi(x) |
| Ai0(s) | s-th zero of the Airy function Ai(x) |
| Bi0(s) | s-th zero of the Airy function Bi(x) |
| Aid0(s) | s-th zero of the Airy function derivative Ai'(x) |
| Bid0(s) | s-th zero of the Airy function derivative Bi'(x) |
| J0(x) | regular cylindrical Bessel function of zeroth order, J0(x) |
| J1(x) | regular cylindrical Bessel function of first order, J1(x) |
| Jn(n,x) | regular cylindrical Bessel function of order n, Jn(x) |
| Y0(x) | irregular cylindrical Bessel function of zeroth order, Y0(x) |
| Y1(x) | irregular cylindrical Bessel function of first order, Y1(x) |
| Yn(n,x) | irregular cylindrical Bessel function of order n, Yn(x) |
| I0(x) | regular modified cylindrical Bessel function of zeroth order, I0(x) |
| I1(x) | regular modified cylindrical Bessel function of first order, I1(x) |
| In(n,x) | regular modified cylindrical Bessel function of order n, In(x) |
| I0s(x) | scaled regular modified cylindrical Bessel function of zeroth order, exp (-|x|) I0(x) |
| I1s(x) | scaled regular modified cylindrical Bessel function of first order, exp(-|x|) I1(x) |
| Ins(n,x) | scaled regular modified cylindrical Bessel function of order n, exp(-|x|) In(x) |
| K0(x) | irregular modified cylindrical Bessel function of zeroth order, K0(x) |
| K1(x) | irregular modified cylindrical Bessel function of first order, K1(x) |
| Kn(n,x) | irregular modified cylindrical Bessel function of order n, Kn(x) |
| K0s(x) | scaled irregular modified cylindrical Bessel function of zeroth order, exp(x) K0(x) |
| K1s(x) | scaled irregular modified cylindrical Bessel function of first order, exp(x) K1(x) |
| Kns(n,x) | scaled irregular modified cylindrical Bessel function of order n, exp(x) Kn(x) |
| j0(x) | regular spherical Bessel function of zeroth order, j0(x) |
| j1(x) | regular spherical Bessel function of first order, j1(x) |
| j2(x) | regular spherical Bessel function of second order, j2(x) |
| jl(l,x) | regular spherical Bessel function of order l, jl(x) |
| y0(x) | irregular spherical Bessel function of zeroth order, y0(x) |
| y1(x) | irregular spherical Bessel function of first order, y1(x) |
| y2(x) | irregular spherical Bessel function of second order, y2(x) |
| yl(l,x) | irregular spherical Bessel function of order l, yl(x) |
| i0s(x) | scaled regular modified spherical Bessel function of zeroth order, exp(-|x|) i0(x) |
| i1s(x) | scaled regular modified spherical Bessel function of first order, exp(-|x|) i1(x) |
| i2s(x) | scaled regular modified spherical Bessel function of second order, exp(-|x|) i2(x) |
| ils(l,x) | scaled regular modified spherical Bessel function of order l, exp(-|x|) il(x) |
| k0s(x) | scaled irregular modified spherical Bessel function of zeroth order, exp(x) k0(x) |
| k1s(x) | scaled irregular modified spherical Bessel function of first order, exp(x) k1(x) |
| k2s(x) | scaled irregular modified spherical Bessel function of second order, exp(x) k2(x) |
| kls(l,x) | scaled irregular modified spherical Bessel function of order l, exp(x) kl(x) |
| Jnu(ν,x) | regular cylindrical Bessel function of fractional order ν, Jν(x) |
| Ynu(ν,x) | irregular cylindrical Bessel function of fractional order ν, Yν(x) |
| Inu(ν,x) | regular modified Bessel function of fractional order ν, Iν(x) |
| Inus(ν,x) | scaled regular modified Bessel function of fractional order ν, exp(-|x|) Iν(x) |
| Knu(ν,x) | irregular modified Bessel function of fractional order ν, Kν(x) |
| lnKnu(ν,x) | logarithm of the irregular modified Bessel function of fractional order ν,ln(Kν(x)) |
| Knus(ν,x) | scaled irregular modified Bessel function of fractional order ν, exp(|x|) Kν(x) |
| J0_0(s) | s-th positive zero of the Bessel function J0(x) |
| J1_0(s) | s-th positive zero of the Bessel function J1(x) |
| Jnu_0(nu,s) | s-th positive zero of the Bessel function Jν(x) |
| clausen(x) | Clausen integral Cl2(x) |
| hydrogenicR_1(Z,R) | lowest-order normalized hydrogenic bound state radial wavefunction R1 := 2Z √Z exp(-Z r) |
| hydrogenicR(n,l,Z,R) | n-th normalized hydrogenic bound state radial wavefunction |
| dawson(x) | Dawson's integral |
| D1(x) | first-order Debye function D1(x) = (1/x) ∫0x(t/(et - 1)) dt |
| D2(x) | second-order Debye function D2(x) = (2/x2) ∫0x (t2/(et - 1)) dt |
| D3(x) | third-order Debye function D3(x) = (3/x3) ∫0x (t3/(et - 1)) dt |
| D4(x) | fourth-order Debye function D4(x) = (4/x4) ∫0x (t4/(et - 1)) dt |
| D5(x) | fifth-order Debye function D5(x) = (5/x5) ∫0x (t5/(et - 1)) dt |
| D6(x) | sixth-order Debye function D6(x) = (6/x6) ∫0x (t6/(et - 1)) dt |
| Li2(x) | dilogarithm |
| Kc(k) | complete elliptic integral K(k) |
| Ec(k) | complete elliptic integral E(k) |
| F(phi,k) | incomplete elliptic integral F(phi,k) |
| E(phi,k) | incomplete elliptic integral E(phi,k) |
| P(phi,k,n) | incomplete elliptic integral P(phi,k,n) |
| D(phi,k,n) | incomplete elliptic integral D(phi,k,n) |
| RC(x,y) | incomplete elliptic integral RC(x,y) |
| RD(x,y,z) | incomplete elliptic integral RD(x,y,z) |
| RF(x,y,z) | incomplete elliptic integral RF(x,y,z) |
| RJ(x,y,z) | incomplete elliptic integral RJ(x,y,z,p) |
| erf(x) | error function erf(x) = 2/√π ∫0x exp(-t2) dt |
| erfc(x) | complementary error function erfc(x) = 1 - erf(x) = 2/√π ∫x∞ exp(-t2) dt |
| log_erfc(x) | logarithm of the complementary error function log(erfc(x)) |
| erf_Z(x) | Gaussian probability function Z(x) = (1/(2π)) exp(-x2/2) |
| erf_Q(x) | upper tail of the Gaussian probability function Q(x) = (1/(2π)) ∫x∞ exp(-t2/2) dt |
| hazard(x) | hazard function for the normal distribution |
| exp(x) | Exponential, base e |
| expm1(x) | exp(x)-1 |
| exp_mult(x,y) | exponentiate x and multiply by the factor y to return the product y exp(x) |
| exprel(x) | (exp(x)-1)/x using an algorithm that is accurate for small x |
| exprel2(x) | 2(exp(x)-1-x)/x2 using an algorithm that is accurate for small x |
| expreln(n,x) | n-relative exponential, which is the n-th generalization of the functions `exprel' |
| E1(x) | exponential integral E1(x), E1(x) := Re ∫1∞ exp(-xt)/t dt |
| E2(x) | second-order exponential integral E2(x), E2(x) := Re ∫1∞ exp(-xt)/t2 dt |
| En(x) | exponential integral E_n(x) of order n, En(x) := Re ∫1∞ exp(-xt)/tn dt) |
| Ei(x) | exponential integral E_i(x), Ei(x) := PV(∫-x∞ exp(-t)/t dt) |
| shi(x) | Shi(x) = ∫0x sinh(t)/t dt |
| chi(x) | integral Chi(x) := Re[ γE + log(x) + ∫0x (cosh[t]-1)/t dt ] |
| Ei3(x) | exponential integral Ei3(x) = ∫0x exp(-t3) dt for x >= 0 |
| si(x) | Sine integral Si(x) = ∫0x sin(t)/t dt |
| ci(x) | Cosine integral Ci(x) = -∫x∞ cos(t)/t dt for x > 0 |
| atanint(x) | Arctangent integral AtanInt(x) = ∫0x arctan(t)/t dt |
| Fm1(x) | complete Fermi-Dirac integral with an index of -1, F-1(x) = ex / (1 + ex) |
| F0(x) | complete Fermi-Dirac integral with an index of 0, F0(x) = ln(1 + ex) |
| F1(x) | complete Fermi-Dirac integral with an index of 1, F1(x) = ∫0∞ (t /(exp(t-x)+1)) dt |
| F2(x) | complete Fermi-Dirac integral with an index of 2, F2(x) = (1/2) ∫0∞ (t2 /(exp(t-x)+1)) dt |
| Fj(j,x) | complete Fermi-Dirac integral with an index of j, Fj(x) = (1/Γ(j+1)) ∫0∞ (tj /(exp(t-x)+1)) dt |
| Fmhalf(x) | complete Fermi-Dirac integral F-1/2(x) |
| Fhalf(x) | complete Fermi-Dirac integral F1/2(x) |
| F3half(x) | complete Fermi-Dirac integral F3/2(x) |
| Finc0(x,b) | incomplete Fermi-Dirac integral with an index of zero, F0(x,b) = ln(1 + eb-x) - (b-x) |
| lngamma(x) | logarithm of the Gamma function |
| gammastar(x) | regulated Gamma Function Γ*(x) for x > 0 |
| gammainv(x) | reciprocal of the gamma function, 1/Γ(x) using the real Lanczos method. |
| fact(n) | factorial n! |
| doublefact(n) | double factorial n!! = n(n-2)(n-4)... |
| lnfact(n) | logarithm of the factorial of n, log(n!) |
| lndoublefact(n) | logarithm of the double factorial log(n!!) |
| choose(n,m) | combinatorial factor `n choose m' = n!/(m!(n-m)!) |
| lnchoose(n,m) | logarithm of `n choose m' |
| taylor(n,x) | Taylor coefficient xn / n! for x >= 0, n >= 0 |
| poch(a,x) | Pochhammer symbol (a)x := Γ(a + x)/Γ(x) |
| lnpoch(a,x) | logarithm of the Pochhammer symbol (a)x := Γ(a + x)/Γ(x) |
| pochrel(a,x) | relative Pochhammer symbol ((a,x) - 1)/x where (a,x) = (a)x := Γ(a + x)/Γ(a) |
| gammainc(a,x) | incomplete Gamma Function Γ(a,x) = ∫x∞ ta-1 exp(-t) dt for a > 0, x >= 0 |
| gammaincQ(a,x) | normalized incomplete Gamma Function P(a,x) = 1/Γ(a) ∫x∞ ta-1 exp(-t) dt for a > 0, x >= 0 |
| gammaincP(a,x) | complementary normalized incomplete Gamma Function P(a,x) = 1/Γ(a) ∫0x ta-1 exp(-t) dt for a > 0, x >= 0 |
| beta(a,b) | Beta Function, B(a,b) = Γ(a) Γ(b)/Γ(a+b) for a > 0, b > 0 |
| lnbeta(a,b) | logarithm of the Beta Function, log(B(a,b)) for a > 0, b > 0 |
| betainc(a,b,x) | normalize incomplete Beta function B_x(a,b)/B(a,b) for a > 0, b > 0 |
| C1(λ,x) | Gegenbauer polynomial Cλ1(x) |
| C2(λ,x) | Gegenbauer polynomial Cλ2(x) |
| C3(λ,x) | Gegenbauer polynomial Cλ3(x) |
| Cn(n,λ,x) | Gegenbauer polynomial Cλn(x) |
| hyperg_0F1(c,x) | hypergeometric function 0F1(c,x) |
| hyperg_1F1i(m,n,x) | confluent hypergeometric function 1F1(m,n,x) = M(m,n,x) for integer parameters m, n |
| hyperg_1F1(a,b,x) | confluent hypergeometric function 1F1(a,b,x) = M(a,b,x) for general parameters a,b |
| hyperg_Ui(m,n,x) | confluent hypergeometric function U(m,n,x) for integer parameters m,n |
| hyperg_U(a,b,x) | confluent hypergeometric function U(a,b,x) |
| hyperg_2F1(a,b,c,x) | Gauss hypergeometric function 2F1(a,b,c,x) |
| hyperg_2F1c(aR,aI,c,x) | Gauss hypergeometric function 2F1(aR + i aI, aR - i aI, c, x) with complex parameters |
| hyperg_2F1r(aR,aI,c,x) | renormalized Gauss hypergeometric function 2F1(a,b,c,x) / Γ(c) |
| hyperg_2F1cr(aR,aI,c,x) | renormalized Gauss hypergeometric function 2F1(aR + i aI, aR - i aI, c, x) / Γ(c) |
| hyperg_2F0(a,b,x) | hypergeometric function 2F0(a,b,x) |
| L1(a,x) | generalized Laguerre polynomials La1(x) |
| L2(a,x) | generalized Laguerre polynomials La2(x) |
| L3(a,x) | generalized Laguerre polynomials La3(x) |
| W0(x) | principal branch of the Lambert W function, W0(x) |
| Wm1(x) | secondary real-valued branch of the Lambert W function, W-1(x) |
| P1(x) | Legendre polynomials P1(x) |
| P2(x) | Legendre polynomials P2(x) |
| P3(x) | Legendre polynomials P3(x) |
| Pl(l,x) | Legendre polynomials Pl(x) |
| Q0(x) | Legendre polynomials Q0(x) |
| Q1(x) | Legendre polynomials Q1(x) |
| Ql(l,x) | Legendre polynomials Ql(x) |
| Plm(l,m,x) | associated Legendre polynomial Plm(x) |
| Pslm(l,m,x) | normalized associated Legendre polynomial √{(2l+1)/(4π)} √{(l-m)!/(l+m)!} Plm(x) suitable for use in spherical harmonics |
| Phalf(λ,x) | irregular Spherical Conical Function P1/2-1/2 + i λ(x) for x > -1 |
| Pmhalf(λ,x) | regular Spherical Conical Function P-1/2-1/2 + i λ(x) for x > -1 |
| Pc0(λ,x) | conical function P0-1/2 + i λ(x) for x > -1 |
| Pc1(λ,x) | conical function P1-1/2 + i λ(x) for x > -1 |
| Psr(l,λ,x) | Regular Spherical Conical Function P-1/2-l-1/2 + i λ(x) for x > -1, l >= -1 |
| Pcr(l,λ,x) | Regular Cylindrical Conical Function P-m-1/2 + i λ(x) for x > -1, m >= -1 |
| H3d0(λ,η) | zeroth radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, LH3d0(λ,,η) := sin(λ η)/(λ sinh(η)) for η >= 0 |
| H3d1(λ,η) | zeroth radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space, LH3d1(λ,η) := 1/√{λ2 + 1} sin(λ η)/(λ sinh(η)) (coth(η) - λ cot(λ η)) for η >= 0 |
| H3d(l,λ,η) | L'th radial eigenfunction of the Laplacian on the 3-dimensional hyperbolic space eta >= 0, l >= 0 |
| logabs(x) | logarithm of the magnitude of X, log(|x|) |
| logp(x) | log(1 + x) for x > -1 using an algorithm that is accurate for small x |
| logm(x) | log(1 + x) - x for x > -1 using an algorithm that is accurate for small x |
| psiint(n) | digamma function ψ(n) for positive integer n |
| psi(x) | digamma function ψ(n) for general x |
| psi1piy(y) | real part of the digamma function on the line 1+i y, Re[ψ(1 + i y)] |
| psi1int(n) | Trigamma function ψ'(n) for positive integer n |
| psi1(n) | Trigamma function ψ'(x) for general x |
| psin(m,x) | polygamma function ψ(m)(x) for m >= 0, x > 0 |
| synchrotron1(x) | first synchrotron function x ∫x∞ K5/3(t) dt for x >= 0 |
| synchrotron2(x) | second synchrotron function x K2/3(x) for x >= 0 |
| J2(x) | transport function J(2,x) |
| J3(x) | transport function J(3,x) |
| J4(x) | transport function J(4,x) |
| J5(x) | transport function J(5,x) |
| zetaint(n) | Riemann zeta function ζ(n) for integer n |
| zeta(s) | Riemann zeta function ζ(s) for arbitrary s |
| zetam1int(n) | Riemann ζ function minus 1 for integer n |
| zetam1(s) | Riemann ζ function minus 1 |
| zetaintm1(s) | Riemann ζ function for integer n minus 1 |
| hzeta(s,q) | Hurwitz zeta function ζ(s,q) for s > 1, q > 0 |
| etaint(n) | eta function η(n) for integer n |
| eta(s) | eta function η(s) for arbitrary s |