Module Gsl.Sf

Special functions

The library includes routines for calculating the values of Airy functions, Bessel functions, Clausen functions, Coulomb functions, Coupling coefficients, The Dawson function, Debye functions, Dilogarithms, Elliptic integrals, Error, Exponential functions, Exponential integrals, Fermi-Dirac function, Gamma function, Gegenbauer functions aka Ultraspherical polynomials, Hypergeometric functions, Laguerre functions, Lambert W functions, Legendre functions and Associated Legendre functions and Spherical Harmonics, log, Power function, Psi (Digamma) function, Synchrotron functions, Transport functions, Trigonometric functions, and Zeta functions.

Each routine also computes an estimate of the numerical error in the calculated value of the function.

Airy functions

val airy_Ai : float -> Fun.mode -> float
val airy_Ai_e : float -> Fun.mode -> Fun.result
val airy_Bi : float -> Fun.mode -> float
val airy_Bi_e : float -> Fun.mode -> Fun.result
val airy_Ai_scaled : float -> Fun.mode -> float
val airy_Ai_scaled_e : float -> Fun.mode -> Fun.result
val airy_Bi_scaled : float -> Fun.mode -> float
val airy_Bi_scaled_e : float -> Fun.mode -> Fun.result
val airy_Ai_deriv : float -> Fun.mode -> float
val airy_Ai_deriv_e : float -> Fun.mode -> Fun.result
val airy_Bi_deriv : float -> Fun.mode -> float
val airy_Bi_deriv_e : float -> Fun.mode -> Fun.result
val airy_Ai_deriv_scaled : float -> Fun.mode -> float
val airy_Ai_deriv_scaled_e : float -> Fun.mode -> Fun.result
val airy_Bi_deriv_scaled : float -> Fun.mode -> float
val airy_Bi_deriv_scaled_e : float -> Fun.mode -> Fun.result
val airy_zero_Ai : int -> float
val airy_zero_Ai_e : int -> Fun.result
val airy_zero_Bi : int -> float
val airy_zero_Bi_e : int -> Fun.result

Bessel functions

val bessel_J0 : float -> float
val bessel_J0_e : float -> Fun.result
val bessel_J1 : float -> float
val bessel_J1_e : float -> Fun.result
val bessel_Jn : int -> float -> float
val bessel_Jn_e : int -> float -> Fun.result
val bessel_Jn_array : int -> float -> float array -> unit
val bessel_Y0 : float -> float
val bessel_Y0_e : float -> Fun.result
val bessel_Y1 : float -> float
val bessel_Y1_e : float -> Fun.result
val bessel_Yn : int -> float -> float
val bessel_Yn_e : int -> float -> Fun.result
val bessel_Yn_array : int -> float -> float array -> unit
val bessel_I0 : float -> float
val bessel_I0_e : float -> Fun.result
val bessel_I1 : float -> float
val bessel_I1_e : float -> Fun.result
val bessel_In : int -> float -> float
val bessel_In_e : int -> float -> Fun.result
val bessel_In_array : int -> float -> float array -> unit
val bessel_K0 : float -> float
val bessel_K0_e : float -> Fun.result
val bessel_K1 : float -> float
val bessel_K1_e : float -> Fun.result
val bessel_Kn : int -> float -> float
val bessel_Kn_e : int -> float -> Fun.result
val bessel_Kn_array : int -> float -> float array -> unit
val bessel_I0_scaled : float -> float
val bessel_I0_scaled_e : float -> Fun.result
val bessel_I1_scaled : float -> float
val bessel_I1_scaled_e : float -> Fun.result
val bessel_In_scaled : int -> float -> float
val bessel_In_scaled_e : int -> float -> Fun.result
val bessel_In_scaled_array : int -> float -> float array -> unit
val bessel_K0_scaled : float -> float
val bessel_K0_scaled_e : float -> Fun.result
val bessel_K1_scaled : float -> float
val bessel_K1_scaled_e : float -> Fun.result
val bessel_Kn_scaled : int -> float -> float
val bessel_Kn_scaled_e : int -> float -> Fun.result
val bessel_Kn_scaled_array : int -> float -> float array -> unit
val bessel_j0 : float -> float
val bessel_j0_e : float -> Fun.result
val bessel_j1 : float -> float
val bessel_j1_e : float -> Fun.result
val bessel_j2 : float -> float
val bessel_j2_e : float -> Fun.result
val bessel_jl : int -> float -> float
val bessel_jl_e : int -> float -> Fun.result
val bessel_jl_array : int -> float -> float array -> unit
val bessel_jl_steed_array : float -> float array -> unit
val bessel_y0 : float -> float
val bessel_y0_e : float -> Fun.result
val bessel_y1 : float -> float
val bessel_y1_e : float -> Fun.result
val bessel_y2 : float -> float
val bessel_y2_e : float -> Fun.result
val bessel_yl : int -> float -> float
val bessel_yl_e : int -> float -> Fun.result
val bessel_yl_array : int -> float -> float array -> unit
val bessel_i0_scaled : float -> float
val bessel_i0_scaled_e : float -> Fun.result
val bessel_i1_scaled : float -> float
val bessel_i1_scaled_e : float -> Fun.result
val bessel_il_scaled : int -> float -> float
val bessel_il_scaled_e : int -> float -> Fun.result
val bessel_il_scaled_array : int -> float -> float array -> unit
val bessel_k0_scaled : float -> float
val bessel_k0_scaled_e : float -> Fun.result
val bessel_k1_scaled : float -> float
val bessel_k1_scaled_e : float -> Fun.result
val bessel_kl_scaled : int -> float -> float
val bessel_kl_scaled_e : int -> float -> Fun.result
val bessel_kl_scaled_array : int -> float -> float array -> unit
val bessel_Jnu : float -> float -> float
val bessel_Jnu_e : float -> float -> Fun.result
val bessel_sequence_Jnu_e : float -> Fun.mode -> float array -> unit
val bessel_Ynu : float -> float -> float
val bessel_Ynu_e : float -> float -> Fun.result
val bessel_Inu : float -> float -> float
val bessel_Inu_e : float -> float -> Fun.result
val bessel_Inu_scaled : float -> float -> float
val bessel_Inu_scaled_e : float -> float -> Fun.result
val bessel_Knu : float -> float -> float
val bessel_Knu_e : float -> float -> Fun.result
val bessel_lnKnu : float -> float -> float
val bessel_lnKnu_e : float -> float -> Fun.result
val bessel_Knu_scaled : float -> float -> float
val bessel_Knu_scaled_e : float -> float -> Fun.result
val bessel_zero_J0 : int -> float
val bessel_zero_J0_e : int -> Fun.result
val bessel_zero_J1 : int -> float
val bessel_zero_J1_e : int -> Fun.result
val bessel_zero_Jnu : float -> int -> float
val bessel_zero_Jnu_e : float -> int -> Fun.result

Clausen functions

val clausen : float -> float
val clausen_e : float -> Fun.result

Coulomb functions

val hydrogenicR_1 : float -> float -> float
val hydrogenicR_1_e : float -> float -> Fun.result
val hydrogenicR : int -> int -> float -> float -> float
val hydrogenicR_e : int -> int -> float -> float -> Fun.result
val coulomb_CL_e : float -> float -> Fun.result
val coulomb_CL_array : float -> float -> float array -> unit

Coupling coefficients

The Dawson function

val dawson : float -> float
val dawson_e : float -> Fun.result

Debye functions

val debye_1 : float -> float
val debye_1_e : float -> Fun.result
val debye_2 : float -> float
val debye_2_e : float -> Fun.result
val debye_3 : float -> float
val debye_3_e : float -> Fun.result
val debye_4 : float -> float
val debye_4_e : float -> Fun.result
val debye_5 : float -> float
val debye_5_e : float -> Fun.result
val debye_6 : float -> float
val debye_6_e : float -> Fun.result

Dilogarithms

val dilog : float -> float
val dilog_e : float -> Fun.result
val complex_dilog_xy_e : float -> float -> Fun.result * Fun.result
val complex_dilog_e : float -> float -> Fun.result * Fun.result
val complex_spence_xy_e : float -> float -> Fun.result * Fun.result

Elementary operations

val multiply_e : float -> float -> Fun.result
val multiply_err_e : x:float -> dx:float -> y:float -> dy:float -> Fun.result

Elliptic integrals

val ellint_Kcomp : float -> Fun.mode -> float
val ellint_Kcomp_e : float -> Fun.mode -> Fun.result
val ellint_Ecomp : float -> Fun.mode -> float
val ellint_Ecomp_e : float -> Fun.mode -> Fun.result
val ellint_Pcomp : float -> float -> Fun.mode -> float
val ellint_Pcomp_e : float -> float -> Fun.mode -> Fun.result
val ellint_Dcomp : float -> Fun.mode -> float
val ellint_Dcomp_e : float -> Fun.mode -> Fun.result
val ellint_F : float -> float -> Fun.mode -> float
val ellint_F_e : float -> float -> Fun.mode -> Fun.result
val ellint_E : float -> float -> Fun.mode -> float
val ellint_E_e : float -> float -> Fun.mode -> Fun.result
val ellint_P : float -> float -> float -> Fun.mode -> float
val ellint_P_e : float -> float -> float -> Fun.mode -> Fun.result
val ellint_D : float -> float -> Fun.mode -> float
val ellint_D_e : float -> float -> Fun.mode -> Fun.result
val ellint_RC : float -> float -> Fun.mode -> float
val ellint_RC_e : float -> float -> Fun.mode -> Fun.result
val ellint_RD : float -> float -> float -> Fun.mode -> float
val ellint_RD_e : float -> float -> float -> Fun.mode -> Fun.result
val ellint_RF : float -> float -> float -> Fun.mode -> float
val ellint_RF_e : float -> float -> float -> Fun.mode -> Fun.result
val ellint_RJ : float -> float -> float -> float -> Fun.mode -> float
val ellint_RJ_e : float -> float -> float -> float -> Fun.mode -> Fun.result

Error function

val erf : float -> float
val erf_e : float -> Fun.result
val erfc : float -> float
val erfc_e : float -> Fun.result
val log_erfc : float -> float
val log_erfc_e : float -> Fun.result
val erf_Z : float -> float
val erf_Z_e : float -> Fun.result
val erf_Q : float -> float
val erf_Q_e : float -> Fun.result

Exponential functions

val exp : float -> float
val exp_e : float -> Fun.result

exp x computes the exponential function eˣ using GSL semantics and error checking.

val exp_e10 : float -> Fun.result_e10

exp_e10 x computes the exponential eˣ and returns a result with extended range. This function may be useful if the value of eˣ would overflow the numeric range of double.

val exp_mult : float -> float -> float
val exp_mult_e : float -> float -> Fun.result

exp_mult x y exponentiate x and multiply by the factor y to return the product y eˣ.

val exp_mult_e10 : float -> float -> Fun.result_e10

Same as exp_e10 but return a result with extended numeric range.

val expm1 : float -> float
val expm1_e : float -> Fun.result

expm1 x compute the quantity eˣ-1 using an algorithm that is accurate for small x.

val exprel : float -> float
val exprel_e : float -> Fun.result

exprel x compute the quantity (eˣ-1)/x using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion (eˣ-1)/x = 1 + x/2 + x²/(2*3) + x³/(2*3*4) + ⋯

val exprel_2 : float -> float
val exprel_2_e : float -> Fun.result

exprel_2 x compute the quantity 2(eˣ-1-x)/x² using an algorithm that is accurate for small x. For small x the algorithm is based on the expansion 2(eˣ-1-x)/x^2 = 1 + x/3 + x²/(3*4) + x³/(3*4*5) + ⋯

val exprel_n : int -> float -> float
val exprel_n_e : int -> float -> Fun.result

exprel_n x compute the n-relative exponential, which is the n-th generalization of the functions exprel and exprel_2. The N-relative exponential is given by,

                         n-1
exprel_n x = n!/xⁿ (aˣ -  ∑ xᵏ/k!)
                         k=0
           = 1 + x/(N+1) + x²/((N+1)(N+2)) + ⋯
val exp_err_e : x:float -> dx:float -> Fun.result
val exp_err_e10 : x:float -> dx:float -> Fun.result_e10
val exp_mult_err_e : x:float -> dx:float -> y:float -> dy:float -> Fun.result
val exp_mult_err_e10_e : x:float -> dx:float -> y:float -> dy:float -> Fun.result_e10

Exponential integrals

val expint_E1 : float -> float
val expint_E1_e : float -> Fun.result
val expint_E2 : float -> float
val expint_E2_e : float -> Fun.result
val expint_E1_scaled : float -> float
val expint_E1_scaled_e : float -> Fun.result
val expint_E2_scaled : float -> float
val expint_E2_scaled_e : float -> Fun.result
val expint_Ei : float -> float
val expint_Ei_e : float -> Fun.result
val expint_Ei_scaled : float -> float
val expint_Ei_scaled_e : float -> Fun.result
val shi : float -> float
val chi : float -> float
val expint_3 : float -> float
val expint_3_e : float -> Fun.result
val si : float -> float
val ci : float -> float
val atanint : float -> float
val atanint_e : float -> Fun.result

Fermi-Dirac function

val fermi_dirac_m1 : float -> float
val fermi_dirac_m1_e : float -> Fun.result
val fermi_dirac_0 : float -> float
val fermi_dirac_0_e : float -> Fun.result
val fermi_dirac_1 : float -> float
val fermi_dirac_1_e : float -> Fun.result
val fermi_dirac_2 : float -> float
val fermi_dirac_2_e : float -> Fun.result
val fermi_dirac_int : int -> float -> float
val fermi_dirac_int_e : int -> float -> Fun.result
val fermi_dirac_mhalf : float -> float
val fermi_dirac_mhalf_e : float -> Fun.result
val fermi_dirac_half : float -> float
val fermi_dirac_half_e : float -> Fun.result
val fermi_dirac_3half : float -> float
val fermi_dirac_3half_e : float -> Fun.result
val fermi_dirac_inc_0 : float -> float -> float
val fermi_dirac_inc_0_e : float -> float -> Fun.result

Gamma function

val gamma : float -> float
val gamma_e : float -> Fun.result
val lngamma : float -> float
val lngamma_e : float -> Fun.result
val lngamma_sgn_e : float -> Fun.result * float
val gammastar : float -> float
val gammastar_e : float -> Fun.result
val gammainv : float -> float
val gammainv_e : float -> Fun.result
val lngamma_complex_e : float -> float -> Fun.result * Fun.result
val taylorcoeff : int -> float -> float
val taylorcoeff_e : int -> float -> Fun.result
val fact : int -> float
val fact_e : int -> Fun.result
val doublefact : int -> float
val doublefact_e : int -> Fun.result
val lnfact : int -> float
val lnfact_e : int -> Fun.result
val lndoublefact : int -> float
val lndoublefact_e : int -> Fun.result
val choose : int -> int -> float
val choose_e : int -> int -> Fun.result
val lnchoose : int -> int -> float
val lnchoose_e : int -> int -> Fun.result
val poch : float -> float -> float
val poch_e : float -> float -> Fun.result
val lnpoch : float -> float -> float
val lnpoch_e : float -> float -> Fun.result
val lnpoch_sgn_e : float -> float -> Fun.result * float
val pochrel : float -> float -> float
val pochrel_e : float -> float -> Fun.result
val gamma_inc_Q : float -> float -> float
val gamma_inc_Q_e : float -> float -> Fun.result
val gamma_inc_P : float -> float -> float
val gamma_inc_P_e : float -> float -> Fun.result
val gamma_inc : float -> float -> float
val gamma_inc_e : float -> float -> Fun.result
val beta : float -> float -> float
val beta_e : float -> float -> Fun.result
val lnbeta : float -> float -> float
val lnbeta_e : float -> float -> Fun.result
val lnbeta_sgn_e : float -> float -> Fun.result * float
val beta_inc : float -> float -> float -> float
val beta_inc_e : float -> float -> float -> Fun.result

Gegenbauer functions aka Ultraspherical polynomials

Gegenbauer functions are defined in DLMF.

val gegenpoly_1 : float -> float -> float
val gegenpoly_1_e : float -> float -> Fun.result

gegenpoly_1 l x = C₁⁽ˡ⁾(x).

val gegenpoly_2 : float -> float -> float
val gegenpoly_2_e : float -> float -> Fun.result

gegenpoly_2 l x = C₂⁽ˡ⁾(x).

val gegenpoly_3 : float -> float -> float
val gegenpoly_3_e : float -> float -> Fun.result

gegenpoly_3 l x = C₃⁽ˡ⁾(x).

val gegenpoly_n : int -> float -> float -> float
val gegenpoly_n_e : int -> float -> float -> Fun.result

gegenpoly_n n l x = Cₙ⁽ˡ⁾(x). Constraints: l > -1/2, n ≥ 0.

val gegenpoly_array : float -> float -> float array -> unit

gegenpoly_array l x c computes an array of Gegenbauer polynomials c.(n) = Cₙ⁽ˡ⁾(x) for n = 0, 1, 2,̣..., Array.length c - 1. Constraints: l > -1/2.

Hypergeometric functions

val hyperg_0F1 : c:float -> float -> float
val hyperg_0F1_e : c:float -> float -> Fun.result

hyperg_0F1 c x computes the hypergeometric function ₀F₁(c; x).

val hyperg_1F1_int : m:int -> n:int -> float -> float
val hyperg_1F1_int_e : m:int -> n:int -> float -> Fun.result

hyperg_1F1_int m n x computes the confluent hypergeometric function ₁F₁(m;n;x) = M(m,n,x) for integer parameters m, n.

val hyperg_1F1 : a:float -> b:float -> float -> float
val hyperg_1F1_e : a:float -> b:float -> float -> Fun.result

hyperg_1F1 a b x computes the confluent hypergeometric function ₁F₁(a;b;x) = M(a,b,x) for general parameters a, b.

val hyperg_U_int : m:int -> n:int -> float -> float
val hyperg_U_int_e : m:int -> n:int -> float -> Fun.result

hyperg_U_int m n x computes the confluent hypergeometric function U(m,n,x) for integer parameters m, n.

val hyperg_U_int_e10 : m:int -> n:int -> float -> Fun.result_e10

hyperg_U_int_e10 m n x computes the confluent hypergeometric function U(m,n,x) for integer parameters m, n with extended range.

val hyperg_U : a:float -> b:float -> float -> float
val hyperg_U_e : a:float -> b:float -> float -> Fun.result

hyperg_U a b x computes the confluent hypergeometric function U(a,b,x).

val hyperg_U_e10 : a:float -> b:float -> float -> Fun.result_e10

hyperg_U_e10 a b x computes the confluent hypergeometric function U(a,b,x) with extended range.

val hyperg_2F1 : a:float -> b:float -> c:float -> float -> float
val hyperg_2F1_e : a:float -> b:float -> c:float -> float -> Fun.result

hyperg_2F1 a b c x computes the Gauss hypergeometric function ₂F₁(a,b,c,x) = F(a,b,c,x) for |x| < 1.

If the arguments (a,b,c,x) are too close to a singularity then the function can raise the exception Error.Gsl_exn(Error.EMAXITER, _) when the series approximation converges too slowly. This occurs in the region of x=1, c - a - b ∈ ℤ.

val hyperg_2F1_conj : aR:float -> aI:float -> c:float -> float -> float
val hyperg_2F1_conj_e : aR:float -> aI:float -> c:float -> float -> Fun.result

hyperg_2F1_conj aR aI c x computes the Gauss hypergeometric function ₂F₁(aR + i aI, aR - i aI, c, x) with complex parameters for |x| < 1.

val hyperg_2F1_renorm : a:float -> b:float -> c:float -> float -> float
val hyperg_2F1_renorm_e : a:float -> b:float -> c:float -> float -> Fun.result

hyperg_2F1_renorm a b c x computes the renormalized Gauss hypergeometric function ₂F₁(a,b,c,x) / Γ(c) for |x| < 1.

val hyperg_2F1_conj_renorm : aR:float -> aI:float -> c:float -> float -> float
val hyperg_2F1_conj_renorm_e : aR:float -> aI:float -> c:float -> float -> Fun.result

hyperg_2F1_conj_renorm aR aI c x computes the renormalized Gauss hypergeometric function ₂F₁(aR + i aI, aR - i aI, c, x) / Γ(c) for |x| < 1.

val hyperg_2F0 : a:float -> b:float -> float -> float
val hyperg_2F0_e : a:float -> b:float -> float -> Fun.result

hyperg_2F0 a b x computes 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)

Laguerre functions

val laguerre_1 : a:float -> float -> float
val laguerre_1_e : a:float -> float -> Fun.result
val laguerre_2 : a:float -> float -> float
val laguerre_2_e : a:float -> float -> Fun.result
val laguerre_3 : a:float -> float -> float
val laguerre_3_e : a:float -> float -> Fun.result
val laguerre_n : n:int -> a:float -> float -> float
val laguerre_n_e : n:int -> a:float -> float -> Fun.result

Lambert W functions

val lambert_W0 : float -> float
val lambert_W0_e : float -> Fun.result
val lambert_Wm1 : float -> float
val lambert_Wm1_e : float -> Fun.result

Legendre functions

val legendre_P1 : float -> float
val legendre_P1_e : float -> Fun.result
val legendre_P2 : float -> float
val legendre_P2_e : float -> Fun.result
val legendre_P3 : float -> float
val legendre_P3_e : float -> Fun.result
val legendre_Pl : int -> float -> float
val legendre_Pl_e : int -> float -> Fun.result
val legendre_Pl_array : float -> float array -> unit
val legendre_Q0 : float -> float
val legendre_Q0_e : float -> Fun.result
val legendre_Q1 : float -> float
val legendre_Q1_e : float -> Fun.result
val legendre_Ql : int -> float -> float
val legendre_Ql_e : int -> float -> Fun.result

Associated Legendre functions and Spherical Harmonics

type legendre_t =
  1. | Schmidt
    (*

    Specifies the computation of the Schmidt semi-normalized associated Legendre polynomials Sₗᵐ(x).

    *)
  2. | Spharm
    (*

    Specifies the computation of the spherical harmonic associated Legendre polynomials Yₗᵐ(x).

    *)
  3. | Full
    (*

    Specifies the computation of the fully normalized associated Legendre polynomials Nₗᵐ(x).

    *)
  4. | None
    (*

    Specifies the computation of the unnormalized associated Legendre polynomials Pₗᵐ(x).

    *)

Normalization of Legendre functions. See the GSL documentation.

val legendre_array : legendre_t -> int -> float -> float array -> unit

legendre_array norm lmax x result calculate all normalized associated Legendre polynomials for 0 ≤ l ≤ lmax and 0 ≤ m ≤ l for |x| ≤ 1. The norm parameter specifies which normalization is used. The normalized Pₗᵐ(x) values are stored in result, whose minimum size can be obtained from calling legendre_array_n. The array index of Pₗᵐ(x) is obtained from calling legendre_array_index(l, m). To include or exclude the Condon-Shortley phase factor of (-1)ᵐ, set the parameter csphase to either -1 or 1 respectively in the _e function. This factor is included by default.

val legendre_array_n : int -> int

legendre_array_n lmax returns the minimum array size for maximum degree lmax needed for the array versions of the associated Legendre functions. Size is calculated as the total number of Pₗᵐ(x) functions, plus extra space for precomputing multiplicative factors used in the recurrence relations.

val legendre_array_index : int -> int -> int

legendre_array_index l m returns the index into the result array of the legendre_array, legendre_deriv_array, legendre_deriv_alt_array, legendre_deriv2_array, and !legendre_deriv2_alt_array corresponding to Pₗᵐ(x), ∂ₓPₗᵐ(x), or ∂ₓ²Pₗₗᵐ(x). The index is given by l(l+1)/2 + m.

val legendre_Plm : int -> int -> float -> float
val legendre_Plm_e : int -> int -> float -> Fun.result

legendre_Plm l m x and legendre_Plm_e l m x compute the associated Legendre polynomial Pₗᵐ(x) for m ≥ 0, l ≥ m, |x| ≤ 1.

val legendre_sphPlm : int -> int -> float -> float
val legendre_sphPlm_e : int -> int -> float -> Fun.result

legendre_sphPlm l m x and legendre_Plm_e compute the normalized associated Legendre polynomial √((2l+1)/(4\pi)) √((l-m)!/(l+m)!) Pₗᵐ(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 Pₗᵐ(x).

Logarithm and related functions

val log : float -> float
val log_e : float -> Fun.result
val log_abs : float -> float
val log_abs_e : float -> Fun.result
val log_complex_e : float -> float -> Fun.result * Fun.result
val log_1plusx : float -> float
val log_1plusx_e : float -> Fun.result
val log_1plusx_mx : float -> float
val log_1plusx_mx_e : float -> Fun.result

Power function

val pow_int : float -> int -> float
val pow_int_e : float -> int -> Fun.result

Psi (Digamma) function

val psi_int : int -> float
val psi_int_e : int -> Fun.result
val psi : float -> float
val psi_e : float -> Fun.result
val psi_1piy : float -> float
val psi_1piy_e : float -> Fun.result
val psi_complex_e : float -> float -> Fun.result * Fun.result
val psi_1_int : int -> float
val psi_1_int_e : int -> Fun.result
val psi_1 : float -> float
val psi_1_e : float -> Fun.result
val psi_n : int -> float -> float
val psi_n_e : int -> float -> Fun.result

Synchrotron functions

val synchrotron_1 : float -> float
val synchrotron_1_e : float -> Fun.result
val synchrotron_2 : float -> float
val synchrotron_2_e : float -> Fun.result

Transport functions

val transport_2 : float -> float
val transport_2_e : float -> Fun.result
val transport_3 : float -> float
val transport_3_e : float -> Fun.result
val transport_4 : float -> float
val transport_4_e : float -> Fun.result
val transport_5 : float -> float
val transport_5_e : float -> Fun.result

Trigonometric functions

val sin : float -> float
val sin_e : float -> Fun.result
val cos : float -> float
val cos_e : float -> Fun.result
val hypot : float -> float -> float
val hypot_e : float -> float -> Fun.result
val sinc : float -> float
val sinc_e : float -> Fun.result
val complex_sin_e : float -> float -> Fun.result * Fun.result
val complex_cos_e : float -> float -> Fun.result * Fun.result
val complex_logsin_e : float -> float -> Fun.result * Fun.result
val lnsinh : float -> float
val lnsinh_e : float -> Fun.result
val lncosh : float -> float
val lncosh_e : float -> Fun.result
val rect_of_polar : r:float -> theta:float -> Fun.result * Fun.result
val polar_of_rect : x:float -> y:float -> Fun.result * Fun.result
val angle_restrict_symm : float -> float
val angle_restrict_pos : float -> float
val sin_err_e : float -> dx:float -> Fun.result
val cos_err_e : float -> dx:float -> Fun.result

Zeta functions

val zeta_int : int -> float
val zeta_int_e : int -> Fun.result
val zeta : float -> float
val zeta_e : float -> Fun.result
val hzeta : float -> float -> float
val hzeta_e : float -> float -> Fun.result
val eta_int : int -> float
val eta_int_e : int -> Fun.result
val eta : float -> float
val eta_e : float -> Fun.result