Addditional special functions (real arguments, double precision)

Addditional special functions (real arguments, double precision)#

General concepts

See also: https://en.wikipedia.org/wiki/Power_of_a_test

See also: https://en.wikipedia.org/wiki/Sample_size_determination

Conversions of parameters of elliptic functions
- Elliptic nome \(q(k)\) (DAMath)
  - math53.elliptic_nome()
- Elliptic modulus \(k(q)\) (DAMath)
  - math53.elliptic_modulus()
- Jacobi amplitude, \(\mathrm{am}(x, k)\) (DAMath)
  - math53.jacobi_amplitude()
Additional elliptic integrals
- Legendre complete elliptic integral \(B(k)\)
  - math53.elliptic_b()
- Legendre complete elliptic integral \(D(k)\)
  - math53.elliptic_d()
- Legendre incomplete elliptic integral \(B(\phi, k)\)
  - math53.elliptic_b_inc()
- Legendre incomplete elliptic integral \(D(\phi, k)\)
  - math53.elliptic_d_inc()
- Heuman’s Lambda function, \(\Lambda_0(\phi,k)\)
  - math53.heuman_lambda()
- Jacobi Zeta function, \(Z(\phi,k)\)
  - math53.jacobi_zeta()
Bulirsch elliptic integrals
- Complete elliptic integral of the 1st kind \(\mathrm{cel1}(k_c)\)
  - math53.cel1()
- Complete elliptic integral of the 2nd kind \(\mathrm{cel2}(k_c, a, b)\)
  - math53.cel2()
- General complete elliptic integral \(\mathrm{cel}(k_c, p, a, b)\)
  - math53.cel()
- Incomplete elliptic integral of the 1st kind \(\mathrm{el1}(x, k_c)\)
  - math53.el1()
- Incomplete elliptic integral of the 2nd kind \(\mathrm{el2}(x, k_c, a, b)\)
  - math53.el2()
- Incomplete elliptic integral of the 3rd kind \(\mathrm{el3}(x, k_c, p)\)
  - math53.el3()
Maple style elliptic integrals
- Complete integral of the 1st kind, \(\mathrm{EllipticK}(k)\)
  - math53.ellipticK()
- Complete integral of the 1st kind for imaginary modulus, \(\mathrm{EllipticKim}(k)\)
  - math53.ellipticKim()
- Complementary complete integral of the 1st kind, \(\mathrm{EllipticCK}(k) = K'(k)\)
  - math53.ellipticCK()
- Complete integral of the 2nd kind, \(\mathrm{EllipticEC}(k)\)
  - math53.ellipticEC()
- Complete integral of the 2nd kind for imaginary modulus, \(\mathrm{EllipticECim}(k)\)
  - math53.ellipticECim()
- Complementary complete integral of the 2nd kind, \(\mathrm{EllipticCE}(k)\)
  - math53.ellipticCE()
- Complete integral of the 3rd kind, \(\mathrm{EllipticPiC}(\nu, k)\)
  - math53.ellipticPiC()
- Complete integral of the 3rd kind for imaginary modulus, \(\mathrm{EllipticPiCim}(\nu, k)\)
  - math53.ellipticPiCim()
- Complementary complete integral of the 3rd kind, \(\mathrm{EllipticCPi}(\nu, k)\)
  - math53.ellipticCPi()
- Incomplete integral of the 1st kind, \(\mathrm{EllipticF}(z, k)\)
  - math53.ellipticF()
- Incomplete integral of the 2nd kind, \(\mathrm{EllipticE}(z, k)\)
  - math53.ellipticE()
- Incomplete integral of the 3rd kind, \(\mathrm{EllipticPi}(z, \nu, k)\)
  - math53.ellipticPi()
Jacobi theta functions at \(x=0\) for \(0 \le q <1\)
- Jacobi theta function \(\theta'_1(q)\)
  - math53.jacobi_theta1p()
- Jacobi theta function \(\theta_2(q)\)
  - math53.jacobi_theta02()
- Jacobi theta function \(\theta_3(q)\)
  - math53.jacobi_theta03()
- Jacobi theta function \(\theta_4(q)\)
  - math53.jacobi_theta04()
Inverse Jacobi elliptic functions
- Inverse Jacobi elliptic function \(\mathrm{arcsn}(x, k)\)
  - math53.jacobi_arcsn()
- Inverse Jacobi elliptic function \(\mathrm{arccn}(x, k)\)
  - math53.jacobi_arccn()
- Inverse Jacobi elliptic function \(\mathrm{arcdn}(x, k)\)
  - math53.jacobi_arcdn()
- Inverse Jacobi elliptic function \(\mathrm{arccd}(x, k)\)
  - math53.jacobi_arccd()
- Inverse Jacobi elliptic function \(\mathrm{arcsd}(x, k)\)
  - math53.jacobi_arcsd()
- Inverse Jacobi elliptic function \(\mathrm{arcnd}(x, k)\)
  - math53.jacobi_arcnd()
- Inverse Jacobi elliptic function \(\mathrm{arcdc}(x, k)\)
  - math53.jacobi_arcdc()
- Inverse Jacobi elliptic function \(\mathrm{arcnc}(x, k)\)
  - math53.jacobi_arcnc()
- Inverse Jacobi elliptic function \(\mathrm{arcsc}(x, k)\)
  - math53.jacobi_arcsc()
- Inverse Jacobi elliptic function \(\mathrm{arcns}(x, k)\)
  - math53.jacobi_arcns()
- Inverse Jacobi elliptic function \(\mathrm{arcds}(x, k)\)
  - math53.jacobi_arcds()
- Inverse Jacobi elliptic function \(\mathrm{arccs}(x, k)\)
  - math53.jacobiArcCS()
Lemniscate functions
- Lemniscate sine function, \(\mathrm{sinlemn}(x)\)
  - math53.sin_lemniscate()
- Lemniscate cosine function, \(\mathrm{coslemn}(x)\)
  - math53.cos_lemniscate()
- Inverse lemniscate cosine function, \(\mathrm{arcsl}(x)\)
  - math53.acos_lemniscate()
- Inverse lemniscate sine function, \(\mathrm{arccl}(x)\)
  - math53.asin_lemniscate()
Neville theta functions
- Neville \(\theta_s(x,k)\)
  - math53.neville_theta_s()
- Neville \(\theta_c(x,k)\)
  - math53.neville_theta_c()
- Neville \(\theta_d(x,k)\)
  - math53.neville_theta_d()
- Neville \(\theta_n(x,k)\)
  - math53.neville_theta_n()
Polygamma, and related functions
- Pentagamma function \(\psi'''(x)\)
  - math53.pentagamma()
- Tetragamma function, \(\psi''(x)\)
  - math53.tetragamma()
- Auxiliary function \(\psi^{*}(x)\)
  - math53.psistar()
- Inverse digamma function \(\psi^{-1}(y)\)
  - math53.psi_inv()
- Bateman function \(G(x)\)
  - math53.bateman_g()
Polylogarithm, and related functions
- Polylogarithm of integer order, \(\mathrm{Li}_n(x)\)
  - math53.polylog_i()
- Fermi-Dirac integrals of integer order, \(F_n(x)\)
  - math53.fermi_dirac_i()
- Fermi-Dirac integral \(F_{-1/2}(x)\)
  - math53.fermi_dirac_m05()
- Fermi-Dirac integral \(F_{1/2}(x)\)
  - math53.fermi_dirac_p05()
- Fermi-Dirac integral \(F_{3/2}(x)\)
  - math53.fermi_dirac_p15()
- Fermi-Dirac integral \(F_{5/2}(x)\)
  - math53.fermi_dirac_p25()
- Inverse tangent integral, \(\mathrm{Ti}_2(x)\)
  - math53.tangent_int_2()
- Lobachevsky’s log-cos integral, \(L(x)\)
  - math53.lobachevsky_c()
- Lobachevsky’s log-sin integral, \(\Lambda(x)\)
  - math53.lobachevsky_s()
- Debye functions, \(\mathrm{D}_n(x)\)
  - math53.debye()
- Transport integral, \(\mathrm{J}_n(x)\)
  - math53.transport_jn()
Riemann zeta, and related functions
- Riemann \(\zeta(n)\) for integer arguments
  - math53.zeta_i()
- Riemann \(\zeta(1+x)\)
  - math53.zeta1p()
- Dirichlet eta function for integer argument, \(\eta(n)\)
  - math53.dirichlet_eta_i()
- Inverse of the Riemann prime counting function, \(R^{-1}(x)\)
  - math53.riemann_r_inv()
- Rogers-Ramanujan continued fraction
  - math53.rogers_ramanujan_cf()
- q-Pochhammer Euler function, \((q)_{\infty}\)
  - math53.euler_q()
Bessel functions of integer order
- Bessel function of the 1st kind, order \(0, J_{0}(x)\)
  - ctx.bessel_j0()
- Bessel function of the 1st kind, order \(1, J_{1}(x)\)
  - ctx.bessel_j1()
- Bessel function of the 1st kind, integer order \(n, J_{n}(x)\)
  - ctx.bessel_jn()
- Bessel function of the 2nd kind, order \(0, Y_{0}(x)\)
  - ctx.bessel_y0()
- Bessel function of the 2nd kind, order \(1, Y_{1}(x)\)
  - ctx.bessel_y1()
- Bessel function of the 2nd kind, integer order \(n, Y_{n}(x)\)
  - ctx.bessel_yn()
- Bessel Lambda function, \(\Lambda(\nu,x)\)
  - math53.bessel_lambda()
Modified Bessel functions of integer order
- Modified Bessel function of the 1st kind, order \(0, I_{0}(x)\)
  - math53.bessel_i0()
- Modified Bessel function of the 1st kind, order \(1, I_{1}(x)\)
  - math53.bessel_i1()
- Modified Bessel function of the 1st kind, integer order \(n, I_{n}(x)\)
  - math53.damath_bessel_in()
- Modified Bessel function of the 2nd kind, order 0, \(K_{0}(x)\)
  - math53.bessel_k0()
- Modified Bessel function of the 2nd kind, order \(1, K_{1}(x)\)
  - math53.bessel_k1()
- Modified Bessel function of the 2nd kind, integer order \(n, K_{n}(x)\)
  - math53.bessel_kn()
- Exponentially scaled modified Bessel function of the 1st kind, order \(0, I_{0,e}(x)\)
  - math53.bessel_i0e()
- Exponentially scaled modified Bessel function of the 1st kind, order \(1, I_{1,e}(x)\)
  - math53.bessel_i1e()
- Exponentially scaled modified Bessel function of the 2nd kind, order 0, \(K_{0,e}(x)\)
  - math53.bessel_k0e()
- Exponentially scaled modified Bessel function of the 2nd kind, order \(1, K_{1,e}(x)\)
  - math53.bessel_k1e()
Integrals of zero-order Bessel functions
- Integral of \(I_0\)
  - math53.bessel_i0_int()
- Integral of \(J_0\)
  - math53.bessel_j0_int()
- Integral of \(K_0\)
  - math53.bessel_k0_int()
- Integral of \(Y_0\)
  - math53.bessel_y0_int()
Kelvin functions of order 0
- Kelvin function \(\mathrm{ber0}(x)\)
  - ctx.kelvin_ber0()
- Kelvin function \(\mathrm{bei0}(x)\)
  - ctx.kelvin_bei0()
- Kelvin function \(\mathrm{ker0}(x)\)
  - ctx.kelvin_ker0()
- Kelvin function \(\mathrm{kei0}(x)\)
  - ctx.kelvin_kei0()
- First derivative of the Kelvin function \(\mathrm{ber0}, \mathrm{ber0}'(x)\)
  - ctx.kelvin_ber0_prime()
- First derivative of the Kelvin function \(\mathrm{bei0}, \mathrm{bei0}'(x)\)
  - ctx.kelvin_bei0_prime()
- First derivative of the Kelvin function \(\mathrm{ker0}, \mathrm{ker0}'(\nu, x)\)
  - ctx.kelvin_ker0_prime()
- First derivative of the Kelvin function \(\mathrm{kei0}, \mathrm{kei0}'(x)\)
  - ctx.kelvin_kei0_prime()
- Derivatives of Kelvin functions
  - math53.kelvin0Prime()
Synchrotron functions
- First synchrotron function \(F(x)\)
  - math53.synchrotron_f()
- Second synchrotron function \(G(x)\)
  - math53.synchrotron_g()
Error function, and related functions
- Generalized Dawson integral, \(F(p, x)\)
  - math53.dawson2()
- Generalized error function, \(\mathrm{erfg}(p, x)\)
  - math53.erfg()
- Expint3, \(\mathrm{erfg}(3, x)\)
  - math53.expint3()
- Exponentially scaled complementary error function, \(\mathrm{erfcx}(x)\)
  - math53.erfcx()
- Scaled repeated integrals of the complementary error function, \(i^n \mathrm{erfc}(x)\)
  - math53.inerfc()
- Fresnel auxiliary function \(\mathrm{f}(x)\)
  - math53.fresnel_f()
- Fresnel auxiliary function \(\mathrm{g}(x)\)
  - math53.fresnel_g()
- Goodwin-Staton integral, \(G(x)\)
  - math53.goodwin_staton()
Exponential integrals, and related functions
- Entire cosine integral \(\mathrm{Cin}(x)\)
  - math53.cin()
- Entire hyperbolic cosine integral \(\mathrm{Cinh}(x)\)
  - math53.cinh()
- Entire exponential integral \(\displaystyle \mathrm{Ein}(x)\)
  - math53.ein()
- Scaled exponential integral \(\mathrm{e1s}(x)\)
  - math53.exp_integral_e1_scaled()
- Scaled exponential integral \(\mathrm{eis}(x)\)
  - math53.exp_integral_ei_scaled()
- Scaled exponential integral \(\mathrm{eisx2}(x)\)
  - math53.eisx2()
- Inverse of the exponential integral, \(\mathrm{Ei}^{-1}(x)\)
  - math53.ei_inv()
- Generalized exponential integral \(E_p(p,x)\)
  - math53.gei()
- Exponential integral \(\displaystyle \beta_n(x)\)
  - math53.eibeta()
- Inverse of the logarithmic integral, \(\mathrm{li}^{-1}(x)\)
  - math53.log_integral_inv()
- Shifted sine integral \(\mathrm{si}(x) = \mathrm{Si}(x) - \pi/2\)
  - math53.shifted_sin_integral()
Coulomb, Whittaker and parabolic cylinder function
- Normalizing Gamow constant for Coulomb wave functions
  - math53.coulomb_cl()
- Coulomb phase shift \(\sigma_L(\eta)\)
  - math53.coulomb_sl()
- Regular Coulomb wave functions \(F_l(\eta, x), F'_l(\eta, x)\)
  - math53.coulomb_f_fprime()
- Regular Coulomb wave function \(F_l(\eta,z)\)
  - math53.coulomb_f()
- Irregular Coulomb wave functions \(G_l(\eta,z), x), G'_l(\eta,z)\)
  - math53.coulomb_g()
Hypergeometric pFq, and related functions
- Struve function \(\mathbf{H}_0(x)\)
  - math53.struve_h0()
- Struve function \(\mathbf{H}_1(x)\)
  - math53.struve_h1()
- Struve function \(\mathbf{L}_0(x)\)
  - math53.struve_l0()
- Struve function \(\mathbf{L}_1(x)\)
  - math53.struve_l1()
Miscellaneous functions
- Log-series distribution, pmf
  - Math53.logseries_pmf()
- Log-series distribution, cdf
  - Math53.logseries_cdf()
- Zeta distribution, pmf
  - Math53.zeta_pmf()
- Zeta distribution, cdf
  - Math53.zeta_cdf()
- Voigt function U
  - math53.voigt_u()
- Voigt function V
  - math53.voigt_v()
- Voigt function H
  - math53.voigt_h()
- Voigt Profile distribution, pdf
  - math53.voigt_profile_pdf()