Real functions related to \(\,_1F_1\)

• Incomplete gamma functions
  • Truncated exponential function, \(e_n(x)\)
    • math53.expn()
  • Relative exponential, \(\mathrm{exprel}_n(x)\)
    • math53.expreln()
• Coulomb wave functions
  • 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()
  • Irregular Coulomb wave functions \(G_l(\eta,z), x), G'_l(\eta,z)\)
    • math53.coulomb_g()
• Error function, and related functions
  • Exponentially scaled complementary error function, \(\mathrm{erfcx}(x)\)
    • math53.erfcx()
  • Difference of error functions, \(\mathrm{erfh}(x,h) = \mathrm{erf}(x+h)-\mathrm{erf}(x-h)\)
    • math53.erfh()
  • Difference of error functions, \(\mathrm{erf2}(x_1,x_2) = \mathrm{erf}(x_2)-\mathrm{erf}(x_1)\)
    • math53.erf2()
  • Probability function \(Q(x) = \Phi(-x)\)
    • math53.erfq()
  • Inverse of the exponentially scaled complementary error function, \(\mathrm{erfcx}^{-1}(x)\)
    • math53.erfcx_inv()
  • Inverse of the imaginary error function, \(\mathrm{erfi}^{-1}(x)\)
    • math53.erfi_inv()
  • Generalized Dawson integral, \(F(p, x)\)
    • math53.dawson2()
  • Generalized error function, \(\mathrm{erfg}(p, x)\)
    • math53.erfg()
  • Expint3, \(\mathrm{erfg}(3, x)\)
    • math53.expint3()
  • 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()
  • 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()
• 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()
• Struve 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()