FewSpecialFunctions.Debye_function — FunctionDebye_function(n,x,min_tol=1e-15)The Debye function(n,x) given by
\[ D_n(x) = \frac{n}{x^n} \int_0^x \frac{t^n}{e^{t}-1} dt\]
Returns the value $D(n,x)$
FewSpecialFunctions.regular_Coulomb — Functionregular_coulomb(ℓ,η,ρ)Regular Coulomb wave function ℓ is the order(non-negative integer), η is the charge (real parameter) and ρ is the radial coordinate (non-negative real variable).
returns the value F_ℓ(η,ρ) given by
\[ F_\ell(\eta,\rho) = \frac{\rho^{\ell+1}2^\ell e^{i\rho-(\pi\eta/2)}}{|\Gamma(\ell+1+i\eta)|} \int_0^1 e^{-2i\rho t}t^{\ell+i\eta}(1-t)^{\ell-i\eta} \, dt\]
FewSpecialFunctions.irregular_Coulomb — Functionirregular_Coulomb(ℓ,η,ρ)Regular Coulomb wave function ℓ is the order(non-negative integer), η is the charge (real parameter) and ρ is the radial coordinate (non-negative real variable).
returns the value G_ℓ(η,ρ)
FewSpecialFunctions.C — FunctionC(ℓ,η)Returns Coulomb normalization constant given by
\[ C_\ell(\eta) = \frac{2^\ell \exp(-\pi \eta/2) |\Gamma(\ell+1+i \eta)|}{(2\ell+1)!}\]
FewSpecialFunctions.θ — Functionθ(ℓ,η,ρ)Returns the phase of the Coulomb functions given by
\[ \theta_\ell(\eta,\rho) = \rho - \eta \ln(2\rho) - \frac{1}{2}\ell \pi + \sigma_\ell(\eta)\]
FewSpecialFunctions.Coulomb_H_minus — FunctionCoulomb_H_minus(ℓ,η,ρ)Complex Coulomb wave function. Infinity handled using the substitution f(t) -> f(u/(1-u)*1/(1-u)^2). Returns Coulomb wave function
\[ H^{-}_\ell = G_\ell - iF_\ell\]
FewSpecialFunctions.Coulomb_H_plus — FunctionReturns Coulomb wave function
\[ H^{+}_\ell = G_\ell + iF_\ell\]
FewSpecialFunctions.Coulomb_cross — FunctionCoulomb_cross(ℓ,η)Wronskian relation / cross product.
\[ F_{\ell-1}G_{\ell}-F_{\ell}G_{\ell-1} = \ell/(\ell^2+\eta^2)^{1/2}\]
FewSpecialFunctions.regular_Coulomb_approx — Functionregular_Coulomb_approx(ℓ,η,ρ)For ρ -> 0 and η fixed approximate the regular Coulomb wave function as
\[ F_\ell(\eta,\rho) \simeq C_\ell(\eta)^{\ell+1}\]
FewSpecialFunctions.irregular_Coulomb_approx — Functionirregular_Coulomb_approx(ℓ,η,ρ)For ρ -> 0 and η fixed approximate the irregular Coulomb wave function as
\[ G_\ell(\eta,\rho) \simeq \frac{\rho^{-\ell}}{(2\ell+1)C_\ell(\eta)}\]
FewSpecialFunctions.regular_Coulomb_limit — Functionregular_Coulomb_limit(ℓ,η,ρ)In the limit ρ -> ∞ with η fixed, returns the regular Coulomb wave as
\[ F_{\ell}(\eta,\rho) \simeq \sin(\theta_\ell(\eta,\rho))\]
FewSpecialFunctions.irregular_Coulomb_limit — Functionirregular_Coulomb_limit(ℓ,η,ρ)In the limit ρ -> ∞ with η fixed, returns the irregular Coulomb wave as
\[ G_{\ell}(\eta,\rho) \simeq \cos(\theta_\ell(\eta,\rho))\]
FewSpecialFunctions.Struve — FunctionStruve(ν,z,min_tol=1e-15)Returns the Struve function given by
\[ \mathbf{H}_\nu(z) = \frac{2(z/2)^\nu}{\sqrt{\pi}\Gamma(\nu+1/2)} \int_0^1 (1-t)^{{\nu-1/2}}\sin(zt) \, \text{d}t\]
FewSpecialFunctions.Fresnel_S_integral_pi — FunctionFresnel_S_integral_pi(x)The Fresnel function S(z) using the definition in Handbook of Mathematical Functions: Abramowitz and Stegun, where
\[ S(z) = \int_0^x \cos(\pi t^2/2) dt\]
Returns the value $S(x)$
FewSpecialFunctions.Fresnel_C_integral_pi — FunctionFresnel_C_integral_pi(x)The Fresnel function C(z) using the definition in Handbook of Mathematical Functions: Abramowitz and Stegun, where
\[ C(z) = \int_0^x \sin(\pi t^2/2) dt\]
Returns the value $C(x)$
FewSpecialFunctions.Fresnel_S_integral — FunctionFresnel_S_integral(x)The Fresnel function S(z) using the definition wiki
\[ S(z) = \int_0^x \sin(t^2) dt\]
Returns the value $S(x)$
FewSpecialFunctions.Fresnel_C_integral — FunctionFresnel_C_integral(x)The Fresnel function C(z) using the definition wiki
\[ C(z) = \int_0^x \cos(t^2) dt\]
Returns the value $C(x)$
FewSpecialFunctions.Fresnel_S_erf — FunctionFresnel_S_erf(x)The Fresnel function S(z) using the definition wiki and the error function.
\[ S(z) = \sqrt{\frac{\pi}{2}}\frac{1+i}{4} \bigg( \text{erf}\big(\frac{1+i}{\sqrt{2}}z \big) - i \text{erf}\big(\frac{1-i}{\sqrt{2}}z \big)\bigg)\]
Returns the value $S(x)$
FewSpecialFunctions.Fresnel_C_erf — FunctionFresnel_C_erf(x)The Fresnel function C(z) using the definition wiki and the error function.
\[ C(z) = \sqrt{\frac{\pi}{2}}\frac{1-i}{4} \bigg( \text{erf}\big(\frac{1+i}{\sqrt{2}}z \big) + i \text{erf}\big(\frac{1-i}{\sqrt{2}}z \big)\bigg)\]
Returns the value $C(x)$
FewSpecialFunctions.hypergeometric_0F1 — Functionhypergeometric_0F1(b,z)Returns the confluent hypergeometric function given by
\[ {}_0 F_1(a,b) = \sum_{k=0}^\infty \frac{z^k}{(b)_k k!}\]
for the parameters $a$ and $b$
FewSpecialFunctions.confluent_hypergeometric_1F1 — Functionconfluent_hypergeometric_1F1(a,b,z)Returns the Kummer confluent hypergeometric function
\[ {}_1 F_1(a,b,z) = \sum_{k=0}^{\infty} \frac{(a)_k z^k}{(b)_k k!}\]
FewSpecialFunctions.confluent_hypergeometric_U — Functionconfluent_hypergeometric_U(a,b,z)Returns the Kummer confluent hypergeometric function
\[ U(a,b,z) = \frac{\Gamma(b-1)}{\Gamma(a)}z^{1-b} {}_1 F_1(a-b+1,2-b,z)+\frac{\Gamma(1-b)}{\Gamma(a-b+1)} {}_1F_1(a,b,z)\]
FewSpecialFunctions.FermiDiracIntegral — FunctionFermiDiracIntegral(j, x)The Fermi-Dirac integral
\[ F_j(x) = \int_0^\infty \frac{t^j}{\exp(t-x)+1} \, dt\]
Returns the value $F_j(x)$
Resources: [1] D. Bednarczyk and J. Bednarczyk, Phys. Lett. A, 64, 409 (1978) [2] J. S. Blakemore, Solid-St. Electron, 25, 1067 (1982) [3] X. Aymerich-Humet, F. Serra-Mestres, and J. Millan, Solid-St. Electron, 24, 981 (1981) [4] X. Aymerich-Humet, F. Serra-Mestres, and J. Millan, J. Appl. Phys., 54, 2850 (1983) [5] H. M. Antia, Rational Function Approximations for Fermi-Dirac Integrals (1993)
https://arxiv.org/abs/0811.0116 https://de.wikipedia.org/wiki/Fermi-Dirac-Integral https://dlmf.nist.gov/25.12#iii
FewSpecialFunctions.FermiDiracIntegralNorm — FunctionFermiDiracIntegralNorm(j,x)The Fermi-Dirac integral
\[ F_j(x) = \frac{1}{\Gamma(j+1)}\int_0^\infty \frac{t^j}{\exp(t-x)+1} \, dt\]
Returns the value $F_j(x)$
Resources: [1] D. Bednarczyk and J. Bednarczyk, Phys. Lett. A, 64, 409 (1978) [2] J. S. Blakemore, Solid-St. Electron, 25, 1067 (1982) [3] X. Aymerich-Humet, F. Serra-Mestres, and J. Millan, Solid-St. Electron, 24, 981 (1981) [4] X. Aymerich-Humet, F. Serra-Mestres, and J. Millan, J. Appl. Phys., 54, 2850 (1983) [5] H. M. Antia, Rational Function Approximations for Fermi-Dirac Integrals (1993)
https://arxiv.org/abs/0811.0116 https://de.wikipedia.org/wiki/Fermi-Dirac-Integral https://dlmf.nist.gov/25.12#iii
FewSpecialFunctions.MarcumQ — FunctionMarcumQ(μ::Float64, a::Float64, b::Float64)Compute the generalized Marcum Q-function of order μ with non-centrality parameter a and threshold b.
Reference: [1] https://arxiv.org/pdf/1311.0681v1
FewSpecialFunctions.dQdb — FunctiondQdb(M, a, b)Derivative ∂Q_M(a,b)/∂b of the (standard) Marcum Q-function of order M. Requires M integer ≥1 and a>0.
FewSpecialFunctions.U — FunctionU(a::Float64, x::Float64)::Float64Compute the parabolic cylinder function U(a,x) of the first kind for real parameters.
S. Zhang and J. Jin, 'Computation of Special functions' (Wiley, 1966), E. Cojocaru, January 2009
FewSpecialFunctions.V — FunctionV(a::Float64, x::Float64)::Float64Compute the parabolic cylinder function V(a,x).
FewSpecialFunctions.W — FunctionW(a::Float64, x::Float64)::Float64Compute the parabolic cylinder function W(a,x) for real parameters.
FewSpecialFunctions.dU — FunctiondU(a::Float64, x::Float64)::Float64Compute the derivative of the parabolic cylinder function U(a,x) for real parameters.
FewSpecialFunctions.dV — FunctiondV(a::Float64, x::Float64)::Float64Compute the derivative of the parabolic cylinder function V(a,x) for real parameters.
FewSpecialFunctions.dW — FunctiondW(a::Float64, x::Float64)::Float64Compute the derivative of the parabolic cylinder function W with parameters a evaluated at x.