η(a::Number, k::Number)
η(ϵ::Number)

Coulomb parameter. For two arguments, returns 1/(a*k). For one argument, returns 1/sqrt(ϵ).

C(ℓ::Number, η::Number)

Coulomb normalization constant.

θ(ℓ::Number, η::Number, ρ::Number)

Coulomb phase function.

F(ℓ::Number, η::Number, ρ::Number)

Regular Coulomb wave function.

References:

• Coulomb wave function
• Implementation paper

D⁺(ℓ::Number, η::Number)

Coulomb D⁺ normalization factor.

D⁻(ℓ::Number, η::Number)

Coulomb D⁻ normalization factor.

H⁺(ℓ::Number, η::Number, ρ::Number)

Outgoing Coulomb wave function.

References:

• Coulomb wave function
• Implementation paper

H⁻(ℓ::Number, η::Number, ρ::Number)

Incoming Coulomb wave function.

References:

• Coulomb wave function
• Implementation paper

F_imag(ℓ::Number, η::Number, ρ::Number)

Imaginary part of the regular Coulomb wave function.

G(ℓ::Number, η::Number, ρ::Number)

Irregular Coulomb wave function.

References:

• Coulomb wave function
• Implementation paper

M_regularized(α::Number, β::Number, γ::Number)

Regularized confluent hypergeometric function.

Φ(ℓ::Number, η::Number, ρ::Number)

Modified Coulomb function Φ.

w(ℓ::Integer, η::Number)
w(ℓ::Number, η::Number)

Auxiliary function for Coulomb wave functions.

w_plus(ℓ::Number, η::Number)

Auxiliary function for Coulomb wave functions.

w_minus(ℓ::Number, η::Number)

Auxiliary function for Coulomb wave functions.

h_plus(ℓ::Number, η::Number)

Auxiliary function for Coulomb wave functions.

h_minus(ℓ::Number, η::Number)

Auxiliary function for Coulomb wave functions.

g(ℓ::Number, η::Number)

Auxiliary function for Coulomb wave functions.

Φ_dot(ℓ::Number, η::Number, ρ::Number; h=1e-6)

Numerical derivative of Φ with respect to ℓ.

F_dot(ℓ::Number, η::Number, ρ::Number; h=1e-6)

Numerical derivative of F with respect to ℓ.

Ψ(ℓ::Number, η::Number, ρ::Number; h=1e-6)

Auxiliary function for Coulomb wave functions.

I(ℓ::Number, η::Number, ρ::Number; h=1e-6)

Auxiliary function for Coulomb wave functions.

debye_function(n::Float64, β::Float64, x::Float64; 
              tol=1e-35, max_terms=2000) -> Float64

Compute the generalized Debye function with parameters n, β, and x.

References:

• Debye function
• Paper

FresnelC(z::Number) -> Number

Computes the Fresnel cosine integral C(z) for the given number z.

Arguments

• z::Number: The input value (can be real or complex).

Returns

• Number: The value of the Fresnel cosine integral at z.

FresnelS(z::Number) -> Number

Computes the Fresnel sine integral S(z) for a given number z.

Arguments

• z::Number: The input value (real or complex) at which to evaluate the Fresnel sine integral.

Returns

• Number: The value of the Fresnel sine integral S(z).

FresnelE(z::Number) -> Number

Computes the Fresnel E integral for the given input z.

Arguments

• z::Number: The input value (real or complex) at which to evaluate the Fresnel E integral.

Returns

• Number: The value of the Fresnel E integral at z.

Clausen(n::Int, θ::Float64; N::Int=10, m::Int=20)

Compute the Clausen function of order n at angle θ.

References:

• Clausen function
• Implementation paper

Ci_complex(z::ComplexF64)

Complex cosine integral function used in Clausen function calculations.

f_n(n::Int, k::Int, θ::Float64)

Compute the Clausen series summand fₙ(k, θ): sin(kθ)/kⁿ for even n, cos(kθ)/kⁿ for odd n.

F_clausen(n::Int, z::ComplexF64, θ::Float64)

Dispatch to the correct primitive function Fₙ(z, θ) for n = 1..6.

FermiDiracIntegral(j, x)

The Fermi-Dirac integral

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://en.wikipedia.org/wiki/CompleteFermi%E2%80%93Diracintegral https://dlmf.nist.gov/25.12#iii

FermiDiracIntegralNorm(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

MarcumQ(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

dQdb(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.

U(a::Float64, x::Float64)::Float64

Compute 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

V(a::Float64, x::Float64)::Float64

Compute the parabolic cylinder function V(a,x).

W(a::Float64, x::Float64)::Float64

Compute the parabolic cylinder function W(a,x) for real parameters.

dU(a::Float64, x::Float64)::Float64

Compute the derivative of the parabolic cylinder function U(a,x) for real parameters.

dV(a::Float64, x::Float64)::Float64

Compute the derivative of the parabolic cylinder function V(a,x) for real parameters.

dW(a::Float64, x::Float64)::Float64

Compute the derivative of the parabolic cylinder function W with parameters a evaluated at x.