API Reference

Fmn(m, n, dr)

Compute the normalization factor Fₘₙ for radial spheroidal wave functions.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• dr::Vector: expansion coefficients

Returns

• Normalization factor Fₘₙ

Nmn(m, n, dr)

Compute the normalization constant Nₘₙ(c) for spheroidal wave functions.

Calculates the normalization constant using the expansion coefficients and factorial recursion relations.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• dr::Vector: expansion coefficients

Returns

• Normalization constant Nₘₙ(c)

Note

The normalization ensures proper orthogonality and norm conditions for the spheroidal wave functions.

Nᵣᵐ1(m, n, c, λ, Nmax)

Compute the backward continued fraction for the characteristic value equation.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter
• λ::Number: trial characteristic value
• Nmax::Integer: maximum number of terms in continued fraction

Returns

• Value of the backward continued fraction

Nᵣᵐ1_prime(m, n, c, λ, Nmax)

Compute the derivative of the backward continued fraction with respect to λ.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter
• λ::Number: trial characteristic value
• Nmax::Integer: maximum number of terms

Returns

• Derivative dNᵣᵐ1/dλ

Nᵣᵐ2(m, n, c, λ, Nmax)

Compute the forward continued fraction for the characteristic value equation.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter
• λ::Number: trial characteristic value
• Nmax::Integer: maximum number of terms in continued fraction

Returns

• Value of the forward continued fraction

Nᵣᵐ2_prime(m, n, c, λ, Nmax)

Compute the derivative of the forward continued fraction with respect to λ.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter
• λ::Number: trial characteristic value
• Nmax::Integer: maximum number of terms

Returns

• Derivative dNᵣᵐ2/dλ

assoc_legendre_Pm(m, N, x)

Compute associated Legendre functions Pₙᵐ(x) and their derivatives for n = m, m+1, ..., N.

Uses upward recurrence relations from DLMF for numerical stability.

Arguments

• m::Integer: order (m ≥ 0)
• N::Integer: maximum degree (N ≥ m)
• x::Number: argument in [-1, 1]

Returns

• P::Vector: values [Pₘᵐ(x), Pₘ₊₁ᵐ(x), ..., Pₙᵐ(x)]
• dP::Vector: derivatives [dPₘᵐ/dx, dPₘ₊₁ᵐ/dx, ..., dPₙᵐ/dx]

Method

• DLMF 14.10.1: Initial value Pₘᵐ(x) = (-1)ᵐ (2m-1)!! (1-x²)^(m/2)
• DLMF 14.10.2: Pₘ₊₁ᵐ(x) = x(2m+1)Pₘᵐ(x)
• DLMF 14.10.3: Upward recurrence for n ≥ m+1
• DLMF 14.10.5: Derivative formula

Note

For large m, numerical overflow may occur. Consider using a specialized library.

asymp_λ_pro(m, n, c)

Compute the characteristic value using asymptotic expansion for large c (oblate case).

Uses the asymptotic expansion for large values of the spheroidal parameter c in the prolate case.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter (should be large)

Returns

• Approximate characteristic value λₘₙ(c) for large c

Reference

Page 754, Abramowitz and Stegun, "Handbook of Mathematical Functions"

build_tridiagonal(m, n, c, N)

Construct the tridiagonal matrix for the three-term recurrence relation of spheroidal wave functions.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter
• N::Integer: matrix dimension

Returns

• Tridiagonal matrix with elements αᵣ, βᵣ, γᵣ

compute_c2k(m, n, dr)

Compute the power series expansion coefficients c₂ₖ from the Legendre expansion coefficients dᵣ.

Transforms the expansion coefficients from the Legendre function basis to the power series basis (1-x²)ᵏ, which is useful for certain computational approaches.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• dr::Vector: Legendre expansion coefficients

Returns

• Vector of power series coefficients c₂ₖ

Notes

The recurrence relations used are:

• (2m+r)!/r! → (2m+r+1)(2m+r+2)/((r+1)(r+2))
• (-r/2)ₖ → (-r/2 + 1) / (-r/2 + k - 1)
• (m+r/2+0.5)ₖ → (m + r/2 + 0.5 + k)/(m + r/2 + 0.5)

compute_dr2_mix(m, n, c, λ, max_terms=25 + div(n - m, 2) + round(Int, abs(c)))

Compute the expansion coefficients dᵣ for the spheroidal wave function using a mixed method.

Combines backward and forward recursion to compute the expansion coefficients in the Legendre function expansion. The coefficients satisfy a three-term recurrence relation.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter
• λ::Number: characteristic value
• max_terms::Int: maximum number of expansion terms (default: adaptive)

Returns

• Vector of normalized expansion coefficients dᵣ

Method

Uses backward recursion from the tail and forward recursion from the start, joining at an optimal point to maintain numerical stability.

compute_normalization_constant(m, n, is_even)

Compute the normalization constant based on the parity of (n-m).

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• is_even::Bool: true if (n-m) is even

Returns

• Normalization constant

compute_normalization_sum(dr, r_values, m, is_even)

Compute the normalization sum for the expansion coefficients.

Arguments

• dr::Vector: expansion coefficients
• r_values::Range: range of r indices
• m::Integer: azimuthal order
• is_even::Bool: true if (n-m) is even

Returns

• Normalization sum value

cv_matrix(m, n, c, N=10 + div(n - m, 2) + round(Int, abs(c)))

Compute the characteristic value using the matrix eigenvalue method.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter
• N::Integer: matrix dimension (default: adaptive based on n, m, c)

Returns

• Characteristic value λₘₙ(c)

cv_matrix_seq(m, n, c, N=10 + div(n - m, 2) + round(Int, abs(c)))

Compute a sequence of characteristic values using the matrix eigenvalue method.

Arguments

• m::Integer: azimuthal order
• n::Integer: maximum mode number
• c::Number: spheroidal parameter
• N::Integer: matrix dimension (default: adaptive based on n, m, c)

Returns

• Vector of characteristic values

expr_1(n)

Compute (2n)!/n! = (n+1)(n+2)⋯(2n).

Arguments

• n::Integer: input value

Returns

• Value of (2n)!/n!

expr_2(n)

Compute (2n+2)!/(2(n+1)!).

Arguments

• n::Integer: input value

Returns

• Value of (2n+2)!/(2(n+1)!)

expr_3(m, n)

Compute the normalization factor for even (n-m) case.

Calculates (n+m)! / ((n-m)/2)! / 2^(n-m).

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number

Returns

• Normalization factor for even (n-m)

expr_4(m, n)

Compute the normalization factor for odd (n-m) case.

Calculates (n+m+1)! / ((n-m-1)/2)! / 2^(n-m).

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number

Returns

• Normalization factor for odd (n-m)

find_eigenvalue(m, n, c; tol=1e-12, max_iter=25, num_terms=100)

Compute the characteristic value using Newton-Raphson iteration with continued fractions.

Uses the matrix method to obtain an initial guess, then refines it using Newton-Raphson iteration on the continued fraction equation.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: mode number (n ≥ m)
• c::Number: spheroidal parameter
• tol::Float64: convergence tolerance (default: 1e-12)
• max_iter::Int: maximum number of iterations (default: 25)
• num_terms::Int: number of terms in continued fraction (default: 100)

Returns

• Characteristic value λₘₙ(c)

Examples

λ = find_eigenvalue(1, 2, 0.5)
λ = find_eigenvalue(1, 2, 0.5, tol=1e-14, max_iter=50)

find_eigenvalue_guess(m, n, c, λ_guess; tol=1e-12, max_iter=25, num_terms=40)

Refine a characteristic value using Newton-Raphson iteration with a given initial guess.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter
• λ_guess::Number: initial guess for the characteristic value
• tol::Float64: convergence tolerance (default: 1e-12)
• max_iter::Int: maximum number of iterations (default: 25)
• num_terms::Int: number of terms in continued fraction (default: 40)

Returns

• Refined characteristic value λₘₙ(c)

find_eigenvalue_seq(m, n, c; max_iter=20, num_terms=30)

Compute a sequence of characteristic values for modes m through n.

Arguments

• m::Integer: azimuthal order
• n::Integer: maximum mode number
• c::Number: spheroidal parameter
• max_iter::Int: maximum iterations per eigenvalue (default: 20)
• num_terms::Int: number of terms in continued fraction (default: 30)

Returns

• Vector of characteristic values [λₘₘ, λₘ₊₁,ₘ, ..., λₙₘ]

kmn1(m, n, c)

Compute the normalization constant kₘₙ for the first radial function.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter

Returns

• Normalization constant kₘₙ

Reference

DLMF: (30.11.10) / (30.11.11)

oblate_angular_leg(m, n, c, x)
oblate_angular_leg(m, n, c, λ, x)
oblate_angular_leg(m, n, c, λ, c2k, x)

Compute the oblate angular spheroidal wave function and its derivative using the Legendre expansion.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: mode number (n ≥ m)
• c::Number: spheroidal parameter
• λ::Number: characteristic value (optional, computed if not provided)
• c2k::Vector: expansion coefficients (optional, computed if not provided)
• x::Number: argument in [-1, 1]

Returns

• S: value of the angular function Sₘₙ(-ic, x)
• dS: derivative dSₘₙ/dx

Examples

S, dS = oblate_angular_leg(1, 2, 0.5, 0.3)

oblate_angular_ps(m, n, c, x)
oblate_angular_ps(m, n, c, λ, x)
oblate_angular_ps(m, n, c, λ, c2k, x)

Compute the oblate angular spheroidal wave function and its derivative using the power series expansion.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: mode number (n ≥ m)
• c::Number: spheroidal parameter
• λ::Number: characteristic value (optional, computed if not provided)
• c2k::Vector: expansion coefficients (optional, computed if not provided)
• x::Number: argument in [-1, 1]

Returns

• S: value of the angular function Sₘₙ(-ic, x)
• dS: derivative dSₘₙ/dx

Examples

S, dS = oblate_angular_ps(1, 2, 0.5, 0.3)

oblate_cv(m, n, c)

Compute the characteristic value (eigenvalue) for oblate spheroidal wave functions.

The characteristic value λₘₙ(-ic) is the eigenvalue of the differential equation for the angular spheroidal wave function in the oblate case.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: mode number (n ≥ m)
• c::Number: spheroidal parameter

Returns

• λ: characteristic value λₘₙ(-ic)

Examples

λ = oblate_cv(1, 2, 0.5)

oblate_cv_seq(m, n, c)

Compute a sequence of characteristic values for oblate spheroidal wave functions.

Returns the characteristic values λₘₘ(-ic), λₘ₊₁,ₘ(-ic), ..., λₙₘ(-ic) for all modes from m to n.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: maximum mode number (n ≥ m)
• c::Number: spheroidal parameter

Returns

• Vector of characteristic values [λₘₘ(-ic), λₘ₊₁,ₘ(-ic), ..., λₙₘ(-ic)]

Examples

λ_seq = oblate_cv_seq(1, 5, 0.5)  # Returns [λ₁₁, λ₂₁, λ₃₁, λ₄₁, λ₅₁]

oblate_radial1(m, n, c, ξ)
oblate_radial1(m, n, c, λ, ξ)
oblate_radial1(m, n, c, λ, dr, ξ)

Compute the oblate radial spheroidal wave function of the first kind and its derivative.

The radial function is regular at the origin and corresponds to the modified spherical Bessel function expansion.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: mode number (n ≥ m)
• c::Number: spheroidal parameter
• λ::Number: characteristic value (optional, computed if not provided)
• dr::Vector: expansion coefficients (optional, computed if not provided)
• ξ::Number: radial argument (ξ ≥ 1)

Returns

• R: value of the radial function Rₘₙ⁽¹⁾(-ic, iξ)
• dR: derivative dRₘₙ⁽¹⁾/dξ

Examples

R, dR = oblate_radial1(1, 2, 0.5, 1.5)

oblate_radial2(m, n, c, ξ)
oblate_radial2(m, n, c, λ, ξ)
oblate_radial2(m, n, c, λ, dr, ξ)

Compute the oblate radial spheroidal wave function of the second kind and its derivative.

The radial function is irregular at the origin and corresponds to the modified spherical Neumann function expansion.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: mode number (n ≥ m)
• c::Number: spheroidal parameter
• λ::Number: characteristic value (optional, computed if not provided)
• dr::Vector: expansion coefficients (optional, computed if not provided)
• ξ::Number: radial argument (ξ ≥ 1)

Returns

• R: value of the radial function Rₘₙ⁽²⁾(-ic, iξ)
• dR: derivative dRₘₙ⁽²⁾/dξ

Examples

R, dR = oblate_radial2(1, 2, 0.5, 1.5)

pochhammer(a, k)

Compute the Pochhammer symbol (rising factorial) (a)ₖ = a(a+1)(a+2)⋯(a+k-1).

Arguments

• a::Number: base value
• k::Integer: number of terms

Returns

• Pochhammer symbol (a)ₖ

prolate_angular_leg(m, n, c, x)
prolate_angular_leg(m, n, c, λ, x)
prolate_angular_leg(m, n, c, λ, dr, x)

Compute the prolate angular spheroidal wave function and its derivative using the Legendre expansion.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: mode number (n ≥ m)
• c::Number: spheroidal parameter
• λ::Number: characteristic value (optional, computed if not provided)
• dr::Vector: expansion coefficients (optional, computed if not provided)
• x::Number: argument in [-1, 1]

Returns

• S: value of the angular function Sₘₙ(c, x)
• dS: derivative dSₘₙ/dx

Examples

S, dS = prolate_angular_leg(1, 2, 0.5, 0.3)

prolate_angular_ps(m, n, c, x)
prolate_angular_ps(m, n, c, λ, x)
prolate_angular_ps(m, n, c, λ, c2k, x)

Compute the prolate angular spheroidal wave function and its derivative using the power series expansion.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: mode number (n ≥ m)
• c::Number: spheroidal parameter
• λ::Number: characteristic value (optional, computed if not provided)
• c2k::Vector: expansion coefficients (optional, computed if not provided)
• x::Number: argument in [-1, 1]

Returns

• S: value of the angular function Sₘₙ(c, x)
• dS: derivative dSₘₙ/dx

Examples

S, dS = prolate_angular_ps(1, 2, 0.5, 0.3)

prolate_cv(m, n, c)

Compute the characteristic value (eigenvalue) for prolate spheroidal wave functions.

The characteristic value λₘₙ(c) is the eigenvalue of the differential equation for the angular spheroidal wave function.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: mode number (n ≥ m)
• c::Number: spheroidal parameter

Returns

• λ: characteristic value λₘₙ(c)

Examples

λ = prolate_cv(1, 2, 0.5)

prolate_cv_seq(m, n, c)

Compute a sequence of characteristic values for prolate spheroidal wave functions.

Returns the characteristic values λₘₘ(c), λₘ₊₁,ₘ(c), ..., λₙₘ(c) for all modes from m to n.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: maximum mode number (n ≥ m)
• c::Number: spheroidal parameter

Returns

• Vector of characteristic values [λₘₘ(c), λₘ₊₁,ₘ(c), ..., λₙₘ(c)]

Examples

λ_seq = prolate_cv_seq(1, 5, 0.5)  # Returns [λ₁₁, λ₂₁, λ₃₁, λ₄₁, λ₅₁]

prolate_radial1(m, n, c, ξ)
prolate_radial1(m, n, c, λ, ξ)
prolate_radial1(m, n, c, λ, dr, ξ)

Compute the prolate radial spheroidal wave function of the first kind and its derivative.

The radial function is regular at the origin and corresponds to the spherical Bessel function expansion.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: mode number (n ≥ m)
• c::Number: spheroidal parameter
• λ::Number: characteristic value (optional, computed if not provided)
• dr::Vector: expansion coefficients (optional, computed if not provided)
• ξ::Number: radial argument (ξ ≥ 1)

Returns

• R: value of the radial function Rₘₙ⁽¹⁾(c, ξ)
• dR: derivative dRₘₙ⁽¹⁾/dξ

Examples

R, dR = prolate_radial1(1, 2, 0.5, 1.5)

prolate_radial2(m, n, c, ξ)
prolate_radial2(m, n, c, λ, ξ)
prolate_radial2(m, n, c, λ, dr, ξ)

Compute the prolate radial spheroidal wave function of the second kind and its derivative.

The radial function is irregular at the origin and corresponds to the spherical Neumann function expansion.

Arguments

• m::Integer: azimuthal order (m ≥ 0)
• n::Integer: mode number (n ≥ m)
• c::Number: spheroidal parameter
• λ::Number: characteristic value (optional, computed if not provided)
• dr::Vector: expansion coefficients (optional, computed if not provided)
• ξ::Number: radial argument (ξ ≥ 1)

Returns

• R: value of the radial function Rₘₙ⁽²⁾(c, ξ)
• dR: derivative dRₘₙ⁽²⁾/dξ

Examples

R, dR = prolate_radial2(1, 2, 0.5, 1.5)

ps_λ(m, n, c)

Compute the characteristic value using power series expansion for small c.

Uses the asymptotic expansion λₘₙ(c) ≈ n(n+1) + ℓ₂c² + ℓ₄c⁴ + ℓ₆c⁶ for small values of the spheroidal parameter c.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter (should be small)

Returns

• Approximate characteristic value λₘₙ(c)

Reference

Page 754, Abramowitz and Stegun, "Handbook of Mathematical Functions"

sphericalbesselj_derivative(ν, z)

Compute the derivative of the spherical Bessel function jᵥ(z) with respect to z.

Uses the recurrence relation: jᵥ'(z) = jᵥ₋₁(z) - (ν+1)/z jᵥ(z).

Arguments

• ν::Number: order
• z::Number: argument

Returns

• Derivative djᵥ/dz

sphericalbessely_derivative(ν, z)

Compute the derivative of the spherical Neumann function yᵥ(z) with respect to z.

Uses the recurrence relation: yᵥ'(z) = yᵥ₋₁(z) - (ν+1)/z yᵥ(z).

Arguments

• ν::Number: order
• z::Number: argument

Returns

• Derivative dyᵥ/dz

spheroidal_ang_1(m, n, dr, x)

Compute the angular spheroidal wave function using Legendre function expansion (method 1).

Evaluates the spheroidal function as:

• Sₘₙ(x, γ²) = ∑ₖ d₂ₖ(c) Pₘ₊₂ₖᵐ(x) for even (n-m)
• Sₘₙ(x, γ²) = ∑ₖ d₂ₖ₊₁(c) Pₘ₊₂ₖ₊₁ᵐ(x) for odd (n-m)

where Pₙᵐ are associated Legendre functions.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• dr::Vector: expansion coefficients dᵣ
• x::Number: argument in [-1, 1]

Returns

• S: value of the angular function Sₘₙ(x)
• dS: derivative dSₘₙ/dx

spheroidal_ang_2(m, n, c2k, x)

Compute the angular spheroidal wave function using power series expansion (method 2).

Evaluates the spheroidal function as:

• Sₘₙ(x, γ²) = (-1)ᵐ (1 - x²)^(m/2) ∑ₖ c₂ₖ (1 - x²)ᵏ for even (n-m)
• Sₘₙ(x, γ²) = (-1)ᵐ x(1 - x²)^(m/2) ∑ₖ c₂ₖ (1 - x²)ᵏ for odd (n-m)

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c2k::Vector: power series coefficients c₂ₖ
• x::Number: argument in [-1, 1]

Returns

• S: value of the angular function Sₘₙ(x)
• dS: derivative dSₘₙ/dx

spheroidal_rad_1(m, n, c, dr, ξ)

Compute the radial spheroidal wave function of the first kind using spherical Bessel functions.

The first kind radial function is regular at ξ = 1 and is expressed as a series of spherical Bessel functions jₙ.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter
• dr::Vector: expansion coefficients
• ξ::Number: radial argument (ξ ≥ 1)

Returns

• R: value of the radial function Rₘₙ⁽¹⁾(c, ξ)
• dR: derivative dRₘₙ⁽¹⁾/dξ

spheroidal_rad_2(m, n, c, dr, ξ)

Compute the radial spheroidal wave function of the second kind using spherical Neumann functions.

The second kind radial function is irregular at ξ = 1 and is expressed as a series of spherical Neumann functions yₙ.

Arguments

• m::Integer: azimuthal order
• n::Integer: mode number
• c::Number: spheroidal parameter
• dr::Vector: expansion coefficients
• ξ::Number: radial argument (ξ ≥ 1)

Returns

• R: value of the radial function Rₘₙ⁽²⁾(c, ξ)
• dR: derivative dRₘₙ⁽²⁾/dξ

symmetrize_tridiagonal(A)

Convert a tridiagonal matrix to a symmetric tridiagonal form for eigenvalue computation.

Arguments

• A::Tridiagonal: input tridiagonal matrix

Returns

• Symmetric tridiagonal matrix with off-diagonal elements √(Aᵢⱼ * Aⱼᵢ)

αᵣ(m, r, γ²)

Compute the upper diagonal element of the three-term recurrence relation.

Arguments

• m::Integer: azimuthal order
• r::Integer: index
• γ²::Number: γ² = c²

Returns

• Coefficient αᵣ

βᵐ(m, r, γ²)

Compute the product γᵣ(m, r, γ²) * αᵣ(m, r-2, γ²) used in the continued fraction.

βᵣ(m, r, γ²)

Compute the diagonal element of the three-term recurrence relation.

Arguments

• m::Integer: azimuthal order
• r::Integer: index
• γ²::Number: γ² = c²

Returns

• Coefficient βᵣ

γᵐ(m, r, γ²)

Alias for βᵣ used in the continued fraction formulation.

γᵣ(m, r, γ²)

Compute the lower diagonal element of the three-term recurrence relation.

Arguments

• m::Integer: azimuthal order
• r::Integer: index
• γ²::Number: γ² = c²

Returns

• Coefficient γᵣ