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Spherical Oscillator

The spherical oscillator (3D isotropic harmonic oscillator) is the most frequently used model in quantum physics. This model uses a spherical coordinate system.

Definitions

Antique.SphericalOscillator — Type

Model

This model is described with the time-independent Schrödinger equation

\[ \hat{H} \psi(\pmb{r}) = E \psi(\pmb{r}),\]

and the Hamiltonian

\[ \hat{H} = - \frac{\hbar^2}{2\mu} \nabla^2 + \frac{1}{2} k r^2.\]

Parameters are specified with the following struct:

SO = SphericalOscillator(k=1.0, μ=1.0, ℏ=1.0)

$k$ is the force constant, $μ$ is the mass of particle and $\hbar$ is the reduced Planck constant (Dirac's constant).

References

source

Potential

Antique.V — Method

V(model::SphericalOscillator, r)

\[V(r) = \frac{1}{2} k r^2 = \frac{1}{2} \mu \omega^2 r^2 = \frac{1}{2} \hbar \omega \xi^2,\]

where $\omega = \sqrt{k/\mu}$ is the angular frequency and $\xi = \sqrt{\frac{\mu\omega}{\hbar}}r$.

source

Eigenvalues

Antique.E — Method

E(model::SphericalOscillator; n::Int=0, l::Int=0)

\[E_{nl} = \left(2n + l + \frac{3}{2}\right)\hbar \omega,\]

where $\omega = \sqrt{k/\mu}$.

source

Eigenfunctions

Antique.ψ — Method

ψ(model::SphericalOscillator, r, θ, φ; n::Int=0, l::Int=0, m::Int=0)

\[\psi_{nlm}(\pmb{r}) = R_{nl}(r) Y_{lm}(\theta,\varphi)\]

The domain is $0\leq r \lt \infty, 0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$.

source

Radial Functions

Antique.R — Method

R(model::SphericalOscillator, r; n=0, l=0)

\[R_{nl}(r) = \sqrt{ \frac{\gamma^{3/2}}{2\sqrt{\pi}}} \sqrt{\frac{2^{n+l+3} n!}{(2n+2l+1)!!}} \xi^l \exp\left(-\xi^2/2\right)L_{n}^{(l+\frac{1}{2})} \left(\xi^2\right),\]

where $\gamma = \mu\omega/\hbar$ and $\xi = \sqrt{\gamma}r = \sqrt{\mu\omega/\hbar}r$ are defined. The generalized Laguerre polynomials are defined as $L_n^{(\alpha)}(x) = \frac{x^{-\alpha} \mathrm{e}^x}{n !} \frac{\mathrm{d}^n}{\mathrm{d} x^n}\left(\mathrm{e}^{-x} x^{n+\alpha}\right)$. The domain is $0\leq r \lt \infty$.

source

Generalized Laguerre Polynomials

Antique.L — Method

L(model::SphericalOscillator, x; n=0, α=0)

Note

The generalized Laguerre polynomials $L_n^{(\alpha)}(x)$, not the associated Laguerre polynomials $L_n^{k}(x)$, are used in this model.

Rodrigues' formula & closed-form:

\[\begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \left(\begin{array}{l} n+\alpha \\ n-k \end{array}\right) \frac{x^k}{k !} \\ &= \sum_{k=0}^n(-1)^k \frac{\Gamma(\alpha+n+1)}{\Gamma(\alpha+k+1)\Gamma(n-k+1)} \frac{x^k}{k !}. \end{aligned}\]

Examples:

\[\begin{aligned} L_0^{(0)}(x) &= 1, \\ L_1^{(0)}(x) &= 1 - x, \\ L_1^{(1)}(x) &= 2 - x, \\ L_2^{(0)}(x) &= 1 - 2 x + 1/2 x^{2}, \\ L_2^{(1)}(x) &= 3 - 3 x + 1/2 x^{2}, \\ L_2^{(2)}(x) &= 6 - 4 x + 1/2 x^{2}, \\ L_3^{(0)}(x) &= 1 - 3 x + 3/2 x^{2} - 1/6 x^{3}, \\ L_3^{(1)}(x) &= 4 - 6 x + 2 x^{2} - 1/6 x^{3}, \\ L_3^{(2)}(x) &= 10 - 10 x + 5/2 x^{2} - 1/6 x^{3}, \\ L_3^{(3)}(x) &= 20 - 15 x + 3 x^{2} - 1/6 x^{3}, \\ L_4^{(0)}(x) &= 1 - 4 x + 3 x^{2} - 2/3 x^{3} + 1/24 x^{4}, \\ L_4^{(1)}(x) &= 5 - 10 x + 5 x^{2} - 5/6 x^{3} + 1/24 x^{4}, \\ L_4^{(2)}(x) &= 15 - 20 x + 15/2 x^{2} - 1 x^{3} + 1/24 x^{4}, \\ L_4^{(3)}(x) &= 35 - 35 x + 21/2 x^{2} - 7/6 x^{3} + 1/24 x^{4}, \\ L_4^{(4)}(x) &= 70 - 56 x + 14 x^{2} - 4/3 x^{3} + 1/24 x^{4}, \\ \vdots \end{aligned}\]

source

Spherical Harmonics

Antique.Y — Method

Y(model::SphericalOscillator, θ, φ; l=0, m=0)

\[Y_{lm}(\theta,\varphi) = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} (\cos\theta) \mathrm{e}^{im\varphi}.\]

The domain is $0\leq \theta \lt \pi, 0\leq \varphi \lt 2\pi$. Note that some variants are connected by

\[i^{|m|+m} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^{\frac{|m|+m}{2}} \sqrt{\frac{(l-|m|)!}{(l+|m|)!}} P_l^{|m|} = (-1)^m \sqrt{\frac{(l-m)!}{(l+m)!}} P_l^{m}.\]

source

Associated Legendre Polynomials

Antique.P — Method

P(model::SphericalOscillator, x; n=0, m=0)

Rodrigues' formula & closed-form:

\[\begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ &= \frac{1}{2^n} (1-x^2)^{m/2} \sum_{j=0}^{\left\lfloor\frac{n-m}{2}\right\rfloor} (-1)^j \frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}. \end{aligned},\]

where Legendre polynomials are defined as $P_n(x) = \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right]$. Note that $P_l^{-m} = (-1)^m \frac{(l-m)!}{(l+m)!} P_l^m$ for $m<0$. (It is not compatible with $P_k^m(t) = (-1)^m\left( 1-t^2 \right)^{m/2} \frac{\mathrm{d}^m P_k(t)}{\mathrm{d}t^m}$ caused by $(-1)^m$.) The specific formulae are given below.

Examples:

\[\begin{aligned} P_{0}^{0}(x) &= 1, \\ P_{1}^{0}(x) &= x, \\ P_{1}^{1}(x) &= \left(+1\right)\sqrt{1-x^2}, \\ P_{2}^{0}(x) &= -1/2 + 3/2 x^{2}, \\ P_{2}^{1}(x) &= \left(-3 x\right)\sqrt{1-x^2}, \\ P_{2}^{2}(x) &= 3 - 6 x, \\ P_{3}^{0}(x) &= -3/2 x + 5/2 x^{3}, \\ P_{3}^{1}(x) &= \left(3/2 - 15/2 x^{2}\right)\sqrt{1-x^2}, \\ P_{3}^{2}(x) &= 15 x - 30 x^{2}, \\ P_{3}^{3}(x) &= \left(15 - 30 x\right)\sqrt{1-x^2}, \\ P_{4}^{0}(x) &= 3/8 - 15/4 x^{2} + 35/8 x^{4}, \\ P_{4}^{1}(x) &= \left(- 15/2 x + 35/2 x^{3}\right)\sqrt{1-x^2}, \\ P_{4}^{2}(x) &= -15/2 + 15 x + 105/2 x^{2} - 105 x^{3}, \\ P_{4}^{3}(x) &= \left(105 x - 210 x^{2}\right)\sqrt{1-x^2}, \\ P_{4}^{4}(x) &= 105 - 420 x + 420 x^{2}, \\ & \vdots \end{aligned}\]

source

Usage & Examples

Install Antique.jl for the first use and run using Antique before each use. The energy E(), wavefunction ψ() and potential V() will be exported. In this system, the model is generated by SphericalOscillator and several parameters k, μ and ℏ are set as optional arguments.

using Antique
SO = SphericalOscillator(k=1.0, μ=1.0, ℏ=1.0)

Parameters:

julia> SO.k1.0
julia> SO.μ1.0
julia> SO.ℏ1.0

Eigenvalues:

julia> E(SO, n=0)1.5
julia> E(SO, n=1)3.5
julia> E(SO, n=2)5.5
julia> E(SO, n=0, l=1)2.5
julia> E(SO, n=1, l=1)4.5
julia> E(SO, n=2, l=1)6.5

Testing

Unit testing and Integration testing were done using computer algebra system (Symbolics.jl) and numerical integration (QuadGK.jl). The test script is here.

Associated Legendre Polynomials $P_n^m(x)$

\[ \begin{aligned} P_n^m(x) &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} P_n(x) \\ &= \left( 1-x^2 \right)^{m/2} \frac{\mathrm{d}^m}{\mathrm{d}x^m} \frac{1}{2^n n!} \frac{\mathrm{d}^n}{\mathrm{d}x ^n} \left[ \left( x^2-1 \right)^n \right] \\ &= \frac{1}{2^n} (1-x^2)^{m/2} \sum_{j=0}^{\left\lfloor\frac{n-m}{2}\right\rfloor} (-1)^j \frac{(2n-2j)!}{j! (n-j)! (n-2j-m)!} x^{(n-2j-m)}. \end{aligned}\]

$n=0, m=0:$ ✔

\[\begin{aligned} P_{0}^{0}(x) = 1 &= 1 \\ &= 1 \end{aligned}\]

$n=1, m=0:$ ✔

\[\begin{aligned} P_{1}^{0}(x) = \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right) &= x \\ &= x \end{aligned}\]

$n=1, m=1:$ ✔

\[\begin{aligned} P_{1}^{1}(x) = \left( 1 - x^{2} \right)^{\frac{1}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right) &= \left( 1 - x^{2} \right)^{\frac{1}{2}} \\ &= \left( 1 - x^{2} \right)^{\frac{1}{2}} \end{aligned}\]

$n=2, m=0:$ ✔

\[\begin{aligned} P_{2}^{0}(x) = \frac{1}{8} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{2} &= \frac{-1}{2} + \frac{3}{2} x^{2} \\ &= \frac{-1}{2} + \frac{3}{2} x^{2} \end{aligned}\]

$n=2, m=1:$ ✔

\[\begin{aligned} P_{2}^{1}(x) = \left( 1 - x^{2} \right)^{\frac{1}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{8} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{2} &= 3 \left( 1 - x^{2} \right)^{\frac{1}{2}} x \\ &= 3 \left( 1 - x^{2} \right)^{\frac{1}{2}} x \end{aligned}\]

$n=2, m=2:$ ✔

\[\begin{aligned} P_{2}^{2}(x) = \left( 1 - x^{2} \right) \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{8} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{2} &= 3 - 3 x^{2} \\ &= 3 - 3 x^{2} \end{aligned}\]

$n=3, m=0:$ ✔

\[\begin{aligned} P_{3}^{0}(x) = \frac{1}{48} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{3} &= - \frac{3}{2} x + \frac{5}{2} x^{3} \\ &= - \frac{3}{2} x + \frac{5}{2} x^{3} \end{aligned}\]

$n=3, m=1:$ ✔

\[\begin{aligned} P_{3}^{1}(x) = \left( 1 - x^{2} \right)^{\frac{1}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{48} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{3} &= - \frac{3}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} + \frac{15}{2} x^{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} \\ &= - \frac{3}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} + \frac{15}{2} x^{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} \end{aligned}\]

$n=3, m=2:$ ✔

\[\begin{aligned} P_{3}^{2}(x) = \left( 1 - x^{2} \right) \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{48} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{3} &= 15 x - 15 x^{3} \\ &= 15 x - 15 x^{3} \end{aligned}\]

$n=3, m=3:$ ✔

\[\begin{aligned} P_{3}^{3}(x) = \left( 1 - x^{2} \right)^{\frac{3}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{48} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{3} &= 15 \left( 1 - x^{2} \right)^{\frac{3}{2}} \\ &= 15 \left( 1 - x^{2} \right)^{\frac{3}{2}} \end{aligned}\]

$n=4, m=0:$ ✔

\[\begin{aligned} P_{4}^{0}(x) = \frac{1}{384} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{4} &= \frac{3}{8} - \frac{15}{4} x^{2} + \frac{35}{8} x^{4} \\ &= \frac{3}{8} - \frac{15}{4} x^{2} + \frac{35}{8} x^{4} \end{aligned}\]

$n=4, m=1:$ ✔

\[\begin{aligned} P_{4}^{1}(x) = \left( 1 - x^{2} \right)^{\frac{1}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{384} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{4} &= - \frac{15}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} x + \frac{35}{2} x^{3} \left( 1 - x^{2} \right)^{\frac{1}{2}} \\ &= - \frac{15}{2} \left( 1 - x^{2} \right)^{\frac{1}{2}} x + \frac{35}{2} x^{3} \left( 1 - x^{2} \right)^{\frac{1}{2}} \end{aligned}\]

$n=4, m=2:$ ✔

\[\begin{aligned} P_{4}^{2}(x) = \left( 1 - x^{2} \right) \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{384} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{4} &= \frac{-15}{2} + 60 x^{2} - \frac{105}{2} x^{4} \\ &= \frac{-15}{2} + 60 x^{2} - \frac{105}{2} x^{4} \end{aligned}\]

$n=4, m=3:$ ✔

\[\begin{aligned} P_{4}^{3}(x) = \left( 1 - x^{2} \right)^{\frac{3}{2}} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{384} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{4} &= 105 \left( 1 - x^{2} \right)^{\frac{3}{2}} x \\ &= 105 \left( 1 - x^{2} \right)^{\frac{3}{2}} x \end{aligned}\]

$n=4, m=4:$ ✔

\[\begin{aligned} P_{4}^{4}(x) = \left( 1 - x^{2} \right)^{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{1}{384} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \left( -1 + x^{2} \right)^{4} &= 105 \left( 1 - x^{2} \right)^{2} \\ &= 105 \left( 1 - x^{2} \right)^{2} \end{aligned}\]

Normalization & Orthogonality of $P_n^m(x)$

\[\int_{-1}^{1} P_i^m(x) P_j^m(x) \mathrm{d}x = \frac{2(j+m)!}{(2j+1)(j-m)!} \delta_{ij}\]

 m |  i |  j |        analytical |         numerical 
-- | -- | -- | ----------------- | ----------------- 
 0 |  0 |  0 |    2.000000000000 |    2.000000000000 ✔
 0 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 0 |  0 |  2 |    0.000000000000 |    0.000000000000 ✔
 0 |  0 |  3 |    0.000000000000 |   -0.000000000000 ✔
 0 |  0 |  4 |    0.000000000000 |    0.000000000000 ✔
 0 |  0 |  5 |    0.000000000000 |    0.000000000000 ✔
 0 |  0 |  6 |    0.000000000000 |    0.000000000000 ✔
 0 |  0 |  7 |    0.000000000000 |    0.000000000000 ✔
 0 |  0 |  8 |    0.000000000000 |    0.000000000000 ✔
 0 |  0 |  9 |    0.000000000000 |   -0.000000000000 ✔
 0 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  1 |  1 |    0.666666666667 |    0.666666666667 ✔
 0 |  1 |  2 |    0.000000000000 |    0.000000000000 ✔
 0 |  1 |  3 |    0.000000000000 |   -0.000000000000 ✔
 0 |  1 |  4 |    0.000000000000 |    0.000000000000 ✔
 0 |  1 |  5 |    0.000000000000 |    0.000000000000 ✔
 0 |  1 |  6 |    0.000000000000 |    0.000000000000 ✔
 0 |  1 |  7 |    0.000000000000 |    0.000000000000 ✔
 0 |  1 |  8 |    0.000000000000 |    0.000000000000 ✔
 0 |  1 |  9 |    0.000000000000 |   -0.000000000000 ✔
 0 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  2 |    0.400000000000 |    0.400000000000 ✔
 0 |  2 |  3 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  4 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  5 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  6 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  7 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  8 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  9 |    0.000000000000 |   -0.000000000000 ✔
 0 |  3 |  0 |    0.000000000000 |   -0.000000000000 ✔
 0 |  3 |  1 |    0.000000000000 |   -0.000000000000 ✔
 0 |  3 |  2 |    0.000000000000 |    0.000000000000 ✔
 0 |  3 |  3 |    0.285714285714 |    0.285714285714 ✔
 0 |  3 |  4 |    0.000000000000 |    0.000000000000 ✔
 0 |  3 |  5 |    0.000000000000 |   -0.000000000000 ✔
 0 |  3 |  6 |    0.000000000000 |    0.000000000000 ✔
 0 |  3 |  7 |    0.000000000000 |    0.000000000000 ✔
 0 |  3 |  8 |    0.000000000000 |   -0.000000000000 ✔
 0 |  3 |  9 |    0.000000000000 |   -0.000000000000 ✔
 0 |  4 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  4 |  1 |    0.000000000000 |    0.000000000000 ✔
 0 |  4 |  2 |    0.000000000000 |    0.000000000000 ✔
 0 |  4 |  3 |    0.000000000000 |    0.000000000000 ✔
 0 |  4 |  4 |    0.222222222222 |    0.222222222222 ✔
 0 |  4 |  5 |    0.000000000000 |   -0.000000000000 ✔
 0 |  4 |  6 |    0.000000000000 |   -0.000000000000 ✔
 0 |  4 |  7 |    0.000000000000 |   -0.000000000000 ✔
 0 |  4 |  8 |    0.000000000000 |    0.000000000000 ✔
 0 |  4 |  9 |    0.000000000000 |    0.000000000000 ✔
 0 |  5 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  5 |  1 |    0.000000000000 |    0.000000000000 ✔
 0 |  5 |  2 |    0.000000000000 |    0.000000000000 ✔
 0 |  5 |  3 |    0.000000000000 |   -0.000000000000 ✔
 0 |  5 |  4 |    0.000000000000 |   -0.000000000000 ✔
 0 |  5 |  5 |    0.181818181818 |    0.181818181818 ✔
 0 |  5 |  6 |    0.000000000000 |    0.000000000000 ✔
 0 |  5 |  7 |    0.000000000000 |    0.000000000000 ✔
 0 |  5 |  8 |    0.000000000000 |   -0.000000000000 ✔
 0 |  5 |  9 |    0.000000000000 |   -0.000000000000 ✔
 0 |  6 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  6 |  1 |    0.000000000000 |    0.000000000000 ✔
 0 |  6 |  2 |    0.000000000000 |    0.000000000000 ✔
 0 |  6 |  3 |    0.000000000000 |    0.000000000000 ✔
 0 |  6 |  4 |    0.000000000000 |   -0.000000000000 ✔
 0 |  6 |  5 |    0.000000000000 |    0.000000000000 ✔
 0 |  6 |  6 |    0.153846153846 |    0.153846153846 ✔
 0 |  6 |  7 |    0.000000000000 |   -0.000000000000 ✔
 0 |  6 |  8 |    0.000000000000 |    0.000000000000 ✔
 0 |  6 |  9 |    0.000000000000 |    0.000000000000 ✔
 0 |  7 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  7 |  1 |    0.000000000000 |    0.000000000000 ✔
 0 |  7 |  2 |    0.000000000000 |    0.000000000000 ✔
 0 |  7 |  3 |    0.000000000000 |    0.000000000000 ✔
 0 |  7 |  4 |    0.000000000000 |   -0.000000000000 ✔
 0 |  7 |  5 |    0.000000000000 |    0.000000000000 ✔
 0 |  7 |  6 |    0.000000000000 |   -0.000000000000 ✔
 0 |  7 |  7 |    0.133333333333 |    0.133333333333 ✔
 0 |  7 |  8 |    0.000000000000 |    0.000000000000 ✔
 0 |  7 |  9 |    0.000000000000 |   -0.000000000000 ✔
 0 |  8 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  8 |  1 |    0.000000000000 |    0.000000000000 ✔
 0 |  8 |  2 |    0.000000000000 |    0.000000000000 ✔
 0 |  8 |  3 |    0.000000000000 |   -0.000000000000 ✔
 0 |  8 |  4 |    0.000000000000 |    0.000000000000 ✔
 0 |  8 |  5 |    0.000000000000 |   -0.000000000000 ✔
 0 |  8 |  6 |    0.000000000000 |    0.000000000000 ✔
 0 |  8 |  7 |    0.000000000000 |    0.000000000000 ✔
 0 |  8 |  8 |    0.117647058824 |    0.117647058824 ✔
 0 |  8 |  9 |    0.000000000000 |    0.000000000000 ✔
 0 |  9 |  0 |    0.000000000000 |   -0.000000000000 ✔
 0 |  9 |  1 |    0.000000000000 |   -0.000000000000 ✔
 0 |  9 |  2 |    0.000000000000 |   -0.000000000000 ✔
 0 |  9 |  3 |    0.000000000000 |   -0.000000000000 ✔
 0 |  9 |  4 |    0.000000000000 |    0.000000000000 ✔
 0 |  9 |  5 |    0.000000000000 |   -0.000000000000 ✔
 0 |  9 |  6 |    0.000000000000 |    0.000000000000 ✔
 0 |  9 |  7 |    0.000000000000 |   -0.000000000000 ✔
 0 |  9 |  8 |    0.000000000000 |    0.000000000000 ✔
 0 |  9 |  9 |    0.105263157895 |    0.105263157895 ✔
 1 |  1 |  1 |    1.333333333333 |    1.333333333333 ✔
 1 |  1 |  2 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  3 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  4 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  5 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  6 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  7 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  8 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  9 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  2 |    2.400000000000 |    2.400000000000 ✔
 1 |  2 |  3 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  4 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  5 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  6 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  7 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  8 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  9 |    0.000000000000 |    0.000000000000 ✔
 1 |  3 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  3 |  2 |    0.000000000000 |    0.000000000000 ✔
 1 |  3 |  3 |    3.428571428571 |    3.428571428571 ✔
 1 |  3 |  4 |    0.000000000000 |    0.000000000000 ✔
 1 |  3 |  5 |    0.000000000000 |   -0.000000000000 ✔
 1 |  3 |  6 |    0.000000000000 |   -0.000000000000 ✔
 1 |  3 |  7 |    0.000000000000 |    0.000000000000 ✔
 1 |  3 |  8 |    0.000000000000 |   -0.000000000000 ✔
 1 |  3 |  9 |    0.000000000000 |   -0.000000000000 ✔
 1 |  4 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  4 |  2 |    0.000000000000 |    0.000000000000 ✔
 1 |  4 |  3 |    0.000000000000 |    0.000000000000 ✔
 1 |  4 |  4 |    4.444444444444 |    4.444444444444 ✔
 1 |  4 |  5 |    0.000000000000 |    0.000000000000 ✔
 1 |  4 |  6 |    0.000000000000 |    0.000000000000 ✔
 1 |  4 |  7 |    0.000000000000 |    0.000000000000 ✔
 1 |  4 |  8 |    0.000000000000 |   -0.000000000000 ✔
 1 |  4 |  9 |    0.000000000000 |    0.000000000000 ✔
 1 |  5 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  5 |  2 |    0.000000000000 |   -0.000000000000 ✔
 1 |  5 |  3 |    0.000000000000 |   -0.000000000000 ✔
 1 |  5 |  4 |    0.000000000000 |    0.000000000000 ✔
 1 |  5 |  5 |    5.454545454545 |    5.454545454545 ✔
 1 |  5 |  6 |    0.000000000000 |    0.000000000000 ✔
 1 |  5 |  7 |    0.000000000000 |   -0.000000000000 ✔
 1 |  5 |  8 |    0.000000000000 |   -0.000000000000 ✔
 1 |  5 |  9 |    0.000000000000 |   -0.000000000000 ✔
 1 |  6 |  1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  6 |  2 |    0.000000000000 |    0.000000000000 ✔
 1 |  6 |  3 |    0.000000000000 |   -0.000000000000 ✔
 1 |  6 |  4 |    0.000000000000 |    0.000000000000 ✔
 1 |  6 |  5 |    0.000000000000 |    0.000000000000 ✔
 1 |  6 |  6 |    6.461538461538 |    6.461538461538 ✔
 1 |  6 |  7 |    0.000000000000 |    0.000000000000 ✔
 1 |  6 |  8 |    0.000000000000 |    0.000000000000 ✔
 1 |  6 |  9 |    0.000000000000 |    0.000000000000 ✔
 1 |  7 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  7 |  2 |    0.000000000000 |    0.000000000000 ✔
 1 |  7 |  3 |    0.000000000000 |    0.000000000000 ✔
 1 |  7 |  4 |    0.000000000000 |    0.000000000000 ✔
 1 |  7 |  5 |    0.000000000000 |   -0.000000000000 ✔
 1 |  7 |  6 |    0.000000000000 |    0.000000000000 ✔
 1 |  7 |  7 |    7.466666666667 |    7.466666666667 ✔
 1 |  7 |  8 |    0.000000000000 |    0.000000000000 ✔
 1 |  7 |  9 |    0.000000000000 |    0.000000000000 ✔
 1 |  8 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  8 |  2 |    0.000000000000 |   -0.000000000000 ✔
 1 |  8 |  3 |    0.000000000000 |   -0.000000000000 ✔
 1 |  8 |  4 |    0.000000000000 |   -0.000000000000 ✔
 1 |  8 |  5 |    0.000000000000 |   -0.000000000000 ✔
 1 |  8 |  6 |    0.000000000000 |    0.000000000000 ✔
 1 |  8 |  7 |    0.000000000000 |    0.000000000000 ✔
 1 |  8 |  8 |    8.470588235294 |    8.470588235294 ✔
 1 |  8 |  9 |    0.000000000000 |   -0.000000000000 ✔
 1 |  9 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  9 |  2 |    0.000000000000 |    0.000000000000 ✔
 1 |  9 |  3 |    0.000000000000 |   -0.000000000000 ✔
 1 |  9 |  4 |    0.000000000000 |    0.000000000000 ✔
 1 |  9 |  5 |    0.000000000000 |   -0.000000000000 ✔
 1 |  9 |  6 |    0.000000000000 |    0.000000000000 ✔
 1 |  9 |  7 |    0.000000000000 |    0.000000000000 ✔
 1 |  9 |  8 |    0.000000000000 |   -0.000000000000 ✔
 1 |  9 |  9 |    9.473684210526 |    9.473684210526 ✔
 2 |  2 |  2 |    9.600000000000 |    9.600000000000 ✔
 2 |  2 |  3 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  4 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  5 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  6 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  7 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  8 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  9 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  3 |   34.285714285714 |   34.285714285714 ✔
 2 |  3 |  4 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  5 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  6 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  7 |    0.000000000000 |   -0.000000000000 ✔
 2 |  3 |  8 |    0.000000000000 |    0.000000000000 ✔
 2 |  3 |  9 |    0.000000000000 |   -0.000000000000 ✔
 2 |  4 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  4 |  3 |    0.000000000000 |    0.000000000000 ✔
 2 |  4 |  4 |   80.000000000000 |   80.000000000000 ✔
 2 |  4 |  5 |    0.000000000000 |    0.000000000000 ✔
 2 |  4 |  6 |    0.000000000000 |   -0.000000000000 ✔
 2 |  4 |  7 |    0.000000000000 |   -0.000000000000 ✔
 2 |  4 |  8 |    0.000000000000 |    0.000000000000 ✔
 2 |  4 |  9 |    0.000000000000 |   -0.000000000000 ✔
 2 |  5 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  5 |  3 |    0.000000000000 |    0.000000000000 ✔
 2 |  5 |  4 |    0.000000000000 |    0.000000000000 ✔
 2 |  5 |  5 |  152.727272727273 |  152.727272727273 ✔
 2 |  5 |  6 |    0.000000000000 |   -0.000000000000 ✔
 2 |  5 |  7 |    0.000000000000 |   -0.000000000000 ✔
 2 |  5 |  8 |    0.000000000000 |    0.000000000000 ✔
 2 |  5 |  9 |    0.000000000000 |   -0.000000000000 ✔
 2 |  6 |  2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  6 |  3 |    0.000000000000 |    0.000000000000 ✔
 2 |  6 |  4 |    0.000000000000 |   -0.000000000000 ✔
 2 |  6 |  5 |    0.000000000000 |   -0.000000000000 ✔
 2 |  6 |  6 |  258.461538461538 |  258.461538461538 ✔
 2 |  6 |  7 |    0.000000000000 |    0.000000000000 ✔
 2 |  6 |  8 |    0.000000000000 |    0.000000000000 ✔
 2 |  6 |  9 |    0.000000000000 |   -0.000000000000 ✔
 2 |  7 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  7 |  3 |    0.000000000000 |   -0.000000000000 ✔
 2 |  7 |  4 |    0.000000000000 |   -0.000000000000 ✔
 2 |  7 |  5 |    0.000000000000 |   -0.000000000000 ✔
 2 |  7 |  6 |    0.000000000000 |    0.000000000000 ✔
 2 |  7 |  7 |  403.200000000000 |  403.200000000000 ✔
 2 |  7 |  8 |    0.000000000000 |   -0.000000000000 ✔
 2 |  7 |  9 |    0.000000000000 |   -0.000000000000 ✔
 2 |  8 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  8 |  3 |    0.000000000000 |    0.000000000000 ✔
 2 |  8 |  4 |    0.000000000000 |    0.000000000000 ✔
 2 |  8 |  5 |    0.000000000000 |    0.000000000000 ✔
 2 |  8 |  6 |    0.000000000000 |    0.000000000000 ✔
 2 |  8 |  7 |    0.000000000000 |   -0.000000000000 ✔
 2 |  8 |  8 |  592.941176470588 |  592.941176470588 ✔
 2 |  8 |  9 |    0.000000000000 |    0.000000000000 ✔
 2 |  9 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  9 |  3 |    0.000000000000 |   -0.000000000000 ✔
 2 |  9 |  4 |    0.000000000000 |   -0.000000000000 ✔
 2 |  9 |  5 |    0.000000000000 |   -0.000000000000 ✔
 2 |  9 |  6 |    0.000000000000 |   -0.000000000000 ✔
 2 |  9 |  7 |    0.000000000000 |   -0.000000000000 ✔
 2 |  9 |  8 |    0.000000000000 |    0.000000000000 ✔
 2 |  9 |  9 |  833.684210526316 |  833.684210526316 ✔
 3 |  3 |  3 |  205.714285714286 |  205.714285714286 ✔
 3 |  3 |  4 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  5 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  6 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  7 |    0.000000000000 |   -0.000000000000 ✔
 3 |  3 |  8 |    0.000000000000 |    0.000000000000 ✔
 3 |  3 |  9 |    0.000000000000 |    0.000000000000 ✔
 3 |  4 |  3 |    0.000000000000 |   -0.000000000000 ✔
 3 |  4 |  4 | 1120.000000000000 | 1120.000000000000 ✔
 3 |  4 |  5 |    0.000000000000 |    0.000000000000 ✔
 3 |  4 |  6 |    0.000000000000 |    0.000000000000 ✔
 3 |  4 |  7 |    0.000000000000 |    0.000000000000 ✔
 3 |  4 |  8 |    0.000000000000 |   -0.000000000000 ✔
 3 |  4 |  9 |    0.000000000000 |   -0.000000000000 ✔
 3 |  5 |  3 |    0.000000000000 |   -0.000000000000 ✔
 3 |  5 |  4 |    0.000000000000 |    0.000000000000 ✔
 3 |  5 |  5 | 3665.454545454545 | 3665.454545454545 ✔
 3 |  5 |  6 |    0.000000000000 |    0.000000000000 ✔
 3 |  5 |  7 |    0.000000000000 |   -0.000000000000 ✔
 3 |  5 |  8 |    0.000000000000 |   -0.000000000000 ✔
 3 |  5 |  9 |    0.000000000000 |   -0.000000000000 ✔
 3 |  6 |  3 |    0.000000000000 |    0.000000000000 ✔
 3 |  6 |  4 |    0.000000000000 |    0.000000000000 ✔
 3 |  6 |  5 |    0.000000000000 |    0.000000000000 ✔
 3 |  6 |  6 | 9304.615384615385 | 9304.615384615387 ✔
 3 |  6 |  7 |    0.000000000000 |   -0.000000000000 ✔
 3 |  6 |  8 |    0.000000000000 |   -0.000000000000 ✔
 3 |  6 |  9 |    0.000000000000 |   -0.000000000002 ✔
 3 |  7 |  3 |    0.000000000000 |   -0.000000000000 ✔
 3 |  7 |  4 |    0.000000000000 |    0.000000000000 ✔
 3 |  7 |  5 |    0.000000000000 |   -0.000000000000 ✔
 3 |  7 |  6 |    0.000000000000 |   -0.000000000000 ✔
 3 |  7 |  7 | 20160.000000000000 | 20160.000000000004 ✔
 3 |  7 |  8 |    0.000000000000 |    0.000000000000 ✔
 3 |  7 |  9 |    0.000000000000 |   -0.000000000003 ✔
 3 |  8 |  3 |    0.000000000000 |    0.000000000000 ✔
 3 |  8 |  4 |    0.000000000000 |   -0.000000000000 ✔
 3 |  8 |  5 |    0.000000000000 |   -0.000000000000 ✔
 3 |  8 |  6 |    0.000000000000 |   -0.000000000000 ✔
 3 |  8 |  7 |    0.000000000000 |    0.000000000000 ✔
 3 |  8 |  8 | 39134.117647058825 | 39134.117647058825 ✔
 3 |  8 |  9 |    0.000000000000 |    0.000000000000 ✔
 3 |  9 |  3 |    0.000000000000 |    0.000000000000 ✔
 3 |  9 |  4 |    0.000000000000 |   -0.000000000000 ✔
 3 |  9 |  5 |    0.000000000000 |   -0.000000000000 ✔
 3 |  9 |  6 |    0.000000000000 |   -0.000000000002 ✔
 3 |  9 |  7 |    0.000000000000 |   -0.000000000003 ✔
 3 |  9 |  8 |    0.000000000000 |    0.000000000000 ✔
 3 |  9 |  9 | 70029.473684210534 | 70029.473684210505 ✔
 4 |  4 |  4 | 8960.000000000000 | 8960.000000000002 ✔
 4 |  4 |  5 |    0.000000000000 |   -0.000000000002 ✔
 4 |  4 |  6 |    0.000000000000 |   -0.000000000001 ✔
 4 |  4 |  7 |    0.000000000000 |   -0.000000000000 ✔
 4 |  4 |  8 |    0.000000000000 |    0.000000000007 ✔
 4 |  4 |  9 |    0.000000000000 |    0.000000000000 ✔
 4 |  5 |  4 |    0.000000000000 |   -0.000000000002 ✔
 4 |  5 |  5 | 65978.181818181823 | 65978.181818181838 ✔
 4 |  5 |  6 |    0.000000000000 |   -0.000000000001 ✔
 4 |  5 |  7 |    0.000000000000 |   -0.000000000058 ✔
 4 |  5 |  8 |    0.000000000000 |   -0.000000000002 ✔
 4 |  5 |  9 |    0.000000000000 |   -0.000000000007 ✔
 4 |  6 |  4 |    0.000000000000 |   -0.000000000001 ✔
 4 |  6 |  5 |    0.000000000000 |   -0.000000000001 ✔
 4 |  6 |  6 | 279138.461538461561 | 279138.461538461503 ✔
 4 |  6 |  7 |    0.000000000000 |   -0.000000000018 ✔
 4 |  6 |  8 |    0.000000000000 |    0.000000000055 ✔
 4 |  6 |  9 |    0.000000000000 |    0.000000000029 ✔
 4 |  7 |  4 |    0.000000000000 |   -0.000000000000 ✔
 4 |  7 |  5 |    0.000000000000 |   -0.000000000058 ✔
 4 |  7 |  6 |    0.000000000000 |   -0.000000000018 ✔
 4 |  7 |  7 | 887040.000000000000 | 887040.000000000000 ✔
 4 |  7 |  8 |    0.000000000000 |    0.000000000031 ✔
 4 |  7 |  9 |    0.000000000000 |    0.000000000104 ✔
 4 |  8 |  4 |    0.000000000000 |    0.000000000007 ✔
 4 |  8 |  5 |    0.000000000000 |   -0.000000000002 ✔
 4 |  8 |  6 |    0.000000000000 |    0.000000000055 ✔
 4 |  8 |  7 |    0.000000000000 |    0.000000000031 ✔
 4 |  8 |  8 | 2348047.058823529165 | 2348047.058823529631 ✔
 4 |  8 |  9 |    0.000000000000 |   -0.000000000015 ✔
 4 |  9 |  4 |    0.000000000000 |    0.000000000000 ✔
 4 |  9 |  5 |    0.000000000000 |   -0.000000000007 ✔
 4 |  9 |  6 |    0.000000000000 |    0.000000000029 ✔
 4 |  9 |  7 |    0.000000000000 |    0.000000000104 ✔
 4 |  9 |  8 |    0.000000000000 |   -0.000000000015 ✔
 4 |  9 |  9 | 5462298.947368421592 | 5462298.947368418798 ✔
 5 |  5 |  5 | 659781.818181818235 | 659781.818181818351 ✔
 5 |  5 |  6 |    0.000000000000 |   -0.000000000002 ✔
 5 |  5 |  7 |    0.000000000000 |    0.000000000233 ✔
 5 |  5 |  8 |    0.000000000000 |    0.000000000567 ✔
 5 |  5 |  9 |    0.000000000000 |    0.000000000000 ✔
 5 |  6 |  5 |    0.000000000000 |   -0.000000000002 ✔
 5 |  6 |  6 | 6141046.153846153989 | 6141046.153846156783 ✔
 5 |  6 |  7 |    0.000000000000 |    0.000000000250 ✔
 5 |  6 |  8 |    0.000000000000 |    0.000000001630 ✔
 5 |  6 |  9 |    0.000000000000 |    0.000000000931 ✔
 5 |  7 |  5 |    0.000000000000 |    0.000000000233 ✔
 5 |  7 |  6 |    0.000000000000 |    0.000000000250 ✔
 5 |  7 |  7 | 31933440.000000000000 | 31933440.000000000000 ✔
 5 |  7 |  8 |    0.000000000000 |    0.000000002503 ✔
 5 |  7 |  9 |    0.000000000000 |    0.000000003725 ✔
 5 |  8 |  5 |    0.000000000000 |    0.000000000567 ✔
 5 |  8 |  6 |    0.000000000000 |    0.000000001630 ✔
 5 |  8 |  7 |    0.000000000000 |    0.000000002503 ✔
 5 |  8 |  8 | 122098447.058823525906 | 122098447.058823525906 ✔
 5 |  8 |  9 |    0.000000000000 |   -0.000000001397 ✔
 5 |  9 |  5 |    0.000000000000 |    0.000000000000 ✔
 5 |  9 |  6 |    0.000000000000 |    0.000000000931 ✔
 5 |  9 |  7 |    0.000000000000 |    0.000000003725 ✔
 5 |  9 |  8 |    0.000000000000 |   -0.000000001397 ✔
 5 |  9 |  9 | 382360926.315789461136 | 382360926.315789461136 ✔

Normalization & Orthogonality of $Y_{lm}(\theta,\varphi)$

\[\int_0^{2\pi} \int_0^\pi Y_{lm}(\theta,\varphi)^* Y_{l'm'}(\theta,\varphi) \sin(\theta) ~\mathrm{d}\theta \mathrm{d}\varphi = \delta_{ll'} \delta_{mm'}\]

l₁ | l₂ | m₁ | m₂ |        analytical |         numerical 
-- | -- | -- | -- | ----------------- | ----------------- 
 0 |  0 |  0 |  0 |    1.000000000000 |    1.000000000000 ✔
 0 |  1 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 0 |  1 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 0 |  1 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  0 | -2 |    0.000000000000 |   -0.000000000000 ✔
 0 |  2 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 0 |  2 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 0 |  2 |  0 |  2 |    0.000000000000 |   -0.000000000000 ✔
 1 |  0 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  0 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  0 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 | -1 | -1 |    1.000000000000 |    1.000000000000 ✔
 1 |  1 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 | -1 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  0 |  0 |    1.000000000000 |    1.000000000000 ✔
 1 |  1 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  1 | -1 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  1 |  1 |    1.000000000000 |    1.000000000000 ✔
 1 |  2 | -1 | -2 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 | -1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 | -1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 | -1 |  2 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  0 | -2 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  2 |  0 |  2 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  1 | -2 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  2 |  1 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  0 | -2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  0 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  0 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  0 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  0 |  2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 | -2 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 | -2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 | -2 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 | -1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 | -1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 |  0 | -1 |    0.000000000000 |    0.000000000000 ✔
 2 |  1 |  0 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  1 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 |  1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  1 |  1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 |  2 | -1 |    0.000000000000 |    0.000000000000 ✔
 2 |  1 |  2 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  1 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 | -2 | -2 |    1.000000000000 |    1.000000000000 ✔
 2 |  2 | -2 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 | -2 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 | -2 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 | -2 |  2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 | -1 | -2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 | -1 | -1 |    1.000000000000 |    1.000000000000 ✔
 2 |  2 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 | -1 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 | -1 |  2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  0 | -2 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  0 |  0 |    1.000000000000 |    1.000000000000 ✔
 2 |  2 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  0 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  1 | -2 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  1 | -1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  1 |  1 |    1.000000000000 |    1.000000000000 ✔
 2 |  2 |  1 |  2 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  2 | -2 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  2 | -1 |    0.000000000000 |   -0.000000000000 ✔
 2 |  2 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
 2 |  2 |  2 |  2 |    1.000000000000 |    1.000000000000 ✔

Generalized Laguerre Polynomials $L_n^{(\alpha)}(x)$

\[ \begin{aligned} L_n^{(\alpha)}(x) &= \frac{x^{-\alpha}e^x}{n!} \frac{d^n}{dx^n}\left(x^{n+\alpha}e^{-x}\right) \\ &= \sum_{k=0}^n(-1)^k \frac{\Gamma(\alpha+n+1)}{\Gamma(\alpha+k+1)\Gamma(n-k+1)} \frac{x^k}{k !}. \end{aligned}\]

$n=0, α=0:$ ✔

\[\begin{aligned} L_{0}^{(0)}(x) = e^{ - x} e^{x} &= 1 \\ &= 1 \end{aligned}\]

$n=1, α=0:$ ✔

\[\begin{aligned} L_{1}^{(0)}(x) = \frac{\mathrm{d}}{\mathrm{d}x} x e^{ - x} e^{x} &= 1 - x \\ &= 1 - x \end{aligned}\]

$n=1, α=1:$ ✔

\[\begin{aligned} L_{1}^{(1)}(x) = \frac{e^{x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x}}{x} &= 2 - x \\ &= 2 - x \end{aligned}\]

$n=2, α=0:$ ✔

\[\begin{aligned} L_{2}^{(0)}(x) = \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{2} e^{ - x} e^{x} &= 1 - 2 x + \frac{1}{2} x^{2} \\ &= 1 - 2 x + \frac{1}{2} x^{2} \end{aligned}\]

$n=2, α=1:$ ✔

\[\begin{aligned} L_{2}^{(1)}(x) = \frac{\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} e^{x}}{x} &= 3 - 3 x + \frac{1}{2} x^{2} \\ &= 3 - 3 x + \frac{1}{2} x^{2} \end{aligned}\]

$n=2, α=2:$ ✔

\[\begin{aligned} L_{2}^{(2)}(x) = \frac{\frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x}}{x^{2}} &= 6 - 4 x + \frac{1}{2} x^{2} \\ &= 6 - 4 x + \frac{1}{2} x^{2} \end{aligned}\]

$n=3, α=0:$ ✔

\[\begin{aligned} L_{3}^{(0)}(x) = \frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{3} e^{ - x} e^{x} &= 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3} \\ &= 1 - 3 x + \frac{3}{2} x^{2} - \frac{1}{6} x^{3} \end{aligned}\]

$n=3, α=1:$ ✔

\[\begin{aligned} L_{3}^{(1)}(x) = \frac{\frac{1}{6} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x}}{x} &= 4 - 6 x + 2 x^{2} - \frac{1}{6} x^{3} \\ &= 4 - 6 x + 2 x^{2} - \frac{1}{6} x^{3} \end{aligned}\]

$n=3, α=2:$ ✔

\[\begin{aligned} L_{3}^{(2)}(x) = \frac{\frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{5} e^{ - x} e^{x}}{x^{2}} &= 10 - 10 x + \frac{5}{2} x^{2} - \frac{1}{6} x^{3} \\ &= 10 - 10 x + \frac{5}{2} x^{2} - \frac{1}{6} x^{3} \end{aligned}\]

$n=3, α=3:$ ✔

\[\begin{aligned} L_{3}^{(3)}(x) = \frac{\frac{1}{6} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{6} e^{ - x} e^{x}}{x^{3}} &= 20 - 15 x + 3 x^{2} - \frac{1}{6} x^{3} \\ &= 20 - 15 x + 3 x^{2} - \frac{1}{6} x^{3} \end{aligned}\]

$n=4, α=0:$ ✔

\[\begin{aligned} L_{4}^{(0)}(x) = \frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{4} e^{ - x} e^{x} &= 1 - 4 x + 3 x^{2} - \frac{2}{3} x^{3} + \frac{1}{24} x^{4} \\ &= 1 - 4 x + 3 x^{2} - \frac{2}{3} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]

$n=4, α=1:$ ✔

\[\begin{aligned} L_{4}^{(1)}(x) = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{5} e^{ - x}}{x} &= 5 - 10 x + 5 x^{2} - \frac{5}{6} x^{3} + \frac{1}{24} x^{4} \\ &= 5 - 10 x + 5 x^{2} - \frac{5}{6} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]

$n=4, α=2:$ ✔

\[\begin{aligned} L_{4}^{(2)}(x) = \frac{\frac{1}{24} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{6} e^{ - x} e^{x}}{x^{2}} &= 15 - 20 x + \frac{15}{2} x^{2} - x^{3} + \frac{1}{24} x^{4} \\ &= 15 - 20 x + \frac{15}{2} x^{2} - x^{3} + \frac{1}{24} x^{4} \end{aligned}\]

$n=4, α=3:$ ✔

\[\begin{aligned} L_{4}^{(3)}(x) = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{7} e^{ - x}}{x^{3}} &= 35 - 35 x + \frac{21}{2} x^{2} - \frac{7}{6} x^{3} + \frac{1}{24} x^{4} \\ &= 35 - 35 x + \frac{21}{2} x^{2} - \frac{7}{6} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]

$n=4, α=4:$ ✔

\[\begin{aligned} L_{4}^{(4)}(x) = \frac{\frac{1}{24} e^{x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} x^{8} e^{ - x}}{x^{4}} &= 70 - 56 x + 14 x^{2} - \frac{4}{3} x^{3} + \frac{1}{24} x^{4} \\ &= 70 - 56 x + 14 x^{2} - \frac{4}{3} x^{3} + \frac{1}{24} x^{4} \end{aligned}\]

Normalization & Orthogonality of $L_n^{(\alpha)}(x)$

\[\int_0^\infty L_i^{(\alpha)}(x) L_j^{(\alpha)}(x) x^\alpha \mathrm{e}^{-x} \mathrm{d}x = \frac{\Gamma(n+\alpha+1)}{n!} \delta_{ij}\]

   α |  i |  j |        analytical |         numerical 
---- | -- | -- | ----------------- | ----------------- 
0.01 |  0 |  0 |    0.994325851192 |    0.994325852936 ✔
0.01 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
0.01 |  0 |  2 |    0.000000000000 |    0.000000000000 ✔
0.01 |  0 |  3 |    0.000000000000 |    0.000000000000 ✔
0.01 |  0 |  4 |    0.000000000000 |    0.000000000000 ✔
0.01 |  0 |  5 |    0.000000000000 |    0.000000000000 ✔
0.01 |  0 |  6 |    0.000000000000 |    0.000000000001 ✔
0.01 |  0 |  7 |    0.000000000000 |    0.000000000001 ✔
0.01 |  0 |  8 |    0.000000000000 |    0.000000000001 ✔
0.01 |  0 |  9 |    0.000000000000 |    0.000000000001 ✔
0.01 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
0.01 |  1 |  1 |    1.004269109703 |    1.004269111483 ✔
0.01 |  1 |  2 |    0.000000000000 |    0.000000000000 ✔
0.01 |  1 |  3 |    0.000000000000 |    0.000000000000 ✔
0.01 |  1 |  4 |    0.000000000000 |    0.000000000001 ✔
0.01 |  1 |  5 |    0.000000000000 |    0.000000000001 ✔
0.01 |  1 |  6 |    0.000000000000 |    0.000000000002 ✔
0.01 |  1 |  7 |    0.000000000000 |    0.000000000002 ✔
0.01 |  1 |  8 |    0.000000000000 |    0.000000000002 ✔
0.01 |  1 |  9 |    0.000000000000 |    0.000000000003 ✔
0.01 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
0.01 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
0.01 |  2 |  2 |    1.009290455252 |    1.009290456144 ✔
0.01 |  2 |  3 |    0.000000000000 |    0.000000000001 ✔
0.01 |  2 |  4 |    0.000000000000 |    0.000000000001 ✔
0.01 |  2 |  5 |    0.000000000000 |    0.000000000003 ✔
0.01 |  2 |  6 |    0.000000000000 |    0.000000000002 ✔
0.01 |  2 |  7 |    0.000000000000 |    0.000000000003 ✔
0.01 |  2 |  8 |    0.000000000000 |    0.000000000003 ✔
0.01 |  2 |  9 |    0.000000000000 |    0.000000000007 ✔
0.01 |  3 |  0 |    0.000000000000 |    0.000000000000 ✔
0.01 |  3 |  1 |    0.000000000000 |    0.000000000000 ✔
0.01 |  3 |  2 |    0.000000000000 |    0.000000000001 ✔
0.01 |  3 |  3 |    1.012654756769 |    1.012654758579 ✔
0.01 |  3 |  4 |    0.000000000000 |    0.000000000003 ✔
0.01 |  3 |  5 |    0.000000000000 |    0.000000000003 ✔
0.01 |  3 |  6 |    0.000000000000 |    0.000000000003 ✔
0.01 |  3 |  7 |    0.000000000000 |    0.000000000007 ✔
0.01 |  3 |  8 |    0.000000000000 |    0.000000000014 ✔
0.01 |  3 |  9 |    0.000000000000 |    0.000000000014 ✔
0.01 |  4 |  0 |    0.000000000000 |    0.000000000000 ✔
0.01 |  4 |  1 |    0.000000000000 |    0.000000000001 ✔
0.01 |  4 |  2 |    0.000000000000 |    0.000000000001 ✔
0.01 |  4 |  3 |    0.000000000000 |    0.000000000003 ✔
0.01 |  4 |  4 |    1.015186393661 |    1.015186394564 ✔
0.01 |  4 |  5 |    0.000000000000 |    0.000000000002 ✔
0.01 |  4 |  6 |    0.000000000000 |    0.000000000007 ✔
0.01 |  4 |  7 |    0.000000000000 |    0.000000000014 ✔
0.01 |  4 |  8 |    0.000000000000 |    0.000000000014 ✔
0.01 |  4 |  9 |    0.000000000000 |    0.000000000014 ✔
0.01 |  5 |  0 |    0.000000000000 |    0.000000000000 ✔
0.01 |  5 |  1 |    0.000000000000 |    0.000000000001 ✔
0.01 |  5 |  2 |    0.000000000000 |    0.000000000003 ✔
0.01 |  5 |  3 |    0.000000000000 |    0.000000000003 ✔
0.01 |  5 |  4 |    0.000000000000 |    0.000000000002 ✔
0.01 |  5 |  5 |    1.017216766449 |    1.017216768275 ✔
0.01 |  5 |  6 |    0.000000000000 |    0.000000000014 ✔
0.01 |  5 |  7 |    0.000000000000 |    0.000000000014 ✔
0.01 |  5 |  8 |    0.000000000000 |    0.000000000014 ✔
0.01 |  5 |  9 |    0.000000000000 |    0.000000000028 ✔
0.01 |  6 |  0 |    0.000000000000 |    0.000000000001 ✔
0.01 |  6 |  1 |    0.000000000000 |    0.000000000002 ✔
0.01 |  6 |  2 |    0.000000000000 |    0.000000000002 ✔
0.01 |  6 |  3 |    0.000000000000 |    0.000000000003 ✔
0.01 |  6 |  4 |    0.000000000000 |    0.000000000007 ✔
0.01 |  6 |  5 |    0.000000000000 |    0.000000000014 ✔
0.01 |  6 |  6 |    1.018912127726 |    1.018912128636 ✔
0.01 |  6 |  7 |    0.000000000000 |    0.000000000014 ✔
0.01 |  6 |  8 |    0.000000000000 |    0.000000000028 ✔
0.01 |  6 |  9 |    0.000000000000 |    0.000000000028 ✔
0.01 |  7 |  0 |    0.000000000000 |    0.000000000001 ✔
0.01 |  7 |  1 |    0.000000000000 |    0.000000000002 ✔
0.01 |  7 |  2 |    0.000000000000 |    0.000000000003 ✔
0.01 |  7 |  3 |    0.000000000000 |    0.000000000007 ✔
0.01 |  7 |  4 |    0.000000000000 |    0.000000000014 ✔
0.01 |  7 |  5 |    0.000000000000 |    0.000000000014 ✔
0.01 |  7 |  6 |    0.000000000000 |    0.000000000014 ✔
0.01 |  7 |  7 |    1.020367716480 |    1.020367717392 ✔
0.01 |  7 |  8 |    0.000000000000 |    0.000000000028 ✔
0.01 |  7 |  9 |    0.000000000000 |    0.000000000028 ✔
0.01 |  8 |  0 |    0.000000000000 |    0.000000000001 ✔
0.01 |  8 |  1 |    0.000000000000 |    0.000000000002 ✔
0.01 |  8 |  2 |    0.000000000000 |    0.000000000003 ✔
0.01 |  8 |  3 |    0.000000000000 |    0.000000000014 ✔
0.01 |  8 |  4 |    0.000000000000 |    0.000000000014 ✔
0.01 |  8 |  5 |    0.000000000000 |    0.000000000014 ✔
0.01 |  8 |  6 |    0.000000000000 |    0.000000000028 ✔
0.01 |  8 |  7 |    0.000000000000 |    0.000000000028 ✔
0.01 |  8 |  8 |    1.021643176126 |    1.021643177967 ✔
0.01 |  8 |  9 |    0.000000000000 |    0.000000000028 ✔
0.01 |  9 |  0 |    0.000000000000 |    0.000000000001 ✔
0.01 |  9 |  1 |    0.000000000000 |    0.000000000003 ✔
0.01 |  9 |  2 |    0.000000000000 |    0.000000000007 ✔
0.01 |  9 |  3 |    0.000000000000 |    0.000000000014 ✔
0.01 |  9 |  4 |    0.000000000000 |    0.000000000014 ✔
0.01 |  9 |  5 |    0.000000000000 |    0.000000000028 ✔
0.01 |  9 |  6 |    0.000000000000 |    0.000000000028 ✔
0.01 |  9 |  7 |    0.000000000000 |    0.000000000028 ✔
0.01 |  9 |  8 |    0.000000000000 |    0.000000000028 ✔
0.01 |  9 |  9 |    1.022778335210 |    1.022778336127 ✔
0.05 |  0 |  0 |    0.973504265563 |    0.973504267703 ✔
0.05 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
0.05 |  0 |  2 |    0.000000000000 |    0.000000000000 ✔
0.05 |  0 |  3 |    0.000000000000 |    0.000000000000 ✔
0.05 |  0 |  4 |    0.000000000000 |    0.000000000000 ✔
0.05 |  0 |  5 |    0.000000000000 |    0.000000000001 ✔
0.05 |  0 |  6 |    0.000000000000 |    0.000000000001 ✔
0.05 |  0 |  7 |    0.000000000000 |    0.000000000002 ✔
0.05 |  0 |  8 |    0.000000000000 |    0.000000000002 ✔
0.05 |  0 |  9 |    0.000000000000 |    0.000000000002 ✔
0.05 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
0.05 |  1 |  1 |    1.022179478841 |    1.022179479980 ✔
0.05 |  1 |  2 |    0.000000000000 |    0.000000000000 ✔
0.05 |  1 |  3 |    0.000000000000 |    0.000000000001 ✔
0.05 |  1 |  4 |    0.000000000000 |    0.000000000002 ✔
0.05 |  1 |  5 |    0.000000000000 |    0.000000000002 ✔
0.05 |  1 |  6 |    0.000000000000 |    0.000000000002 ✔
0.05 |  1 |  7 |    0.000000000000 |    0.000000000002 ✔
0.05 |  1 |  8 |    0.000000000000 |    0.000000000004 ✔
0.05 |  1 |  9 |    0.000000000000 |    0.000000000007 ✔
0.05 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
0.05 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
0.05 |  2 |  2 |    1.047733965812 |    1.047733966390 ✔
0.05 |  2 |  3 |    0.000000000000 |    0.000000000002 ✔
0.05 |  2 |  4 |    0.000000000000 |    0.000000000002 ✔
0.05 |  2 |  5 |    0.000000000000 |    0.000000000004 ✔
0.05 |  2 |  6 |    0.000000000000 |    0.000000000004 ✔
0.05 |  2 |  7 |    0.000000000000 |    0.000000000008 ✔
0.05 |  2 |  8 |    0.000000000000 |    0.000000000008 ✔
0.05 |  2 |  9 |    0.000000000000 |    0.000000000008 ✔
0.05 |  3 |  0 |    0.000000000000 |    0.000000000000 ✔
0.05 |  3 |  1 |    0.000000000000 |    0.000000000001 ✔
0.05 |  3 |  2 |    0.000000000000 |    0.000000000002 ✔
0.05 |  3 |  3 |    1.065196198575 |    1.065196199813 ✔
0.05 |  3 |  4 |    0.000000000000 |    0.000000000004 ✔
0.05 |  3 |  5 |    0.000000000000 |    0.000000000004 ✔
0.05 |  3 |  6 |    0.000000000000 |    0.000000000008 ✔
0.05 |  3 |  7 |    0.000000000000 |    0.000000000008 ✔
0.05 |  3 |  8 |    0.000000000000 |    0.000000000016 ✔
0.05 |  3 |  9 |    0.000000000000 |    0.000000000015 ✔
0.05 |  4 |  0 |    0.000000000000 |    0.000000000000 ✔
0.05 |  4 |  1 |    0.000000000000 |    0.000000000002 ✔
0.05 |  4 |  2 |    0.000000000000 |    0.000000000002 ✔
0.05 |  4 |  3 |    0.000000000000 |    0.000000000004 ✔
0.05 |  4 |  4 |    1.078511151058 |    1.078511152326 ✔
0.05 |  4 |  5 |    0.000000000000 |    0.000000000008 ✔
0.05 |  4 |  6 |    0.000000000000 |    0.000000000008 ✔
0.05 |  4 |  7 |    0.000000000000 |    0.000000000017 ✔
0.05 |  4 |  8 |    0.000000000000 |    0.000000000017 ✔
0.05 |  4 |  9 |    0.000000000000 |    0.000000000036 ✔
0.05 |  5 |  0 |    0.000000000000 |    0.000000000001 ✔
0.05 |  5 |  1 |    0.000000000000 |    0.000000000002 ✔
0.05 |  5 |  2 |    0.000000000000 |    0.000000000004 ✔
0.05 |  5 |  3 |    0.000000000000 |    0.000000000004 ✔
0.05 |  5 |  4 |    0.000000000000 |    0.000000000008 ✔
0.05 |  5 |  5 |    1.089296262568 |    1.089296263862 ✔
0.05 |  5 |  6 |    0.000000000000 |    0.000000000017 ✔
0.05 |  5 |  7 |    0.000000000000 |    0.000000000034 ✔
0.05 |  5 |  8 |    0.000000000000 |    0.000000000035 ✔
0.05 |  5 |  9 |    0.000000000000 |    0.000000000034 ✔
0.05 |  6 |  0 |    0.000000000000 |    0.000000000001 ✔
0.05 |  6 |  1 |    0.000000000000 |    0.000000000002 ✔
0.05 |  6 |  2 |    0.000000000000 |    0.000000000004 ✔
0.05 |  6 |  3 |    0.000000000000 |    0.000000000008 ✔
0.05 |  6 |  4 |    0.000000000000 |    0.000000000008 ✔
0.05 |  6 |  5 |    0.000000000000 |    0.000000000017 ✔
0.05 |  6 |  6 |    1.098373731423 |    1.098373732739 ✔
0.05 |  6 |  7 |    0.000000000000 |    0.000000000035 ✔
0.05 |  6 |  8 |    0.000000000000 |    0.000000000035 ✔
0.05 |  6 |  9 |    0.000000000000 |    0.000000000035 ✔
0.05 |  7 |  0 |    0.000000000000 |    0.000000000002 ✔
0.05 |  7 |  1 |    0.000000000000 |    0.000000000002 ✔
0.05 |  7 |  2 |    0.000000000000 |    0.000000000008 ✔
0.05 |  7 |  3 |    0.000000000000 |    0.000000000008 ✔
0.05 |  7 |  4 |    0.000000000000 |    0.000000000017 ✔
0.05 |  7 |  5 |    0.000000000000 |    0.000000000034 ✔
0.05 |  7 |  6 |    0.000000000000 |    0.000000000035 ✔
0.05 |  7 |  7 |    1.106219258076 |    1.106219258720 ✔
0.05 |  7 |  8 |    0.000000000000 |    0.000000000035 ✔
0.05 |  7 |  9 |    0.000000000000 |    0.000000000036 ✔
0.05 |  8 |  0 |    0.000000000000 |    0.000000000002 ✔
0.05 |  8 |  1 |    0.000000000000 |    0.000000000004 ✔
0.05 |  8 |  2 |    0.000000000000 |    0.000000000008 ✔
0.05 |  8 |  3 |    0.000000000000 |    0.000000000016 ✔
0.05 |  8 |  4 |    0.000000000000 |    0.000000000017 ✔
0.05 |  8 |  5 |    0.000000000000 |    0.000000000035 ✔
0.05 |  8 |  6 |    0.000000000000 |    0.000000000035 ✔
0.05 |  8 |  7 |    0.000000000000 |    0.000000000035 ✔
0.05 |  8 |  8 |    1.113133128439 |    1.113133129790 ✔
0.05 |  8 |  9 |    0.000000000000 |    0.000000000074 ✔
0.05 |  9 |  0 |    0.000000000000 |    0.000000000002 ✔
0.05 |  9 |  1 |    0.000000000000 |    0.000000000007 ✔
0.05 |  9 |  2 |    0.000000000000 |    0.000000000008 ✔
0.05 |  9 |  3 |    0.000000000000 |    0.000000000015 ✔
0.05 |  9 |  4 |    0.000000000000 |    0.000000000036 ✔
0.05 |  9 |  5 |    0.000000000000 |    0.000000000034 ✔
0.05 |  9 |  6 |    0.000000000000 |    0.000000000035 ✔
0.05 |  9 |  7 |    0.000000000000 |    0.000000000036 ✔
0.05 |  9 |  8 |    0.000000000000 |    0.000000000074 ✔
0.05 |  9 |  9 |    1.119317201375 |    1.119317202034 ✔
0.10 |  0 |  0 |    0.951350769867 |    0.951350771636 ✔
0.10 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
0.10 |  0 |  2 |    0.000000000000 |    0.000000000000 ✔
0.10 |  0 |  3 |    0.000000000000 |    0.000000000000 ✔
0.10 |  0 |  4 |    0.000000000000 |    0.000000000000 ✔
0.10 |  0 |  5 |    0.000000000000 |    0.000000000001 ✔
0.10 |  0 |  6 |    0.000000000000 |    0.000000000001 ✔
0.10 |  0 |  7 |    0.000000000000 |    0.000000000001 ✔
0.10 |  0 |  8 |    0.000000000000 |    0.000000000001 ✔
0.10 |  0 |  9 |    0.000000000000 |    0.000000000001 ✔
0.10 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
0.10 |  1 |  1 |    1.046485846854 |    1.046485847852 ✔
0.10 |  1 |  2 |    0.000000000000 |    0.000000000001 ✔
0.10 |  1 |  3 |    0.000000000000 |    0.000000000001 ✔
0.10 |  1 |  4 |    0.000000000000 |    0.000000000001 ✔
0.10 |  1 |  5 |    0.000000000000 |    0.000000000001 ✔
0.10 |  1 |  6 |    0.000000000000 |    0.000000000001 ✔
0.10 |  1 |  7 |    0.000000000000 |    0.000000000003 ✔
0.10 |  1 |  8 |    0.000000000000 |    0.000000000003 ✔
0.10 |  1 |  9 |    0.000000000000 |    0.000000000006 ✔
0.10 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
0.10 |  2 |  1 |    0.000000000000 |    0.000000000001 ✔
0.10 |  2 |  2 |    1.098810139196 |    1.098810140297 ✔
0.10 |  2 |  3 |    0.000000000000 |    0.000000000001 ✔
0.10 |  2 |  4 |    0.000000000000 |    0.000000000001 ✔
0.10 |  2 |  5 |    0.000000000000 |    0.000000000003 ✔
0.10 |  2 |  6 |    0.000000000000 |    0.000000000003 ✔
0.10 |  2 |  7 |    0.000000000000 |    0.000000000006 ✔
0.10 |  2 |  8 |    0.000000000000 |    0.000000000006 ✔
0.10 |  2 |  9 |    0.000000000000 |    0.000000000006 ✔
0.10 |  3 |  0 |    0.000000000000 |    0.000000000000 ✔
0.10 |  3 |  1 |    0.000000000000 |    0.000000000001 ✔
0.10 |  3 |  2 |    0.000000000000 |    0.000000000001 ✔
0.10 |  3 |  3 |    1.135437143836 |    1.135437145012 ✔
0.10 |  3 |  4 |    0.000000000000 |    0.000000000006 ✔
0.10 |  3 |  5 |    0.000000000000 |    0.000000000003 ✔
0.10 |  3 |  6 |    0.000000000000 |    0.000000000006 ✔
0.10 |  3 |  7 |    0.000000000000 |    0.000000000006 ✔
0.10 |  3 |  8 |    0.000000000000 |    0.000000000013 ✔
0.10 |  3 |  9 |    0.000000000000 |    0.000000000013 ✔
0.10 |  4 |  0 |    0.000000000000 |    0.000000000000 ✔
0.10 |  4 |  1 |    0.000000000000 |    0.000000000001 ✔
0.10 |  4 |  2 |    0.000000000000 |    0.000000000001 ✔
0.10 |  4 |  3 |    0.000000000000 |    0.000000000006 ✔
0.10 |  4 |  4 |    1.163823072432 |    1.163823073667 ✔
0.10 |  4 |  5 |    0.000000000000 |    0.000000000006 ✔
0.10 |  4 |  6 |    0.000000000000 |    0.000000000013 ✔
0.10 |  4 |  7 |    0.000000000000 |    0.000000000013 ✔
0.10 |  4 |  8 |    0.000000000000 |    0.000000000014 ✔
0.10 |  4 |  9 |    0.000000000000 |    0.000000000029 ✔
0.10 |  5 |  0 |    0.000000000000 |    0.000000000001 ✔
0.10 |  5 |  1 |    0.000000000000 |    0.000000000001 ✔
0.10 |  5 |  2 |    0.000000000000 |    0.000000000003 ✔
0.10 |  5 |  3 |    0.000000000000 |    0.000000000003 ✔
0.10 |  5 |  4 |    0.000000000000 |    0.000000000006 ✔
0.10 |  5 |  5 |    1.187099533881 |    1.187099535166 ✔
0.10 |  5 |  6 |    0.000000000000 |    0.000000000013 ✔
0.10 |  5 |  7 |    0.000000000000 |    0.000000000029 ✔
0.10 |  5 |  8 |    0.000000000000 |    0.000000000030 ✔
0.10 |  5 |  9 |    0.000000000000 |    0.000000000030 ✔
0.10 |  6 |  0 |    0.000000000000 |    0.000000000001 ✔
0.10 |  6 |  1 |    0.000000000000 |    0.000000000001 ✔
0.10 |  6 |  2 |    0.000000000000 |    0.000000000003 ✔
0.10 |  6 |  3 |    0.000000000000 |    0.000000000006 ✔
0.10 |  6 |  4 |    0.000000000000 |    0.000000000013 ✔
0.10 |  6 |  5 |    0.000000000000 |    0.000000000013 ✔
0.10 |  6 |  6 |    1.206884526112 |    1.206884527440 ✔
0.10 |  6 |  7 |    0.000000000000 |    0.000000000030 ✔
0.10 |  6 |  8 |    0.000000000000 |    0.000000000030 ✔
0.10 |  6 |  9 |    0.000000000000 |    0.000000000065 ✔
0.10 |  7 |  0 |    0.000000000000 |    0.000000000001 ✔
0.10 |  7 |  1 |    0.000000000000 |    0.000000000003 ✔
0.10 |  7 |  2 |    0.000000000000 |    0.000000000006 ✔
0.10 |  7 |  3 |    0.000000000000 |    0.000000000006 ✔
0.10 |  7 |  4 |    0.000000000000 |    0.000000000013 ✔
0.10 |  7 |  5 |    0.000000000000 |    0.000000000029 ✔
0.10 |  7 |  6 |    0.000000000000 |    0.000000000030 ✔
0.10 |  7 |  7 |    1.224125733628 |    1.224125734265 ✔
0.10 |  7 |  8 |    0.000000000000 |    0.000000000066 ✔
0.10 |  7 |  9 |    0.000000000000 |    0.000000000031 ✔
0.10 |  8 |  0 |    0.000000000000 |    0.000000000001 ✔
0.10 |  8 |  1 |    0.000000000000 |    0.000000000003 ✔
0.10 |  8 |  2 |    0.000000000000 |    0.000000000006 ✔
0.10 |  8 |  3 |    0.000000000000 |    0.000000000013 ✔
0.10 |  8 |  4 |    0.000000000000 |    0.000000000014 ✔
0.10 |  8 |  5 |    0.000000000000 |    0.000000000030 ✔
0.10 |  8 |  6 |    0.000000000000 |    0.000000000030 ✔
0.10 |  8 |  7 |    0.000000000000 |    0.000000000066 ✔
0.10 |  8 |  8 |    1.239427305298 |    1.239427306699 ✔
0.10 |  8 |  9 |    0.000000000000 |    0.000000000067 ✔
0.10 |  9 |  0 |    0.000000000000 |    0.000000000001 ✔
0.10 |  9 |  1 |    0.000000000000 |    0.000000000006 ✔
0.10 |  9 |  2 |    0.000000000000 |    0.000000000006 ✔
0.10 |  9 |  3 |    0.000000000000 |    0.000000000013 ✔
0.10 |  9 |  4 |    0.000000000000 |    0.000000000029 ✔
0.10 |  9 |  5 |    0.000000000000 |    0.000000000030 ✔
0.10 |  9 |  6 |    0.000000000000 |    0.000000000065 ✔
0.10 |  9 |  7 |    0.000000000000 |    0.000000000031 ✔
0.10 |  9 |  8 |    0.000000000000 |    0.000000000067 ✔
0.10 |  9 |  9 |    1.253198719802 |    1.253198721234 ✔
0.50 |  0 |  0 |    0.886226925453 |    0.886226925863 ✔
0.50 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
0.50 |  0 |  2 |    0.000000000000 |    0.000000000000 ✔
0.50 |  0 |  3 |    0.000000000000 |    0.000000000000 ✔
0.50 |  0 |  4 |    0.000000000000 |    0.000000000000 ✔
0.50 |  0 |  5 |    0.000000000000 |    0.000000000000 ✔
0.50 |  0 |  6 |    0.000000000000 |    0.000000000000 ✔
0.50 |  0 |  7 |    0.000000000000 |    0.000000000000 ✔
0.50 |  0 |  8 |    0.000000000000 |    0.000000000000 ✔
0.50 |  0 |  9 |    0.000000000000 |   -0.000000000000 ✔
0.50 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
0.50 |  1 |  1 |    1.329340388179 |    1.329340389103 ✔
0.50 |  1 |  2 |    0.000000000000 |    0.000000000000 ✔
0.50 |  1 |  3 |    0.000000000000 |    0.000000000000 ✔
0.50 |  1 |  4 |    0.000000000000 |    0.000000000000 ✔
0.50 |  1 |  5 |    0.000000000000 |    0.000000000000 ✔
0.50 |  1 |  6 |    0.000000000000 |    0.000000000000 ✔
0.50 |  1 |  7 |    0.000000000000 |    0.000000000000 ✔
0.50 |  1 |  8 |    0.000000000000 |    0.000000000000 ✔
0.50 |  1 |  9 |    0.000000000000 |    0.000000000000 ✔
0.50 |  2 |  0 |    0.000000000000 |    0.000000000000 ✔
0.50 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
0.50 |  2 |  2 |    1.661675485224 |    1.661675485734 ✔
0.50 |  2 |  3 |    0.000000000000 |    0.000000000000 ✔
0.50 |  2 |  4 |    0.000000000000 |    0.000000000000 ✔
0.50 |  2 |  5 |    0.000000000000 |    0.000000000000 ✔
0.50 |  2 |  6 |    0.000000000000 |    0.000000000000 ✔
0.50 |  2 |  7 |    0.000000000000 |    0.000000000000 ✔
0.50 |  2 |  8 |    0.000000000000 |    0.000000000000 ✔
0.50 |  2 |  9 |    0.000000000000 |   -0.000000000000 ✔
0.50 |  3 |  0 |    0.000000000000 |    0.000000000000 ✔
0.50 |  3 |  1 |    0.000000000000 |    0.000000000000 ✔
0.50 |  3 |  2 |    0.000000000000 |    0.000000000000 ✔
0.50 |  3 |  3 |    1.938621399428 |    1.938621400123 ✔
0.50 |  3 |  4 |    0.000000000000 |    0.000000000000 ✔
0.50 |  3 |  5 |    0.000000000000 |    0.000000000001 ✔
0.50 |  3 |  6 |    0.000000000000 |    0.000000000001 ✔
0.50 |  3 |  7 |    0.000000000000 |    0.000000000001 ✔
0.50 |  3 |  8 |    0.000000000000 |    0.000000000001 ✔
0.50 |  3 |  9 |    0.000000000000 |    0.000000000001 ✔
0.50 |  4 |  0 |    0.000000000000 |    0.000000000000 ✔
0.50 |  4 |  1 |    0.000000000000 |    0.000000000000 ✔
0.50 |  4 |  2 |    0.000000000000 |    0.000000000000 ✔
0.50 |  4 |  3 |    0.000000000000 |    0.000000000000 ✔
0.50 |  4 |  4 |    2.180949074356 |    2.180949075236 ✔
0.50 |  4 |  5 |    0.000000000000 |    0.000000000001 ✔
0.50 |  4 |  6 |    0.000000000000 |    0.000000000001 ✔
0.50 |  4 |  7 |    0.000000000000 |    0.000000000001 ✔
0.50 |  4 |  8 |    0.000000000000 |    0.000000000001 ✔
0.50 |  4 |  9 |    0.000000000000 |    0.000000000003 ✔
0.50 |  5 |  0 |    0.000000000000 |    0.000000000000 ✔
0.50 |  5 |  1 |    0.000000000000 |    0.000000000000 ✔
0.50 |  5 |  2 |    0.000000000000 |    0.000000000000 ✔
0.50 |  5 |  3 |    0.000000000000 |    0.000000000001 ✔
0.50 |  5 |  4 |    0.000000000000 |    0.000000000001 ✔
0.50 |  5 |  5 |    2.399043981792 |    2.399043982856 ✔
0.50 |  5 |  6 |    0.000000000000 |    0.000000000001 ✔
0.50 |  5 |  7 |    0.000000000000 |    0.000000000001 ✔
0.50 |  5 |  8 |    0.000000000000 |    0.000000000003 ✔
0.50 |  5 |  9 |    0.000000000000 |    0.000000000001 ✔
0.50 |  6 |  0 |    0.000000000000 |    0.000000000000 ✔
0.50 |  6 |  1 |    0.000000000000 |    0.000000000000 ✔
0.50 |  6 |  2 |    0.000000000000 |    0.000000000000 ✔
0.50 |  6 |  3 |    0.000000000000 |    0.000000000001 ✔
0.50 |  6 |  4 |    0.000000000000 |    0.000000000001 ✔
0.50 |  6 |  5 |    0.000000000000 |    0.000000000001 ✔
0.50 |  6 |  6 |    2.598964313608 |    2.598964314050 ✔
0.50 |  6 |  7 |    0.000000000000 |    0.000000000003 ✔
0.50 |  6 |  8 |    0.000000000000 |    0.000000000003 ✔
0.50 |  6 |  9 |    0.000000000000 |    0.000000000008 ✔
0.50 |  7 |  0 |    0.000000000000 |    0.000000000000 ✔
0.50 |  7 |  1 |    0.000000000000 |    0.000000000000 ✔
0.50 |  7 |  2 |    0.000000000000 |    0.000000000000 ✔
0.50 |  7 |  3 |    0.000000000000 |    0.000000000001 ✔
0.50 |  7 |  4 |    0.000000000000 |    0.000000000001 ✔
0.50 |  7 |  5 |    0.000000000000 |    0.000000000001 ✔
0.50 |  7 |  6 |    0.000000000000 |    0.000000000003 ✔
0.50 |  7 |  7 |    2.784604621723 |    2.784604622230 ✔
0.50 |  7 |  8 |    0.000000000000 |    0.000000000008 ✔
0.50 |  7 |  9 |    0.000000000000 |    0.000000000009 ✔
0.50 |  8 |  0 |    0.000000000000 |    0.000000000000 ✔
0.50 |  8 |  1 |    0.000000000000 |    0.000000000000 ✔
0.50 |  8 |  2 |    0.000000000000 |    0.000000000000 ✔
0.50 |  8 |  3 |    0.000000000000 |    0.000000000001 ✔
0.50 |  8 |  4 |    0.000000000000 |    0.000000000001 ✔
0.50 |  8 |  5 |    0.000000000000 |    0.000000000003 ✔
0.50 |  8 |  6 |    0.000000000000 |    0.000000000003 ✔
0.50 |  8 |  7 |    0.000000000000 |    0.000000000008 ✔
0.50 |  8 |  8 |    2.958642410581 |    2.958642412199 ✔
0.50 |  8 |  9 |    0.000000000000 |    0.000000000009 ✔
0.50 |  9 |  0 |    0.000000000000 |   -0.000000000000 ✔
0.50 |  9 |  1 |    0.000000000000 |    0.000000000000 ✔
0.50 |  9 |  2 |    0.000000000000 |   -0.000000000000 ✔
0.50 |  9 |  3 |    0.000000000000 |    0.000000000001 ✔
0.50 |  9 |  4 |    0.000000000000 |    0.000000000003 ✔
0.50 |  9 |  5 |    0.000000000000 |    0.000000000001 ✔
0.50 |  9 |  6 |    0.000000000000 |    0.000000000008 ✔
0.50 |  9 |  7 |    0.000000000000 |    0.000000000009 ✔
0.50 |  9 |  8 |    0.000000000000 |    0.000000000009 ✔
0.50 |  9 |  9 |    3.123011433391 |    3.123011435194 ✔
1.00 |  0 |  0 |    1.000000000000 |    1.000000000000 ✔
1.00 |  0 |  1 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  0 |  2 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  0 |  3 |    0.000000000000 |    0.000000000000 ✔
1.00 |  0 |  4 |    0.000000000000 |    0.000000000000 ✔
1.00 |  0 |  5 |    0.000000000000 |    0.000000000000 ✔
1.00 |  0 |  6 |    0.000000000000 |    0.000000000000 ✔
1.00 |  0 |  7 |    0.000000000000 |    0.000000000000 ✔
1.00 |  0 |  8 |    0.000000000000 |    0.000000000000 ✔
1.00 |  0 |  9 |    0.000000000000 |    0.000000000000 ✔
1.00 |  1 |  0 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  1 |  1 |    2.000000000000 |    2.000000000000 ✔
1.00 |  1 |  2 |    0.000000000000 |    0.000000000000 ✔
1.00 |  1 |  3 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  1 |  4 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  1 |  5 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  1 |  6 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  1 |  7 |    0.000000000000 |    0.000000000000 ✔
1.00 |  1 |  8 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  1 |  9 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  2 |  0 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  2 |  1 |    0.000000000000 |    0.000000000000 ✔
1.00 |  2 |  2 |    3.000000000000 |    3.000000000000 ✔
1.00 |  2 |  3 |    0.000000000000 |    0.000000000000 ✔
1.00 |  2 |  4 |    0.000000000000 |    0.000000000000 ✔
1.00 |  2 |  5 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  2 |  6 |    0.000000000000 |    0.000000000000 ✔
1.00 |  2 |  7 |    0.000000000000 |    0.000000000000 ✔
1.00 |  2 |  8 |    0.000000000000 |    0.000000000000 ✔
1.00 |  2 |  9 |    0.000000000000 |    0.000000000000 ✔
1.00 |  3 |  0 |    0.000000000000 |    0.000000000000 ✔
1.00 |  3 |  1 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  3 |  2 |    0.000000000000 |    0.000000000000 ✔
1.00 |  3 |  3 |    4.000000000000 |    4.000000000000 ✔
1.00 |  3 |  4 |    0.000000000000 |    0.000000000000 ✔
1.00 |  3 |  5 |    0.000000000000 |    0.000000000000 ✔
1.00 |  3 |  6 |    0.000000000000 |    0.000000000000 ✔
1.00 |  3 |  7 |    0.000000000000 |    0.000000000000 ✔
1.00 |  3 |  8 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  3 |  9 |    0.000000000000 |   -0.000000000001 ✔
1.00 |  4 |  0 |    0.000000000000 |    0.000000000000 ✔
1.00 |  4 |  1 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  4 |  2 |    0.000000000000 |    0.000000000000 ✔
1.00 |  4 |  3 |    0.000000000000 |    0.000000000000 ✔
1.00 |  4 |  4 |    5.000000000000 |    4.999999999999 ✔
1.00 |  4 |  5 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  4 |  6 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  4 |  7 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  4 |  8 |    0.000000000000 |    0.000000000000 ✔
1.00 |  4 |  9 |    0.000000000000 |    0.000000000000 ✔
1.00 |  5 |  0 |    0.000000000000 |    0.000000000000 ✔
1.00 |  5 |  1 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  5 |  2 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  5 |  3 |    0.000000000000 |    0.000000000000 ✔
1.00 |  5 |  4 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  5 |  5 |    6.000000000000 |    6.000000000000 ✔
1.00 |  5 |  6 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  5 |  7 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  5 |  8 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  5 |  9 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  6 |  0 |    0.000000000000 |    0.000000000000 ✔
1.00 |  6 |  1 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  6 |  2 |    0.000000000000 |    0.000000000000 ✔
1.00 |  6 |  3 |    0.000000000000 |    0.000000000000 ✔
1.00 |  6 |  4 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  6 |  5 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  6 |  6 |    7.000000000000 |    7.000000000000 ✔
1.00 |  6 |  7 |    0.000000000000 |    0.000000000000 ✔
1.00 |  6 |  8 |    0.000000000000 |    0.000000000000 ✔
1.00 |  6 |  9 |    0.000000000000 |    0.000000000000 ✔
1.00 |  7 |  0 |    0.000000000000 |    0.000000000000 ✔
1.00 |  7 |  1 |    0.000000000000 |    0.000000000000 ✔
1.00 |  7 |  2 |    0.000000000000 |    0.000000000000 ✔
1.00 |  7 |  3 |    0.000000000000 |    0.000000000000 ✔
1.00 |  7 |  4 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  7 |  5 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  7 |  6 |    0.000000000000 |    0.000000000000 ✔
1.00 |  7 |  7 |    8.000000000000 |    8.000000000000 ✔
1.00 |  7 |  8 |    0.000000000000 |    0.000000000000 ✔
1.00 |  7 |  9 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  8 |  0 |    0.000000000000 |    0.000000000000 ✔
1.00 |  8 |  1 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  8 |  2 |    0.000000000000 |    0.000000000000 ✔
1.00 |  8 |  3 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  8 |  4 |    0.000000000000 |    0.000000000000 ✔
1.00 |  8 |  5 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  8 |  6 |    0.000000000000 |    0.000000000000 ✔
1.00 |  8 |  7 |    0.000000000000 |    0.000000000000 ✔
1.00 |  8 |  8 |    9.000000000000 |    9.000000000000 ✔
1.00 |  8 |  9 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  9 |  0 |    0.000000000000 |    0.000000000000 ✔
1.00 |  9 |  1 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  9 |  2 |    0.000000000000 |    0.000000000000 ✔
1.00 |  9 |  3 |    0.000000000000 |   -0.000000000001 ✔
1.00 |  9 |  4 |    0.000000000000 |    0.000000000000 ✔
1.00 |  9 |  5 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  9 |  6 |    0.000000000000 |    0.000000000000 ✔
1.00 |  9 |  7 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  9 |  8 |    0.000000000000 |   -0.000000000000 ✔
1.00 |  9 |  9 |   10.000000000000 |   10.000000000002 ✔

Normalization of $R_{nl}(r)$

\[\int |R_{nl}(r)|^2 r^2 \mathrm{d}r = 1\]

 n |  l |        analytical |         numerical 
-- | -- | ----------------- | ----------------- 
 0 |  0 |    1.000000000000 |    1.000000000000 ✔
 1 |  0 |    1.000000000000 |    1.000000000000 ✔
 1 |  1 |    1.000000000000 |    1.000000000000 ✔
 2 |  0 |    1.000000000000 |    1.000000000000 ✔
 2 |  1 |    1.000000000000 |    1.000000000000 ✔
 2 |  2 |    1.000000000000 |    1.000000000000 ✔
 3 |  0 |    1.000000000000 |    1.000000000000 ✔
 3 |  1 |    1.000000000000 |    1.000000000000 ✔
 3 |  2 |    1.000000000000 |    1.000000000000 ✔
 3 |  3 |    1.000000000000 |    1.000000000000 ✔
 4 |  0 |    1.000000000000 |    1.000000000000 ✔
 4 |  1 |    1.000000000000 |    1.000000000000 ✔
 4 |  2 |    1.000000000000 |    1.000000000000 ✔
 4 |  3 |    1.000000000000 |    1.000000000000 ✔
 4 |  4 |    1.000000000000 |    1.000000000000 ✔
 5 |  0 |    1.000000000000 |    1.000000000000 ✔
 5 |  1 |    1.000000000000 |    1.000000000000 ✔
 5 |  2 |    1.000000000000 |    1.000000000000 ✔
 5 |  3 |    1.000000000000 |    1.000000000000 ✔
 5 |  4 |    1.000000000000 |    1.000000000000 ✔
 5 |  5 |    1.000000000000 |    1.000000000000 ✔

Virial Theorem

The virial theorem $\langle T \rangle = \langle V \rangle$ and the definition of Hamiltonian $\langle H \rangle = \langle T \rangle + \langle V \rangle$ derive $\langle H \rangle = 2 \langle V \rangle$.

\[2 \langle V \rangle = 2 \times \int \psi_i^\ast(r,\theta,\varphi) V(r) \psi_j(r,\theta,\varphi) r^2 \sin(\theta) \mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi = 2 \times \int V(r) |R_{nl}(r)|^2 r^2 \mathrm{d}r = E_n\]

 n |  l |  m |        analytical |         numerical 
-- | -- | -- | ----------------- | ----------------- 
 0 |  0 |  0 |    1.500000000000 |    1.500000000122 ✔
 1 |  0 |  0 |    3.500000000000 |    3.499999870039 ✔
 1 |  1 | -1 |    4.500000000000 |    4.499999498869 ✔
 1 |  1 |  0 |    4.500000000000 |    4.499999498869 ✔
 1 |  1 |  1 |    4.500000000000 |    4.499999498869 ✔
 2 |  0 |  0 |    5.500000000000 |    5.500004237451 ✔
 2 |  1 | -1 |    6.500000000000 |    6.500046125526 ✔
 2 |  1 |  0 |    6.500000000000 |    6.500046125526 ✔
 2 |  1 |  1 |    6.500000000000 |    6.500046125526 ✔
 2 |  2 | -2 |    7.500000000000 |    7.500128111476 ✔
 2 |  2 | -1 |    7.500000000000 |    7.500128111427 ✔
 2 |  2 |  0 |    7.500000000000 |    7.500128111526 ✔
 2 |  2 |  1 |    7.500000000000 |    7.500128111427 ✔
 2 |  2 |  2 |    7.500000000000 |    7.500128111476 ✔
 3 |  0 |  0 |    7.500000000000 |    7.500149606525 ✔
 3 |  1 | -1 |    8.500000000000 |    8.499578115951 ✔
 3 |  1 |  0 |    8.500000000000 |    8.499578115951 ✔
 3 |  1 |  1 |    8.500000000000 |    8.499578115951 ✔
 3 |  2 | -2 |    9.500000000000 |    9.496427079822 ✔
 3 |  2 | -1 |    9.500000000000 |    9.496427079760 ✔
 3 |  2 |  0 |    9.500000000000 |    9.496427079884 ✔
 3 |  2 |  1 |    9.500000000000 |    9.496427079760 ✔
 3 |  2 |  2 |    9.500000000000 |    9.496427079822 ✔
 3 |  3 | -3 |   10.500000000000 |   10.492207413354 ✔
 3 |  3 | -2 |   10.500000000000 |   10.492207366556 ✔
 3 |  3 | -1 |   10.500000000000 |   10.492207506758 ✔
 3 |  3 |  0 |   10.500000000000 |   10.492207273248 ✔
 3 |  3 |  1 |   10.500000000000 |   10.492207506758 ✔
 3 |  3 |  2 |   10.500000000000 |   10.492207366556 ✔
 3 |  3 |  3 |   10.500000000000 |   10.492207413354 ✔

 n |  l |        analytical |         numerical 
-- | -- | ----------------- | ----------------- 
 0 |  0 |    1.500000000000 |    1.500000000000 ✔
 1 |  0 |    3.500000000000 |    3.500000000000 ✔
 1 |  1 |    4.500000000000 |    4.500000000000 ✔
 2 |  0 |    5.500000000000 |    5.500000000000 ✔
 2 |  1 |    6.500000000000 |    6.500000000000 ✔
 2 |  2 |    7.500000000000 |    7.500000000000 ✔
 3 |  0 |    7.500000000000 |    7.500000000000 ✔
 3 |  1 |    8.500000000000 |    8.500000000000 ✔
 3 |  2 |    9.500000000000 |    9.500000000001 ✔
 3 |  3 |   10.500000000000 |   10.500000000002 ✔
 4 |  0 |    9.500000000000 |    9.500000000001 ✔
 4 |  1 |   10.500000000000 |   10.500000000003 ✔
 4 |  2 |   11.500000000000 |   11.500000000005 ✔
 4 |  3 |   12.500000000000 |   12.500000000000 ✔
 4 |  4 |   13.500000000000 |   13.500000000000 ✔
 5 |  0 |   11.500000000000 |   11.500000000006 ✔
 5 |  1 |   12.500000000000 |   12.500000000000 ✔
 5 |  2 |   13.500000000000 |   13.500000000000 ✔
 5 |  3 |   14.500000000000 |   14.500000000000 ✔
 5 |  4 |   15.500000000000 |   15.500000000000 ✔
 5 |  5 |   16.500000000000 |   16.500000000000 ✔

Normalization & Orthogonality of $\psi_n(r,\theta,\varphi)$

\[\int \psi_i^\ast(r,\theta,\varphi) \psi_j(r,\theta,\varphi) r^2 \sin(\theta) \mathrm{d}r \mathrm{d}\theta \mathrm{d}\varphi = \delta_{ij}\]

n₁ | n₂ | l₁ | l₂ | m₁ | m₂ |        analytical |         numerical 
-- | -- | -- | -- | -- | -- | ----------------- | ----------------- 
 0 |  0 |  0 |  0 |  0 |  0 |    1.000000000000 |    1.000000000026 ✔
 0 |  1 |  0 |  0 |  0 |  0 |    0.000000000000 |   -0.000000000068 ✔
 0 |  1 |  0 |  1 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 0 |  1 |  0 |  1 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 0 |  1 |  0 |  1 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  0 |  0 |  0 |  0 |  0 |    0.000000000000 |   -0.000000000068 ✔
 1 |  0 |  1 |  0 | -1 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  0 |  1 |  0 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  0 |  1 |  0 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  0 |  0 |  0 |  0 |    1.000000000000 |    0.999999995268 ✔
 1 |  1 |  0 |  1 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  0 |  1 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  0 |  1 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  1 |  0 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  1 |  0 |  0 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  1 |  0 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  1 |  1 | -1 | -1 |    1.000000000000 |    0.999999951768 ✔
 1 |  1 |  1 |  1 | -1 |  0 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  1 |  1 | -1 |  1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  1 |  1 |  0 | -1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  1 |  1 |  0 |  0 |    1.000000000000 |    0.999999951768 ✔
 1 |  1 |  1 |  1 |  0 |  1 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  1 |  1 |  1 | -1 |    0.000000000000 |   -0.000000000000 ✔
 1 |  1 |  1 |  1 |  1 |  0 |    0.000000000000 |    0.000000000000 ✔
 1 |  1 |  1 |  1 |  1 |  1 |    1.000000000000 |    0.999999951768 ✔