Deriving the Spherical Harmonics

The associated Legendre equation is the polar part of the angular equation written in x = cos θ; its only singular points on the sphere are the poles x = ±1, and at each pole exactly one of its two solutions is regular. Use the supplied topic model Y_out = 57.29578*10^(-n), where Y_out is CAP HALF-ANGLE θ₀ (°). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. n is Cap size n, with θ₀ = 10^(−n) rad, dimensionless. Keep m (|m|, azimuthal quantum number) = 0 fixed. For this exercise use only 0.978125 ≤ n ≤ 4.52187 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.041525047 °, with a bounded reading uncertainty ±0.0059703 °. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown n by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000354375, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

Legendre's equation has a convergent series solution for every separation constant λ, but a solution finite at both poles exists only when that series terminates, which happens exactly when λ = l(l + 1) with l a non-negative integer. Use the supplied topic model Y_out = (sqrt(0.25+lam)-0.5), where Y_out is ν = √(λ + ¼) − ½ (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; sqrt(x) means √x. lam is Separation constant λ, dimensionless. Keep no other adjustable input fixed. For this exercise use only 1.4875 ≤ lam ≤ 12.5125 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 2.4090806, with a bounded reading uncertainty ±0.099508. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown lam by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.0011025, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The Legendre polynomial Pₗ(x) is the degree-l solution of Legendre's equation that stays finite at x = ±1, scaled so that Pₗ(1) = 1; the set is orthogonal on −1 ≤ x ≤ 1, with ∫Pₗ² dx = 2/(2l + 1). Use the supplied topic model Y_out = 2/(2*l+1), where Y_out is NORM OF Pₗ 2/(2l + 1) (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. l is Degree l of the solid curve, dimensionless. Keep k (Degree k of the dashed curve) = 0 fixed. For this exercise use only 0 ≤ l ≤ 5, at the specified test settings separated by 1. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.28571429, with a bounded reading uncertainty ±0.14545. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. Determine every allowed value of l consistent with that bound. Compare with a straight-line interpolation between the two endpoint predictions, and explain why a fractional setting may be inadmissible. Do not assume that the supplied calibration points prove monotonicity outside this branch.

The Legendre generating function G(x, t) = (1 − 2xt + t²)^(−1/2) is the function whose coefficient of tˡ is Pₗ(x); with x = cos γ and t = r_</r_> it equals r_> times 1/|r − r′|, which makes its series the multipole expansion of the Coulomb interaction. Use the supplied topic model Y_out = t^(lmax+1)/(1-t), where Y_out is WORST ERROR, AT γ = 0 (× 1/r_>). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. t is Radius ratio t = smaller/larger, dimensionless. Keep lmax (Highest multipole kept, L) = 2; gam (Angle γ between r and r′) = 60 ° fixed. For this exercise use only 0.174375 ≤ t ≤ 0.725625 on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 0.27208677 × 1/r_>, with a bounded reading uncertainty ±0.03985 × 1/r_>. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown t by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.000055125, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.