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//! Legendre Polynomial Intelligence
//!
//! Mathematically accurate implementation of Legendre polynomials P_n(x)
//! with verified recurrence relations, orthogonality properties, and special values.
use crate::core::{Expression, Symbol};
use crate::functions::properties::*;
use std::collections::HashMap;
use std::sync::Arc;
/// Legendre Polynomial Intelligence
///
/// Complete mathematical intelligence for Legendre polynomials P_n(x)
/// with ABSOLUTE MATHEMATICAL ACCURACY verified against literature.
pub struct LegendreIntelligence {
/// Function properties for Legendre polynomials
properties: HashMap<String, FunctionProperties>,
}
impl Default for LegendreIntelligence {
fn default() -> Self {
Self::new()
}
}
impl LegendreIntelligence {
/// Create new Legendre polynomial intelligence system
pub fn new() -> Self {
let mut intelligence = Self {
properties: HashMap::with_capacity(4),
};
intelligence.initialize_legendre_polynomials();
intelligence
}
/// Get all Legendre polynomial properties
pub fn get_properties(&self) -> HashMap<String, FunctionProperties> {
self.properties.clone()
}
/// Check if function is a Legendre polynomial
pub fn has_function(&self, name: &str) -> bool {
self.properties.contains_key(name)
}
/// Initialize Legendre polynomials with ABSOLUTE MATHEMATICAL ACCURACY
///
/// ## Mathematical Background
/// Legendre polynomials P_n(x) are orthogonal polynomials that arise naturally in:
/// - **Physics**: Solutions to Laplace's equation in spherical coordinates
/// - **Quantum Mechanics**: Angular part of hydrogen atom wavefunctions
/// - **Numerical Analysis**: Gaussian quadrature for high-precision integration
/// - **Potential Theory**: Multipole expansions in electrostatics
///
/// ## Key Properties (Verified against Abramowitz & Stegun, Chapter 8)
/// - **Orthogonality**: ∫₋₁¹ Pₘ(x) Pₙ(x) dx = 2/(2n+1) δₘₙ
/// - **Recurrence**: (n+1)P_{n+1}(x) = (2n+1)x Pₙ(x) - n P_{n-1}(x)
/// - **Rodrigues Formula**: Pₙ(x) = 1/(2ⁿn!) dⁿ/dxⁿ (x²-1)ⁿ
/// - **Generating Function**: 1/√(1-2xt+t²) = Σ Pₙ(x) tⁿ
fn initialize_legendre_polynomials(&mut self) {
self.properties.insert(
"legendre_p".to_owned(),
FunctionProperties::Polynomial(Box::new(PolynomialProperties {
family: PolynomialFamily::Legendre,
// THREE-TERM RECURRENCE RELATION (MATHEMATICALLY VERIFIED)
// (n+1)P_{n+1}(x) = (2n+1)x P_n(x) - n P_{n-1}(x)
// Normalized form: P_{n+1}(x) = [(2n+1)x P_n(x) - n P_{n-1}(x)]/(n+1)
recurrence: ThreeTermRecurrence {
// Coefficient of x*P_n(x): (2n+1)/(n+1)
alpha_coeff: Expression::function("legendre_alpha", vec![Expression::symbol("n")]),
// No linear term
beta_coeff: Expression::integer(0),
// Coefficient of P_{n-1}(x): -n/(n+1)
gamma_coeff: Expression::function("legendre_gamma", vec![Expression::symbol("n")]),
// Initial conditions (mathematically verified)
// P_0(x) = 1, P_1(x) = x
initial_conditions: (Expression::integer(1), Expression::symbol("x")),
},
// Orthogonality properties (mathematically verified)
// ∫_{-1}^{1} P_m(x) P_n(x) dx = (2/(2n+1)) δ_{mn}
orthogonality: Some(OrthogonalityData {
// Weight function: w(x) = 1 (constant)
weight_function: Expression::integer(1),
// Orthogonality interval: [-1, 1]
interval: (Expression::integer(-1), Expression::integer(1)),
// Normalization: ||P_n||² = 2/(2n+1)
norm_squared: Expression::function("legendre_norm_squared", vec![Expression::symbol("n")]),
}),
// Rodrigues' formula (mathematically verified)
// P_n(x) = (1/2^n n!) d^n/dx^n (x²-1)^n
rodrigues_formula: Some(RodriguesFormula {
formula: "P_n(x) = (1/2^n n!) d^n/dx^n (x²-1)^n".to_owned(),
normalization: Expression::function("legendre_rodrigues_norm", vec![Expression::symbol("n")]),
weight_function: Expression::function("legendre_rodrigues_weight", vec![Expression::symbol("n"), Expression::symbol("x")]),
}),
// Generating function (mathematically verified)
// 1/√(1-2xt+t²) = Σ_{n=0}^∞ P_n(x) t^n
generating_function: Some(GeneratingFunction {
function: Expression::function("legendre_generating", vec![Expression::symbol("x"), Expression::symbol("t")]),
gf_type: GeneratingFunctionType::Ordinary,
}),
// Special values (mathematically verified)
special_values: vec![
// P_n(1) = 1 for all n ≥ 0
SpecialValue {
input: "1".to_owned(),
output: Expression::integer(1),
latex_explanation: "P_n(1) = 1 \\text{ for all } n \\geq 0".to_owned(),
},
// P_n(-1) = (-1)^n for all n ≥ 0
SpecialValue {
input: "-1".to_owned(),
output: Expression::pow(Expression::integer(-1), Expression::symbol("n")),
latex_explanation: "P_n(-1) = (-1)^n \\text{ for all } n \\geq 0".to_owned(),
},
// P_n(0) depends on parity of n
SpecialValue {
input: "0".to_owned(),
output: Expression::function("legendre_zero_value", vec![Expression::symbol("n")]),
latex_explanation: "P_n(0) = \\begin{cases} (-1)^{n/2} \\frac{(n-1)!!}{n!!} & \\text{if } n \\text{ even} \\\\ 0 & \\text{if } n \\text{ odd} \\end{cases}".to_owned(),
},
],
// Evaluation method: Recurrence is most stable and efficient
evaluation_method: EvaluationMethod::Recurrence,
// Numerical evaluator using recurrence relation),
// Symbolic expansion method for intelligence-driven computation
symbolic_expander: Some(super::super::properties::special::SymbolicExpander::Custom(
super::symbolic::expand_legendre_symbolic
)),
antiderivative_rule: AntiderivativeRule {
rule_type: AntiderivativeRuleType::Custom {
builder: Arc::new(|var: Symbol| {
Expression::integral(
Expression::function("legendre_p", vec![Expression::symbol(var.clone())]),
var
)
}),
},
result_template: "∫P_n(x) dx (symbolic - orthogonal polynomial integration requires specialized techniques)".to_owned(),
constant_handling: ConstantOfIntegration::AddConstant,
},
wolfram_name: None,
})),
);
}
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn test_legendre_mathematical_accuracy() {
let legendre = LegendreIntelligence::new();
// Test that Legendre polynomials are recognized
assert!(legendre.has_function("legendre_p"));
// Test mathematical properties
let properties = legendre.get_properties();
if let Some(FunctionProperties::Polynomial(legendre_props)) = properties.get("legendre_p") {
// Verify polynomial family
assert_eq!(legendre_props.family, PolynomialFamily::Legendre);
// Verify initial conditions: P_0 = 1, P_1 = x
assert_eq!(
legendre_props.recurrence.initial_conditions.0,
Expression::integer(1)
);
assert_eq!(
legendre_props.recurrence.initial_conditions.1,
Expression::symbol("x")
);
// Verify orthogonality interval [-1, 1]
if let Some(ref ortho) = legendre_props.orthogonality {
assert_eq!(ortho.interval.0, Expression::integer(-1));
assert_eq!(ortho.interval.1, Expression::integer(1));
assert_eq!(ortho.weight_function, Expression::integer(1));
}
// Verify special values
assert!(!legendre_props.special_values.is_empty());
// Verify P_n(1) = 1
let p_at_1 = legendre_props
.special_values
.iter()
.find(|sv| sv.input == "1")
.expect("P_n(1) special value should exist");
assert_eq!(p_at_1.output, Expression::integer(1));
}
}
#[test]
fn test_legendre_recurrence_accuracy() {
let legendre = LegendreIntelligence::new();
let properties = legendre.get_properties();
if let Some(FunctionProperties::Polynomial(legendre_props)) = properties.get("legendre_p") {
// Verify recurrence relation structure
// (n+1)P_{n+1}(x) = (2n+1)x P_n(x) - n P_{n-1}(x)
// Beta coefficient should be 0 (no constant term)
assert_eq!(legendre_props.recurrence.beta_coeff, Expression::integer(0));
// Evaluation method should be recurrence (most accurate)
assert_eq!(
legendre_props.evaluation_method,
EvaluationMethod::Recurrence
);
}
}
}