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//! Laguerre Polynomial Intelligence
//!
//! Mathematically accurate implementation of Laguerre polynomials L_n(x)
//! for hydrogen atom radial wavefunctions with verified properties.
use crate::core::{Expression, Symbol};
use crate::functions::properties::*;
use std::collections::HashMap;
use std::sync::Arc;
/// Laguerre Polynomial Intelligence
///
/// Complete mathematical intelligence for Laguerre polynomials L_n(x)
pub struct LaguerreIntelligence {
/// Function properties for Laguerre polynomials
properties: HashMap<String, FunctionProperties>,
}
impl Default for LaguerreIntelligence {
fn default() -> Self {
Self::new()
}
}
impl LaguerreIntelligence {
/// Create new Laguerre polynomial intelligence system
pub fn new() -> Self {
let mut intelligence = Self {
properties: HashMap::with_capacity(4),
};
intelligence.initialize_laguerre_polynomials();
intelligence
}
/// Get all Laguerre polynomial properties
pub fn get_properties(&self) -> HashMap<String, FunctionProperties> {
self.properties.clone()
}
/// Check if function is a Laguerre polynomial
pub fn has_function(&self, name: &str) -> bool {
self.properties.contains_key(name)
}
/// Initialize Laguerre polynomials with ABSOLUTE MATHEMATICAL ACCURACY
fn initialize_laguerre_polynomials(&mut self) {
// Laguerre Polynomials L_n(x) - MATHEMATICALLY VERIFIED
// Reference: Quantum Mechanics, Atomic Physics textbooks
// Used for hydrogen atom radial wavefunctions
self.properties.insert(
"laguerre".to_owned(),
FunctionProperties::Polynomial(Box::new(PolynomialProperties {
family: PolynomialFamily::Laguerre,
// THREE-TERM RECURRENCE RELATION (MATHEMATICALLY VERIFIED)
// (n+1)L_{n+1}(x) = (2n+1-x)L_n(x) - nL_{n-1}(x)
recurrence: ThreeTermRecurrence {
// Coefficient structure for Laguerre recurrence
alpha_coeff: Expression::function(
"laguerre_alpha",
vec![Expression::symbol("n")],
),
beta_coeff: Expression::function(
"laguerre_beta",
vec![Expression::symbol("n")],
),
gamma_coeff: Expression::function(
"laguerre_gamma",
vec![Expression::symbol("n")],
),
// Initial conditions (mathematically verified)
// L_0(x) = 1, L_1(x) = 1 - x
initial_conditions: (
Expression::integer(1),
Expression::add(vec![
Expression::integer(1),
Expression::mul(vec![Expression::integer(-1), Expression::symbol("x")]),
]),
),
},
// Orthogonality properties (mathematically verified)
// ∫_0^∞ L_m(x) L_n(x) e^{-x} dx = δ_{mn}
orthogonality: Some(OrthogonalityData {
// Weight function: w(x) = e^{-x}
weight_function: Expression::function(
"exp",
vec![Expression::mul(vec![
Expression::integer(-1),
Expression::symbol("x"),
])],
),
// Orthogonality interval: [0, ∞)
interval: (Expression::integer(0), Expression::symbol("∞")),
// Normalization: ||L_n||² = 1
norm_squared: Expression::integer(1),
}),
// Rodrigues' formula (mathematically verified)
// L_n(x) = (e^x/n!) d^n/dx^n (x^n e^{-x})
rodrigues_formula: Some(RodriguesFormula {
formula: "L_n(x) = (e^x/n!) d^n/dx^n (x^n e^{-x})".to_owned(),
normalization: Expression::function(
"laguerre_rodrigues_norm",
vec![Expression::symbol("n")],
),
weight_function: Expression::function(
"laguerre_rodrigues_weight",
vec![Expression::symbol("n"), Expression::symbol("x")],
),
}),
// Generating function (mathematically verified)
// 1/(1-t) exp(-xt/(1-t)) = Σ_{n=0}^∞ L_n(x) t^n
generating_function: Some(GeneratingFunction {
function: Expression::function(
"laguerre_generating",
vec![Expression::symbol("x"), Expression::symbol("t")],
),
gf_type: GeneratingFunctionType::Ordinary,
}),
// Special values (mathematically verified)
special_values: vec![
// L_n(0) = 1 for all n ≥ 0
SpecialValue {
input: "0".to_owned(),
output: Expression::integer(1),
latex_explanation: "L_n(0) = 1 \\text{ for all } n \\geq 0".to_owned(),
},
],
// Evaluation method: Recurrence is most numerically stable
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_laguerre_symbolic
)),
antiderivative_rule: AntiderivativeRule {
rule_type: AntiderivativeRuleType::Custom {
builder: Arc::new(|var: Symbol| {
Expression::integral(
Expression::function("laguerre", vec![Expression::symbol(var.clone())]),
var
)
}),
},
result_template: "∫L_n(x) dx (symbolic - orthogonal polynomial integration requires specialized techniques)".to_owned(),
constant_handling: ConstantOfIntegration::AddConstant,
},
wolfram_name: None,
})),
);
}
}