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+# MathJS Scientific Computing Improvement Plan
+
+## Executive Summary
+
+This document outlines a comprehensive plan to transform mathjs into a full-featured scientific computing library comparable to NumPy/SciPy, MATLAB, and Mathematica. The plan is organized into 15 sprints with 5-10 tasks each, designed for parallel agent implementation.
+
+### Current State Analysis
+- **Total Functions Implemented**: ~214 functions across 18 categories
+- **TypeScript Conversion**: 9% complete (61/673 files)
+- **WASM Integration**: Established infrastructure with matrix operations, FFT, and basic numeric functions
+- **Overall Scientific Computing Completeness**: ~35%
+
+### Target State
+- **Goal**: 95%+ scientific computing capability parity with SciPy
+- **New Functions to Add**: ~200+ functions across all domains
+- **Performance**: WASM-accelerated for computationally intensive operations
+
+---
+
+## Sprint Organization
+
+| Sprint | Focus Area | Tasks | Priority | Complexity |
+|--------|-----------|-------|----------|------------|
+| 1 | SVD & Core Linear Algebra | 8 | CRITICAL | High |
+| 2 | Bessel & Airy Functions | 10 | CRITICAL | Medium |
+| 3 | Probability Distributions (Part 1) | 10 | CRITICAL | Medium |
+| 4 | Probability Distributions (Part 2) | 10 | CRITICAL | Medium |
+| 5 | Root Finding & Optimization | 8 | CRITICAL | High |
+| 6 | Numerical Integration | 8 | HIGH | Medium |
+| 7 | Interpolation & Curve Fitting | 8 | HIGH | Medium |
+| 8 | Elliptic & Hypergeometric Functions | 10 | HIGH | High |
+| 9 | Advanced Statistics | 10 | HIGH | Medium |
+| 10 | Signal Processing Filters | 10 | MEDIUM-HIGH | Medium |
+| 11 | Window Functions & Spectral | 8 | MEDIUM-HIGH | Low |
+| 12 | ODE/PDE Enhancements | 8 | MEDIUM | High |
+| 13 | Orthogonal Polynomials | 8 | MEDIUM | Medium |
+| 14 | Additional Special Functions | 10 | MEDIUM | Medium |
+| 15 | Symbolic Computation Enhancements | 8 | MEDIUM | High |
+
+---
+
+## Sprint 1: SVD & Core Linear Algebra Decompositions
+**Priority**: CRITICAL
+**Estimated Complexity**: High
+**Dependencies**: None (foundational)
+
+### Background
+SVD (Singular Value Decomposition) is the most critical missing function. It's fundamental for:
+- Least squares solutions
+- Matrix rank computation
+- Principal Component Analysis (PCA)
+- Image compression
+- Solving ill-conditioned systems
+- Pseudoinverse computation (current pinv.ts has TODO: "Use SVD instead")
+
+### Tasks
+
+#### Task 1.1: Implement SVD (Singular Value Decomposition)
+**File**: `src/function/algebra/decomposition/svd.ts`
+**Description**: Implement full SVD decomposition A = U * S * V^T
+**Algorithms**:
+- Golub-Kahan bidiagonalization
+- QR iteration for singular values
+- Option for thin vs full SVD
+**Interface**:
+```typescript
+svd(A: Matrix): { U: Matrix, S: Vector, V: Matrix }
+svd(A: Matrix, full: boolean): { U: Matrix, S: Matrix, V: Matrix }
+```
+**Tests**: Reconstruct A from U*S*V^T, orthogonality of U and V, singular values ordering
+**WASM**: Yes - implement in `src-wasm/algebra/svd.ts`
+
+#### Task 1.2: Implement Cholesky Decomposition
+**File**: `src/function/algebra/decomposition/cholesky.ts`
+**Description**: Decompose symmetric positive definite matrix A = L * L^T
+**Interface**:
+```typescript
+cholesky(A: Matrix): Matrix  // Returns lower triangular L
+cholesky(A: Matrix, 'upper'): Matrix  // Returns upper triangular U
+```
+**Tests**: Symmetry check, positive definiteness check, reconstruction
+**WASM**: Yes - often 2x faster than LU for symmetric systems
+
+#### Task 1.3: Implement Matrix Rank via SVD
+**File**: `src/function/matrix/rank.ts`
+**Description**: Compute numerical rank using SVD singular value thresholding
+**Interface**:
+```typescript
+rank(A: Matrix): number
+rank(A: Matrix, tol: number): number  // Custom tolerance
+```
+**Tests**: Full rank, rank deficient, numerical tolerance
+
+#### Task 1.4: Implement Null Space (Kernel)
+**File**: `src/function/matrix/nullspace.ts`
+**Description**: Compute orthonormal basis for null space using SVD
+**Interface**:
+```typescript
+nullspace(A: Matrix): Matrix  // Columns are null space basis vectors
+nullspace(A: Matrix, tol: number): Matrix
+```
+**Tests**: A * nullspace(A) ≈ 0, orthonormality of result
+
+#### Task 1.5: Implement Column Space (Range)
+**File**: `src/function/matrix/orth.ts`
+**Description**: Compute orthonormal basis for column space using SVD
+**Interface**:
+```typescript
+orth(A: Matrix): Matrix  // Orthonormal basis for range(A)
+```
+**Tests**: Orthonormality, span verification
+
+#### Task 1.6: Implement Least Squares Solver (lstsq)
+**File**: `src/function/algebra/solver/lstsq.ts`
+**Description**: Solve overdetermined systems using SVD
+**Interface**:
+```typescript
+lstsq(A: Matrix, b: Vector): { x: Vector, residuals: Vector, rank: number, s: Vector }
+```
+**Tests**: Exact solutions, overdetermined, underdetermined, rank-deficient
+
+#### Task 1.7: Implement Polar Decomposition
+**File**: `src/function/algebra/decomposition/polar.ts`
+**Description**: Decompose A = U * P where U is unitary and P is positive semidefinite
+**Interface**:
+```typescript
+polar(A: Matrix): { U: Matrix, P: Matrix }
+```
+**Tests**: Reconstruction, U unitary, P positive semidefinite
+
+#### Task 1.8: Update pinv to Use SVD
+**File**: `src/function/matrix/pinv.ts` (modification)
+**Description**: Refactor pseudoinverse to use SVD implementation
+**Current Issue**: Line 20 has TODO: "Use SVD instead (may improve precision)"
+**Tests**: Compare with current implementation, precision improvement verification
+
+---
+
+## Sprint 2: Bessel & Airy Functions
+**Priority**: CRITICAL
+**Estimated Complexity**: Medium
+**Dependencies**: None
+
+### Background
+Bessel functions are essential for physics and engineering applications:
+- Wave propagation in cylindrical coordinates
+- Heat conduction
+- Electromagnetic theory
+- Quantum mechanics
+- Vibration analysis
+
+### Tasks
+
+#### Task 2.1: Implement Bessel J (First Kind)
+**File**: `src/function/special/besselj.ts`
+**Description**: Bessel functions of the first kind J_n(x)
+**Algorithms**:
+- Miller's backward recurrence for integer orders
+- Series expansion for small x
+- Asymptotic expansion for large x
+**Interface**:
+```typescript
+besselj(n: number, x: number): number  // J_n(x)
+besselj(n: number, x: Matrix): Matrix  // Vectorized
+```
+**Special cases**: j0, j1 optimized implementations
+**Tests**: Known values, recurrence relations, limiting behaviors
+
+#### Task 2.2: Implement Bessel Y (Second Kind)
+**File**: `src/function/special/bessely.ts`
+**Description**: Bessel functions of the second kind Y_n(x) (Neumann functions)
+**Interface**:
+```typescript
+bessely(n: number, x: number): number  // Y_n(x)
+```
+**Tests**: Wronskian relation with J, singularity at x=0
+
+#### Task 2.3: Implement Modified Bessel I
+**File**: `src/function/special/besseli.ts`
+**Description**: Modified Bessel functions of the first kind I_n(x)
+**Interface**:
+```typescript
+besseli(n: number, x: number): number  // I_n(x)
+```
+**Tests**: Exponential growth, relation to J
+
+#### Task 2.4: Implement Modified Bessel K
+**File**: `src/function/special/besselk.ts`
+**Description**: Modified Bessel functions of the second kind K_n(x)
+**Interface**:
+```typescript
+besselk(n: number, x: number): number  // K_n(x)
+```
+**Tests**: Exponential decay, Wronskian with I
+
+#### Task 2.5: Implement Spherical Bessel j
+**File**: `src/function/special/sphericalbesselj.ts`
+**Description**: Spherical Bessel functions j_n(x)
+**Relation**: j_n(x) = sqrt(π/2x) * J_{n+1/2}(x)
+**Interface**:
+```typescript
+sphericalbesselj(n: number, x: number): number
+```
+**Tests**: Known values, recurrence relations
+
+#### Task 2.6: Implement Spherical Bessel y
+**File**: `src/function/special/sphericalbessely.ts`
+**Description**: Spherical Bessel functions y_n(x)
+**Interface**:
+```typescript
+sphericalbessely(n: number, x: number): number
+```
+
+#### Task 2.7: Implement Hankel Functions
+**File**: `src/function/special/hankel.ts`
+**Description**: Hankel functions H1_n(x) and H2_n(x)
+**Interface**:
+```typescript
+hankel1(n: number, x: number): Complex  // H1_n(x) = J_n(x) + i*Y_n(x)
+hankel2(n: number, x: number): Complex  // H2_n(x) = J_n(x) - i*Y_n(x)
+```
+
+#### Task 2.8: Implement Airy Ai Function
+**File**: `src/function/special/airy.ts`
+**Description**: Airy function Ai(x) and derivative Ai'(x)
+**Interface**:
+```typescript
+airy(x: number): { ai: number, aip: number, bi: number, bip: number }
+airyai(x: number): number
+airyaip(x: number): number  // Derivative
+```
+**Tests**: Differential equation, asymptotic behavior
+
+#### Task 2.9: Implement Airy Bi Function
+**File**: `src/function/special/airy.ts` (extension)
+**Description**: Airy function Bi(x) and derivative Bi'(x)
+**Interface**:
+```typescript
+airybi(x: number): number
+airybip(x: number): number  // Derivative
+```
+
+#### Task 2.10: WASM Acceleration for Bessel Functions
+**File**: `src-wasm/special/bessel.ts`
+**Description**: High-performance WASM implementations of Bessel functions
+**Focus**: j0, j1, y0, y1, i0, i1, k0, k1 (most commonly used)
+**Expected speedup**: 3-5x for large arrays
+
+---
+
+## Sprint 3: Probability Distributions (Part 1 - Continuous)
+**Priority**: CRITICAL
+**Estimated Complexity**: Medium
+**Dependencies**: erf (exists), gamma (exists)
+
+### Background
+Probability distributions are fundamental for statistics, Monte Carlo methods, and data science. Each distribution needs: pdf, cdf, quantile (ppf/icdf), and random sampling.
+
+### Tasks
+
+#### Task 3.1: Implement Normal Distribution
+**File**: `src/function/probability/distributions/normal.ts`
+**Description**: Gaussian/Normal distribution N(μ, σ²)
+**Interface**:
+```typescript
+normalPdf(x: number, mu?: number, sigma?: number): number
+normalCdf(x: number, mu?: number, sigma?: number): number
+normalInv(p: number, mu?: number, sigma?: number): number  // Quantile
+normalRandom(mu?: number, sigma?: number): number
+normalRandom(mu?: number, sigma?: number, size: number[]): Matrix
+```
+**Algorithms**:
+- PDF: Gaussian formula
+- CDF: Uses existing erf
+- Inverse CDF: Rational approximation (Abramowitz & Stegun)
+- Random: Box-Muller or Ziggurat algorithm
+**Tests**: Mean/variance of samples, CDF(quantile(p)) = p
+
+#### Task 3.2: Implement Student's t Distribution
+**File**: `src/function/probability/distributions/studentt.ts`
+**Description**: Student's t distribution for small sample statistics
+**Interface**:
+```typescript
+tPdf(x: number, df: number): number
+tCdf(x: number, df: number): number
+tInv(p: number, df: number): number
+tRandom(df: number, size?: number[]): number | Matrix
+```
+**Tests**: Approaches normal as df → ∞
+
+#### Task 3.3: Implement Chi-Square Distribution
+**File**: `src/function/probability/distributions/chi2.ts`
+**Description**: Chi-square distribution for variance estimation and hypothesis testing
+**Interface**:
+```typescript
+chi2Pdf(x: number, df: number): number
+chi2Cdf(x: number, df: number): number
+chi2Inv(p: number, df: number): number
+chi2Random(df: number, size?: number[]): number | Matrix
+```
+**Relation**: Uses gamma distribution (χ² = Gamma(df/2, 2))
+
+#### Task 3.4: Implement F Distribution
+**File**: `src/function/probability/distributions/fdist.ts`
+**Description**: F distribution for ANOVA and variance ratio tests
+**Interface**:
+```typescript
+fPdf(x: number, df1: number, df2: number): number
+fCdf(x: number, df1: number, df2: number): number
+fInv(p: number, df1: number, df2: number): number
+fRandom(df1: number, df2: number, size?: number[]): number | Matrix
+```
+
+#### Task 3.5: Implement Beta Distribution
+**File**: `src/function/probability/distributions/beta.ts`
+**Description**: Beta distribution for proportions and Bayesian analysis
+**Interface**:
+```typescript
+betaPdf(x: number, alpha: number, beta: number): number
+betaCdf(x: number, alpha: number, beta: number): number
+betaInv(p: number, alpha: number, beta: number): number
+betaRandom(alpha: number, beta: number, size?: number[]): number | Matrix
+```
+**Requires**: Beta function and incomplete beta function
+
+#### Task 3.6: Implement Exponential Distribution
+**File**: `src/function/probability/distributions/exponential.ts`
+**Description**: Exponential distribution for waiting times
+**Interface**:
+```typescript
+exponentialPdf(x: number, lambda: number): number
+exponentialCdf(x: number, lambda: number): number
+exponentialInv(p: number, lambda: number): number
+exponentialRandom(lambda: number, size?: number[]): number | Matrix
+```
+
+#### Task 3.7: Implement Gamma Distribution
+**File**: `src/function/probability/distributions/gammadist.ts`
+**Description**: Full gamma distribution (extends existing gamma function)
+**Interface**:
+```typescript
+gammaPdf(x: number, shape: number, scale: number): number
+gammaCdf(x: number, shape: number, scale: number): number
+gammaInv(p: number, shape: number, scale: number): number
+gammaRandom(shape: number, scale: number, size?: number[]): number | Matrix
+```
+**Requires**: Incomplete gamma function (gammainc)
+
+#### Task 3.8: Implement Incomplete Gamma Function
+**File**: `src/function/special/gammainc.ts`
+**Description**: Lower and upper incomplete gamma functions
+**Interface**:
+```typescript
+gammainc(a: number, x: number): number  // P(a, x) = γ(a,x)/Γ(a)
+gammaincc(a: number, x: number): number  // Q(a, x) = Γ(a,x)/Γ(a)
+```
+**Algorithms**: Series expansion, continued fraction
+
+#### Task 3.9: Implement Weibull Distribution
+**File**: `src/function/probability/distributions/weibull.ts`
+**Description**: Weibull distribution for reliability analysis
+**Interface**:
+```typescript
+weibullPdf(x: number, shape: number, scale: number): number
+weibullCdf(x: number, shape: number, scale: number): number
+weibullInv(p: number, shape: number, scale: number): number
+weibullRandom(shape: number, scale: number, size?: number[]): number | Matrix
+```
+
+#### Task 3.10: Implement Lognormal Distribution
+**File**: `src/function/probability/distributions/lognormal.ts`
+**Description**: Lognormal distribution for multiplicative processes
+**Interface**:
+```typescript
+lognormalPdf(x: number, mu: number, sigma: number): number
+lognormalCdf(x: number, mu: number, sigma: number): number
+lognormalInv(p: number, mu: number, sigma: number): number
+lognormalRandom(mu: number, sigma: number, size?: number[]): number | Matrix
+```
+
+---
+
+## Sprint 4: Probability Distributions (Part 2 - Discrete & Additional)
+**Priority**: CRITICAL
+**Estimated Complexity**: Medium
+**Dependencies**: Sprint 3 (some shared utilities)
+
+### Tasks
+
+#### Task 4.1: Implement Poisson Distribution
+**File**: `src/function/probability/distributions/poisson.ts`
+**Description**: Poisson distribution for count data
+**Interface**:
+```typescript
+poissonPmf(k: number, lambda: number): number  // P(X = k)
+poissonCdf(k: number, lambda: number): number  // P(X ≤ k)
+poissonInv(p: number, lambda: number): number
+poissonRandom(lambda: number, size?: number[]): number | Matrix
+```
+
+#### Task 4.2: Implement Binomial Distribution
+**File**: `src/function/probability/distributions/binomial.ts`
+**Description**: Binomial distribution for success/failure trials
+**Interface**:
+```typescript
+binomialPmf(k: number, n: number, p: number): number
+binomialCdf(k: number, n: number, p: number): number
+binomialInv(q: number, n: number, p: number): number
+binomialRandom(n: number, p: number, size?: number[]): number | Matrix
+```
+
+#### Task 4.3: Implement Negative Binomial Distribution
+**File**: `src/function/probability/distributions/negativebinomial.ts`
+**Description**: Negative binomial for number of trials until r successes
+**Interface**:
+```typescript
+negativeBinomialPmf(k: number, r: number, p: number): number
+negativeBinomialCdf(k: number, r: number, p: number): number
+negativeBinomialRandom(r: number, p: number, size?: number[]): number | Matrix
+```
+
+#### Task 4.4: Implement Geometric Distribution
+**File**: `src/function/probability/distributions/geometric.ts`
+**Description**: Geometric distribution for waiting time to first success
+**Interface**:
+```typescript
+geometricPmf(k: number, p: number): number
+geometricCdf(k: number, p: number): number
+geometricRandom(p: number, size?: number[]): number | Matrix
+```
+
+#### Task 4.5: Implement Hypergeometric Distribution
+**File**: `src/function/probability/distributions/hypergeometric.ts`
+**Description**: Hypergeometric for sampling without replacement
+**Interface**:
+```typescript
+hypergeometricPmf(k: number, N: number, K: number, n: number): number
+hypergeometricCdf(k: number, N: number, K: number, n: number): number
+hypergeometricRandom(N: number, K: number, n: number, size?: number[]): number | Matrix
+```
+
+#### Task 4.6: Implement Uniform Distribution (Continuous)
+**File**: `src/function/probability/distributions/uniform.ts`
+**Description**: Continuous uniform distribution
+**Interface**:
+```typescript
+uniformPdf(x: number, a: number, b: number): number
+uniformCdf(x: number, a: number, b: number): number
+uniformInv(p: number, a: number, b: number): number
+uniformRandom(a: number, b: number, size?: number[]): number | Matrix
+```
+
+#### Task 4.7: Implement Cauchy Distribution
+**File**: `src/function/probability/distributions/cauchy.ts`
+**Description**: Cauchy/Lorentz distribution (heavy tails)
+**Interface**:
+```typescript
+cauchyPdf(x: number, x0: number, gamma: number): number
+cauchyCdf(x: number, x0: number, gamma: number): number
+cauchyInv(p: number, x0: number, gamma: number): number
+cauchyRandom(x0: number, gamma: number, size?: number[]): number | Matrix
+```
+
+#### Task 4.8: Implement Logistic Distribution
+**File**: `src/function/probability/distributions/logistic.ts`
+**Description**: Logistic distribution for growth modeling
+**Interface**:
+```typescript
+logisticPdf(x: number, mu: number, s: number): number
+logisticCdf(x: number, mu: number, s: number): number
+logisticInv(p: number, mu: number, s: number): number
+logisticRandom(mu: number, s: number, size?: number[]): number | Matrix
+```
+
+#### Task 4.9: Implement Incomplete Beta Function
+**File**: `src/function/special/betainc.ts`
+**Description**: Regularized incomplete beta function (needed for many distributions)
+**Interface**:
+```typescript
+betainc(a: number, b: number, x: number): number  // I_x(a, b)
+betaincinv(a: number, b: number, y: number): number  // Inverse
+```
+**Algorithms**: Continued fraction, series expansion
+
+#### Task 4.10: Implement Multivariate Normal Distribution
+**File**: `src/function/probability/distributions/mvnormal.ts`
+**Description**: Multivariate normal distribution for correlated random vectors
+**Interface**:
+```typescript
+mvnormalPdf(x: Vector, mean: Vector, cov: Matrix): number
+mvnormalRandom(mean: Vector, cov: Matrix, size?: number): Matrix
+```
+**Requires**: Cholesky decomposition (Sprint 1)
+
+---
+
+## Sprint 5: Root Finding & Optimization
+**Priority**: CRITICAL
+**Estimated Complexity**: High
+**Dependencies**: Sprint 1 (linear algebra for some methods)
+
+### Background
+Root finding and optimization are fundamental to scientific computing - used in solving equations, parameter estimation, machine learning, and engineering design.
+
+### Tasks
+
+#### Task 5.1: Implement Brent's Method (Root Finding)
+**File**: `src/function/numeric/roots/brentq.ts`
+**Description**: Brent's method for scalar root finding (bracketed)
+**Interface**:
+```typescript
+brentq(f: Function, a: number, b: number, options?: { tol?: number, maxIter?: number }): number
+```
+**Algorithm**: Combines bisection, secant, and inverse quadratic interpolation
+**Tests**: Polynomial roots, transcendental equations
+
+#### Task 5.2: Implement Newton-Raphson Method
+**File**: `src/function/numeric/roots/newton.ts`
+**Description**: Newton's method for root finding
+**Interface**:
+```typescript
+newton(f: Function, x0: number, options?: { fprime?: Function, tol?: number }): number
+```
+**Features**: Automatic differentiation if fprime not provided
+**Tests**: Convergence rate, starting point sensitivity
+
+#### Task 5.3: Implement fzero (General Root Finder)
+**File**: `src/function/numeric/fzero.ts`
+**Description**: General-purpose scalar root finding (like MATLAB's fzero)
+**Interface**:
+```typescript
+fzero(f: Function, x0: number | [number, number]): number
+```
+**Strategy**: Bracketing if interval given, otherwise combination of methods
+
+#### Task 5.4: Implement fsolve (Nonlinear System Solver)
+**File**: `src/function/numeric/fsolve.ts`
+**Description**: Solve system of nonlinear equations F(x) = 0
+**Interface**:
+```typescript
+fsolve(F: Function, x0: Vector, options?: { jacobian?: Function, tol?: number }): Vector
+```
+**Algorithms**: Newton-Raphson with line search, Broyden's method
+**Requires**: Linear system solver (lusolve)
+
+#### Task 5.5: Implement Golden Section Search
+**File**: `src/function/numeric/optimize/golden.ts`
+**Description**: Golden section search for 1D minimization (no derivatives)
+**Interface**:
+```typescript
+goldenSection(f: Function, a: number, b: number, options?: { tol?: number }): number
+```
+
+#### Task 5.6: Implement fminbnd (Bounded 1D Optimization)
+**File**: `src/function/numeric/fminbnd.ts`
+**Description**: Find minimum of single-variable function on bounded interval
+**Interface**:
+```typescript
+fminbnd(f: Function, a: number, b: number, options?: { tol?: number }): { x: number, fval: number }
+```
+**Algorithm**: Brent's method for optimization
+
+#### Task 5.7: Implement Nelder-Mead Simplex
+**File**: `src/function/numeric/optimize/neldermead.ts`
+**Description**: Derivative-free multidimensional optimization
+**Interface**:
+```typescript
+nelderMead(f: Function, x0: Vector, options?: NelderMeadOptions): OptimizeResult
+```
+**Tests**: Rosenbrock function, quadratic functions
+
+#### Task 5.8: Implement BFGS (Quasi-Newton)
+**File**: `src/function/numeric/optimize/bfgs.ts`
+**Description**: BFGS quasi-Newton method for unconstrained optimization
+**Interface**:
+```typescript
+bfgs(f: Function, x0: Vector, options?: { grad?: Function, tol?: number }): OptimizeResult
+```
+**Features**: Numerical gradient if not provided
+**Tests**: Quadratic convergence rate
+
+---
+
+## Sprint 6: Numerical Integration (Quadrature)
+**Priority**: HIGH
+**Estimated Complexity**: Medium
+**Dependencies**: None
+
+### Background
+Numerical integration is essential for computing areas, expected values, solving differential equations, and many scientific applications.
+
+### Tasks
+
+#### Task 6.1: Implement Trapezoidal Rule
+**File**: `src/function/numeric/integration/trapz.ts`
+**Description**: Trapezoidal rule for numerical integration
+**Interface**:
+```typescript
+trapz(y: Vector, x?: Vector): number  // Integrate y over x
+trapz(y: Matrix, dim?: number): Vector  // Along dimension
+cumtrapz(y: Vector, x?: Vector): Vector  // Cumulative
+```
+**Tests**: Compare with analytical integrals
+
+#### Task 6.2: Implement Simpson's Rule
+**File**: `src/function/numeric/integration/simps.ts`
+**Description**: Simpson's rule (higher accuracy than trapezoidal)
+**Interface**:
+```typescript
+simps(y: Vector, x?: Vector): number
+```
+**Requires**: Odd number of points (uses composite rule)
+
+#### Task 6.3: Implement Adaptive Quadrature (quad)
+**File**: `src/function/numeric/integration/quad.ts`
+**Description**: Adaptive quadrature for definite integrals
+**Interface**:
+```typescript
+quad(f: Function, a: number, b: number, options?: QuadOptions): { result: number, error: number }
+```
+**Algorithm**: Gauss-Kronrod adaptive
+**Features**: Error estimation, subdivision
+
+#### Task 6.4: Implement Gauss-Legendre Quadrature
+**File**: `src/function/numeric/integration/gausslegendre.ts`
+**Description**: Fixed-order Gauss-Legendre quadrature
+**Interface**:
+```typescript
+gausslegendre(f: Function, a: number, b: number, n?: number): number
+gausslegendreNodes(n: number): { nodes: Vector, weights: Vector }
+```
+**Tests**: Exact for polynomials up to degree 2n-1
+
+#### Task 6.5: Implement Romberg Integration
+**File**: `src/function/numeric/integration/romberg.ts`
+**Description**: Romberg integration using Richardson extrapolation
+**Interface**:
+```typescript
+romberg(f: Function, a: number, b: number, options?: { tol?: number }): number
+```
+
+#### Task 6.6: Implement Infinite Interval Integration
+**File**: `src/function/numeric/integration/quadinf.ts`
+**Description**: Integration over infinite intervals
+**Interface**:
+```typescript
+quadinf(f: Function, a: number, b: number): number  // a or b can be ±Infinity
+```
+**Algorithm**: Variable transformation + adaptive quadrature
+
+#### Task 6.7: Implement 2D Integration (dblquad)
+**File**: `src/function/numeric/integration/dblquad.ts`
+**Description**: Double integration over rectangular or general regions
+**Interface**:
+```typescript
+dblquad(f: Function, ax: number, bx: number, ay: number | Function, by: number | Function): number
+```
+
+#### Task 6.8: Implement Monte Carlo Integration
+**File**: `src/function/numeric/integration/montecarlo.ts`
+**Description**: Monte Carlo integration for high-dimensional integrals
+**Interface**:
+```typescript
+montecarloIntegrate(f: Function, bounds: Array<[number, number]>, n?: number): { result: number, error: number }
+```
+
+---
+
+## Sprint 7: Interpolation & Curve Fitting
+**Priority**: HIGH
+**Estimated Complexity**: Medium
+**Dependencies**: Sprint 1 (linear algebra for splines)
+
+### Tasks
+
+#### Task 7.1: Implement Linear Interpolation (interp1)
+**File**: `src/function/numeric/interpolate/interp1.ts`
+**Description**: 1D interpolation with multiple methods
+**Interface**:
+```typescript
+interp1(x: Vector, y: Vector, xq: number | Vector, method?: 'linear' | 'nearest' | 'spline' | 'pchip'): number | Vector
+```
+**Tests**: Exact at data points, boundary behavior
+
+#### Task 7.2: Implement Cubic Spline Interpolation
+**File**: `src/function/numeric/interpolate/spline.ts`
+**Description**: Natural cubic spline interpolation
+**Interface**:
+```typescript
+spline(x: Vector, y: Vector): SplineFunction
+// SplineFunction.evaluate(xq: number): number
+// SplineFunction.derivative(xq: number): number
+// SplineFunction.integral(a: number, b: number): number
+```
+**Algorithm**: Tridiagonal system for spline coefficients
+
+#### Task 7.3: Implement PCHIP (Monotonic Interpolation)
+**File**: `src/function/numeric/interpolate/pchip.ts`
+**Description**: Piecewise Cubic Hermite Interpolating Polynomial
+**Interface**:
+```typescript
+pchip(x: Vector, y: Vector): PchipFunction
+```
+**Advantage**: Preserves monotonicity, no overshooting
+
+#### Task 7.4: Implement 2D Interpolation (interp2)
+**File**: `src/function/numeric/interpolate/interp2.ts`
+**Description**: Bilinear and bicubic interpolation
+**Interface**:
+```typescript
+interp2(x: Vector, y: Vector, z: Matrix, xq: number, yq: number, method?: 'linear' | 'cubic'): number
+```
+
+#### Task 7.5: Implement Polynomial Fitting (polyfit)
+**File**: `src/function/numeric/fitting/polyfit.ts`
+**Description**: Least squares polynomial fitting
+**Interface**:
+```typescript
+polyfit(x: Vector, y: Vector, n: number): Vector  // Coefficients [a_n, ..., a_1, a_0]
+polyval(p: Vector, x: number | Vector): number | Vector  // Evaluate polynomial
+```
+**Uses**: Linear least squares
+
+#### Task 7.6: Implement General Curve Fitting
+**File**: `src/function/numeric/fitting/curvefit.ts`
+**Description**: Nonlinear least squares curve fitting
+**Interface**:
+```typescript
+curvefit(f: Function, x: Vector, y: Vector, p0: Vector, options?: CurveFitOptions): { params: Vector, residuals: Vector }
+```
+**Algorithm**: Levenberg-Marquardt
+
+#### Task 7.7: Implement Regression (Linear)
+**File**: `src/function/statistics/regression/linearRegression.ts`
+**Description**: Linear regression with statistics
+**Interface**:
+```typescript
+linearRegression(x: Vector | Matrix, y: Vector): {
+  coefficients: Vector,
+  rSquared: number,
+  standardErrors: Vector,
+  pValues: Vector
+}
+```
+
+#### Task 7.8: Implement Polynomial Root Finding (roots)
+**File**: `src/function/algebra/roots.ts`
+**Description**: Find all roots of a polynomial (companion matrix method)
+**Interface**:
+```typescript
+roots(coefficients: Vector): Vector<Complex>  // All roots including complex
+```
+**Algorithm**: Eigenvalues of companion matrix
+**Extends**: Existing polynomialRoot (limited to degree 3)
+
+---
+
+## Sprint 8: Elliptic & Hypergeometric Functions
+**Priority**: HIGH
+**Estimated Complexity**: High
+**Dependencies**: None (but builds on Sprint 2 patterns)
+
+### Background
+Elliptic functions are essential for:
+- Pendulum motion
+- Elliptical orbits
+- Electromagnetic field calculations
+- Cryptography (elliptic curves)
+
+Hypergeometric functions generalize many special functions.
+
+### Tasks
+
+#### Task 8.1: Implement Complete Elliptic Integral K
+**File**: `src/function/special/elliptic.ts`
+**Description**: Complete elliptic integral of the first kind K(m)
+**Interface**:
+```typescript
+ellipk(m: number): number  // K(m), m = k²
+```
+**Algorithm**: Arithmetic-geometric mean (AGM)
+**Tests**: Known values, limiting behaviors
+
+#### Task 8.2: Implement Complete Elliptic Integral E
+**File**: `src/function/special/elliptic.ts` (extension)
+**Description**: Complete elliptic integral of the second kind E(m)
+**Interface**:
+```typescript
+ellipe(m: number): number  // E(m)
+```
+
+#### Task 8.3: Implement Incomplete Elliptic Integrals
+**File**: `src/function/special/elliptic.ts` (extension)
+**Description**: Incomplete elliptic integrals F(φ, m) and E(φ, m)
+**Interface**:
+```typescript
+ellipkinc(phi: number, m: number): number  // F(φ, m)
+ellipeinc(phi: number, m: number): number  // E(φ, m)
+```
+**Algorithm**: Carlson's forms or Legendre reduction
+
+#### Task 8.4: Implement Elliptic Integral of Third Kind
+**File**: `src/function/special/elliptic.ts` (extension)
+**Description**: Π(n, φ, m) - third kind
+**Interface**:
+```typescript
+ellippi(n: number, phi: number, m: number): number
+```
+
+#### Task 8.5: Implement Jacobi Elliptic Functions
+**File**: `src/function/special/jacobi.ts`
+**Description**: Jacobi elliptic functions sn, cn, dn
+**Interface**:
+```typescript
+ellipj(u: number, m: number): { sn: number, cn: number, dn: number }
+jacobisn(u: number, m: number): number
+jacobicn(u: number, m: number): number
+jacobidn(u: number, m: number): number
+```
+**Algorithm**: Arithmetic-geometric mean descent
+
+#### Task 8.6: Implement Confluent Hypergeometric 1F1
+**File**: `src/function/special/hypergeometric.ts`
+**Description**: Kummer's confluent hypergeometric function M(a, b, z)
+**Interface**:
+```typescript
+hyp1f1(a: number, b: number, z: number): number  // ₁F₁(a; b; z)
+```
+**Algorithm**: Series expansion, asymptotic expansion, connection formulas
+
+#### Task 8.7: Implement Gauss Hypergeometric 2F1
+**File**: `src/function/special/hypergeometric.ts` (extension)
+**Description**: Gauss hypergeometric function F(a, b; c; z)
+**Interface**:
+```typescript
+hyp2f1(a: number, b: number, c: number, z: number): number  // ₂F₁(a, b; c; z)
+```
+**Algorithm**: Series for |z| < 1, analytic continuation otherwise
+**Importance**: Generalizes many special functions
+
+#### Task 8.8: Implement Tricomi's Function U
+**File**: `src/function/special/hypergeometric.ts` (extension)
+**Description**: Tricomi's confluent hypergeometric U(a, b, z)
+**Interface**:
+```typescript
+hypU(a: number, b: number, z: number): number
+```
+
+#### Task 8.9: Implement Generalized Hypergeometric pFq
+**File**: `src/function/special/hypergeometric.ts` (extension)
+**Description**: Generalized hypergeometric function
+**Interface**:
+```typescript
+hyppfq(a: number[], b: number[], z: number): number  // pFq
+```
+
+#### Task 8.10: Implement Carlson Elliptic Integrals
+**File**: `src/function/special/carlson.ts`
+**Description**: Carlson symmetric forms RF, RD, RJ, RC
+**Interface**:
+```typescript
+carlsonRF(x: number, y: number, z: number): number
+carlsonRD(x: number, y: number, z: number): number
+carlsonRJ(x: number, y: number, z: number, p: number): number
+carlsonRC(x: number, y: number): number
+```
+**Use**: More numerically stable for elliptic integrals
+
+---
+
+## Sprint 9: Advanced Statistics
+**Priority**: HIGH
+**Estimated Complexity**: Medium
+**Dependencies**: Sprint 3-4 (probability distributions)
+
+### Tasks
+
+#### Task 9.1: Implement T-Test
+**File**: `src/function/statistics/hypothesis/ttest.ts`
+**Description**: Student's t-test for hypothesis testing
+**Interface**:
+```typescript
+ttest(x: Vector, mu?: number): { t: number, pValue: number, ci: [number, number] }  // One-sample
+ttest2(x: Vector, y: Vector, options?: { paired?: boolean }): TTestResult  // Two-sample
+```
+
+#### Task 9.2: Implement Chi-Square Test
+**File**: `src/function/statistics/hypothesis/chi2test.ts`
+**Description**: Chi-square goodness of fit and independence tests
+**Interface**:
+```typescript
+chi2gof(observed: Vector, expected?: Vector): { chi2: number, pValue: number, df: number }
+chi2test(contingencyTable: Matrix): { chi2: number, pValue: number }
+```
+
+#### Task 9.3: Implement F-Test
+**File**: `src/function/statistics/hypothesis/ftest.ts`
+**Description**: F-test for variance equality
+**Interface**:
+```typescript
+ftest(x: Vector, y: Vector): { f: number, pValue: number }
+```
+
+#### Task 9.4: Implement ANOVA
+**File**: `src/function/statistics/anova.ts`
+**Description**: One-way and two-way ANOVA
+**Interface**:
+```typescript
+anova1(groups: Vector[]): AnovaResult
+anova2(data: Matrix, factorA: Vector, factorB: Vector): Anova2Result
+```
+
+#### Task 9.5: Implement Kolmogorov-Smirnov Test
+**File**: `src/function/statistics/hypothesis/kstest.ts`
+**Description**: KS test for distribution comparison
+**Interface**:
+```typescript
+kstest(x: Vector, cdf: Function): { d: number, pValue: number }
+kstest2(x: Vector, y: Vector): { d: number, pValue: number }  // Two-sample
+```
+
+#### Task 9.6: Implement Skewness and Kurtosis
+**File**: `src/function/statistics/moments.ts`
+**Description**: Higher-order moments
+**Interface**:
+```typescript
+skewness(x: Vector): number
+kurtosis(x: Vector): number
+moment(x: Vector, n: number): number  // nth central moment
+```
+
+#### Task 9.7: Implement Covariance Matrix
+**File**: `src/function/statistics/cov.ts`
+**Description**: Covariance and correlation matrices
+**Interface**:
+```typescript
+cov(X: Matrix): Matrix  // Sample covariance
+corrcoef(X: Matrix): Matrix  // Correlation matrix
+```
+
+#### Task 9.8: Implement Bootstrap Methods
+**File**: `src/function/statistics/bootstrap.ts`
+**Description**: Bootstrap confidence intervals and standard errors
+**Interface**:
+```typescript
+bootstrap(data: Vector, statistic: Function, n?: number): BootstrapResult
+```
+
+#### Task 9.9: Implement Kernel Density Estimation
+**File**: `src/function/statistics/kde.ts`
+**Description**: Nonparametric density estimation
+**Interface**:
+```typescript
+kde(data: Vector, bandwidth?: number): KDEFunction
+// KDEFunction.evaluate(x: number): number
+// KDEFunction.pdf(x: Vector): Vector
+```
+
+#### Task 9.10: Implement Quantile and Percentile
+**File**: `src/function/statistics/quantile.ts`
+**Description**: Enhanced quantile computation (extend quantileSeq)
+**Interface**:
+```typescript
+quantile(data: Vector, q: number | Vector, method?: number): number | Vector
+percentile(data: Vector, p: number | Vector): number | Vector
+iqr(data: Vector): number  // Interquartile range
+```
+
+---
+
+## Sprint 10: Signal Processing Filters
+**Priority**: MEDIUM-HIGH
+**Estimated Complexity**: Medium
+**Dependencies**: Sprint 1 (linear algebra), existing fft/ifft
+
+### Tasks
+
+#### Task 10.1: Implement Butterworth Filter Design
+**File**: `src/function/signal/filters/butter.ts`
+**Description**: Butterworth IIR filter design
+**Interface**:
+```typescript
+butter(n: number, Wn: number | [number, number], btype?: 'low' | 'high' | 'band' | 'stop'): { b: Vector, a: Vector }
+```
+**Tests**: Maximally flat passband
+
+#### Task 10.2: Implement Chebyshev Type I Filter
+**File**: `src/function/signal/filters/cheby1.ts`
+**Description**: Chebyshev Type I filter design
+**Interface**:
+```typescript
+cheby1(n: number, rp: number, Wn: number | [number, number], btype?: string): { b: Vector, a: Vector }
+```
+**rp**: Passband ripple in dB
+
+#### Task 10.3: Implement Chebyshev Type II Filter
+**File**: `src/function/signal/filters/cheby2.ts`
+**Description**: Chebyshev Type II filter design
+**Interface**:
+```typescript
+cheby2(n: number, rs: number, Wn: number | [number, number], btype?: string): { b: Vector, a: Vector }
+```
+**rs**: Stopband attenuation in dB
+
+#### Task 10.4: Implement Elliptic Filter
+**File**: `src/function/signal/filters/ellip.ts`
+**Description**: Elliptic (Cauer) filter design
+**Interface**:
+```typescript
+ellip(n: number, rp: number, rs: number, Wn: number | [number, number], btype?: string): { b: Vector, a: Vector }
+```
+**Dependencies**: Elliptic functions (Sprint 8)
+
+#### Task 10.5: Implement FIR Filter Design (fir1)
+**File**: `src/function/signal/filters/fir1.ts`
+**Description**: Window-based FIR filter design
+**Interface**:
+```typescript
+fir1(n: number, Wn: number | [number, number], ftype?: string, window?: string | Vector): Vector
+```
+
+#### Task 10.6: Implement filter Function
+**File**: `src/function/signal/filter.ts`
+**Description**: Apply digital filter to data
+**Interface**:
+```typescript
+filter(b: Vector, a: Vector, x: Vector): Vector
+filtfilt(b: Vector, a: Vector, x: Vector): Vector  // Zero-phase filtering
+```
+
+#### Task 10.7: Implement Convolution
+**File**: `src/function/signal/conv.ts`
+**Description**: 1D and 2D convolution
+**Interface**:
+```typescript
+conv(u: Vector, v: Vector, shape?: 'full' | 'same' | 'valid'): Vector
+conv2(A: Matrix, K: Matrix, shape?: 'full' | 'same' | 'valid'): Matrix
+```
+**WASM**: Yes - use FFT-based convolution for large arrays
+
+#### Task 10.8: Implement Correlation
+**File**: `src/function/signal/xcorr.ts`
+**Description**: Cross-correlation
+**Interface**:
+```typescript
+xcorr(x: Vector, y?: Vector, maxlag?: number): { c: Vector, lags: Vector }
+```
+
+#### Task 10.9: Implement Second-Order Sections
+**File**: `src/function/signal/sos.ts`
+**Description**: Second-order sections filter representation
+**Interface**:
+```typescript
+tf2sos(b: Vector, a: Vector): Matrix  // Convert to SOS
+sos2tf(sos: Matrix): { b: Vector, a: Vector }
+sosfilt(sos: Matrix, x: Vector): Vector  // Filter using SOS
+```
+**Advantage**: Better numerical stability than tf
+
+#### Task 10.10: Implement Transfer Function Conversions
+**File**: `src/function/signal/tfconvert.ts`
+**Description**: Convert between filter representations
+**Interface**:
+```typescript
+tf2ss(b: Vector, a: Vector): { A: Matrix, B: Matrix, C: Matrix, D: Matrix }
+ss2tf(A: Matrix, B: Matrix, C: Matrix, D: Matrix): { b: Vector, a: Vector }
+zpk2sos(z: Vector, p: Vector, k: number): Matrix
+```
+
+---
+
+## Sprint 11: Window Functions & Spectral Analysis
+**Priority**: MEDIUM-HIGH
+**Estimated Complexity**: Low
+**Dependencies**: Sprint 10
+
+### Tasks
+
+#### Task 11.1: Implement Hamming Window
+**File**: `src/function/signal/windows/hamming.ts`
+**Description**: Hamming window function
+**Interface**:
+```typescript
+hamming(n: number, symmetric?: boolean): Vector
+```
+
+#### Task 11.2: Implement Hanning Window
+**File**: `src/function/signal/windows/hanning.ts`
+**Description**: Hanning (Hann) window function
+**Interface**:
+```typescript
+hanning(n: number, symmetric?: boolean): Vector
+```
+
+#### Task 11.3: Implement Blackman Window
+**File**: `src/function/signal/windows/blackman.ts`
+**Description**: Blackman window function
+**Interface**:
+```typescript
+blackman(n: number, symmetric?: boolean): Vector
+```
+
+#### Task 11.4: Implement Kaiser Window
+**File**: `src/function/signal/windows/kaiser.ts`
+**Description**: Kaiser window with adjustable parameter
+**Interface**:
+```typescript
+kaiser(n: number, beta: number): Vector
+```
+**Dependencies**: Bessel I0 (Sprint 2)
+
+#### Task 11.5: Implement Additional Windows
+**File**: `src/function/signal/windows/windows.ts`
+**Description**: Collection of additional windows
+**Interface**:
+```typescript
+bartlett(n: number): Vector
+triang(n: number): Vector
+boxcar(n: number): Vector
+tukey(n: number, alpha?: number): Vector
+parzen(n: number): Vector
+```
+
+#### Task 11.6: Implement Periodogram
+**File**: `src/function/signal/spectral/periodogram.ts`
+**Description**: Power spectral density estimation
+**Interface**:
+```typescript
+periodogram(x: Vector, fs?: number, window?: Vector): { psd: Vector, f: Vector }
+```
+
+#### Task 11.7: Implement Welch's Method
+**File**: `src/function/signal/spectral/pwelch.ts`
+**Description**: Welch's method for PSD estimation
+**Interface**:
+```typescript
+pwelch(x: Vector, window?: number | Vector, noverlap?: number, nfft?: number, fs?: number): { psd: Vector, f: Vector }
+```
+
+#### Task 11.8: Implement Short-Time Fourier Transform
+**File**: `src/function/signal/spectral/stft.ts`
+**Description**: STFT for time-frequency analysis
+**Interface**:
+```typescript
+stft(x: Vector, window: number | Vector, noverlap?: number, nfft?: number): { S: Matrix, f: Vector, t: Vector }
+istft(S: Matrix, window: number | Vector, noverlap?: number): Vector
+```
+
+---
+
+## Sprint 12: ODE/PDE Enhancements
+**Priority**: MEDIUM
+**Estimated Complexity**: High
+**Dependencies**: Sprint 1 (linear algebra)
+
+### Tasks
+
+#### Task 12.1: Implement Backward Differentiation Formula (BDF)
+**File**: `src/function/numeric/ode/bdf.ts`
+**Description**: BDF method for stiff ODEs
+**Interface**:
+```typescript
+solveBDF(odefun: Function, tspan: [number, number], y0: Vector, options?: BDFOptions): ODESolution
+```
+**Use**: Stiff systems where RK methods fail
+
+#### Task 12.2: Implement Adams-Bashforth-Moulton
+**File**: `src/function/numeric/ode/adams.ts`
+**Description**: Adams-Bashforth-Moulton predictor-corrector
+**Interface**:
+```typescript
+solveABM(odefun: Function, tspan: [number, number], y0: Vector, options?: ABMOptions): ODESolution
+```
+
+#### Task 12.3: Implement Event Detection
+**File**: `src/function/numeric/ode/events.ts`
+**Description**: Event detection and handling for ODEs
+**Interface**:
+```typescript
+// Extended solveODE options
+{
+  events: (t: number, y: Vector) => number[],  // Zero-crossing functions
+  terminal: boolean[],  // Stop on event?
+  direction: number[]   // Direction of crossing
+}
+```
+
+#### Task 12.4: Implement DAE Solver
+**File**: `src/function/numeric/dae/solveDAE.ts`
+**Description**: Differential-algebraic equation solver
+**Interface**:
+```typescript
+solveDAE(F: Function, tspan: [number, number], y0: Vector, yp0: Vector): DAESolution
+```
+**Algorithm**: BDF-based method
+
+#### Task 12.5: Implement Boundary Value Problem Solver
+**File**: `src/function/numeric/bvp/bvp4c.ts`
+**Description**: Solve two-point BVPs
+**Interface**:
+```typescript
+bvp4c(odefun: Function, bcfun: Function, solinit: BVPInit): BVPSolution
+```
+**Algorithm**: Collocation method
+
+#### Task 12.6: Implement Finite Difference PDE Solver
+**File**: `src/function/numeric/pde/parabolic.ts`
+**Description**: Solve parabolic PDEs (heat equation type)
+**Interface**:
+```typescript
+pdepe(m: number, pdefun: Function, icfun: Function, bcfun: Function, xmesh: Vector, tspan: Vector): PDESolution
+```
+
+#### Task 12.7: Implement Method of Lines
+**File**: `src/function/numeric/pde/mol.ts`
+**Description**: Method of lines for PDE → ODE conversion
+**Interface**:
+```typescript
+methodOfLines(pde: PDESpec, xmesh: Vector): (t: number, y: Vector) => Vector  // Returns ODE function
+```
+
+#### Task 12.8: Implement Delay Differential Equations
+**File**: `src/function/numeric/dde/solveDDE.ts`
+**Description**: Solve DDEs with constant delays
+**Interface**:
+```typescript
+solveDDE(ddefun: Function, delays: Vector, tspan: [number, number], history: Function): DDESolution
+```
+
+---
+
+## Sprint 13: Orthogonal Polynomials
+**Priority**: MEDIUM
+**Estimated Complexity**: Medium
+**Dependencies**: None
+
+### Tasks
+
+#### Task 13.1: Implement Legendre Polynomials
+**File**: `src/function/special/polynomials/legendre.ts`
+**Description**: Legendre polynomials P_n(x)
+**Interface**:
+```typescript
+legendre(n: number, x: number): number  // P_n(x)
+legendreP(n: number, x: number): number  // Same as legendre
+assocLegendre(l: number, m: number, x: number): number  // Associated P_l^m(x)
+```
+**Tests**: Orthogonality, recurrence relations
+
+#### Task 13.2: Implement Hermite Polynomials
+**File**: `src/function/special/polynomials/hermite.ts`
+**Description**: Hermite polynomials (physicist's and probabilist's)
+**Interface**:
+```typescript
+hermite(n: number, x: number): number  // Physicist's H_n(x)
+hermiteProb(n: number, x: number): number  // Probabilist's He_n(x)
+```
+**Use**: Quantum harmonic oscillator, Gaussian quadrature
+
+#### Task 13.3: Implement Laguerre Polynomials
+**File**: `src/function/special/polynomials/laguerre.ts`
+**Description**: Laguerre polynomials L_n(x)
+**Interface**:
+```typescript
+laguerre(n: number, x: number): number  // L_n(x)
+assocLaguerre(n: number, alpha: number, x: number): number  // L_n^α(x)
+```
+**Use**: Hydrogen atom wavefunctions
+
+#### Task 13.4: Implement Chebyshev Polynomials
+**File**: `src/function/special/polynomials/chebyshev.ts`
+**Description**: Chebyshev polynomials T_n(x) and U_n(x)
+**Interface**:
+```typescript
+chebyshevT(n: number, x: number): number  // First kind
+chebyshevU(n: number, x: number): number  // Second kind
+```
+**Use**: Numerical approximation, spectral methods
+
+#### Task 13.5: Implement Jacobi Polynomials
+**File**: `src/function/special/polynomials/jacobi.ts`
+**Description**: Jacobi polynomials P_n^(α,β)(x)
+**Interface**:
+```typescript
+jacobi(n: number, alpha: number, beta: number, x: number): number
+```
+**Note**: Generalizes Legendre, Chebyshev, Gegenbauer
+
+#### Task 13.6: Implement Gegenbauer Polynomials
+**File**: `src/function/special/polynomials/gegenbauer.ts`
+**Description**: Gegenbauer (ultraspherical) polynomials
+**Interface**:
+```typescript
+gegenbauer(n: number, alpha: number, x: number): number  // C_n^α(x)
+```
+
+#### Task 13.7: Implement Spherical Harmonics
+**File**: `src/function/special/sphericalharmonics.ts`
+**Description**: Spherical harmonics Y_l^m(θ, φ)
+**Interface**:
+```typescript
+sphericalHarmonic(l: number, m: number, theta: number, phi: number): Complex
+sphericalHarmonicReal(l: number, m: number, theta: number, phi: number): number
+```
+**Dependencies**: Associated Legendre (Task 13.1)
+
+#### Task 13.8: Implement Polynomial Roots and Weights
+**File**: `src/function/special/polynomials/quadrature.ts`
+**Description**: Compute quadrature nodes and weights from orthogonal polynomials
+**Interface**:
+```typescript
+legendreRoots(n: number): { roots: Vector, weights: Vector }
+hermiteRoots(n: number): { roots: Vector, weights: Vector }
+laguerreRoots(n: number): { roots: Vector, weights: Vector }
+chebyshevRoots(n: number): { roots: Vector, weights: Vector }
+```
+**Use**: Gaussian quadrature
+
+---
+
+## Sprint 14: Additional Special Functions
+**Priority**: MEDIUM
+**Estimated Complexity**: Medium
+**Dependencies**: Sprints 2, 8
+
+### Tasks
+
+#### Task 14.1: Implement erfc, erfinv, erfcinv
+**File**: `src/function/special/erf.ts` (extension)
+**Description**: Complementary error function and inverses
+**Interface**:
+```typescript
+erfc(x: number): number  // 1 - erf(x)
+erfinv(y: number): number  // Inverse of erf
+erfcinv(y: number): number  // Inverse of erfc
+```
+**Algorithm**: Rational approximation for inverses
+
+#### Task 14.2: Implement Digamma and Polygamma
+**File**: `src/function/special/digamma.ts`
+**Description**: Psi function and derivatives
+**Interface**:
+```typescript
+digamma(x: number): number  // ψ(x) = Γ'(x)/Γ(x)
+trigamma(x: number): number  // ψ'(x)
+polygamma(n: number, x: number): number  // ψ^(n)(x)
+```
+
+#### Task 14.3: Implement Lambert W Function
+**File**: `src/function/special/lambertw.ts`
+**Description**: Lambert W function W(z) where W(z)*e^W(z) = z
+**Interface**:
+```typescript
+lambertw(z: number, k?: number): number  // Branch k (default 0)
+```
+**Algorithm**: Halley's method
+
+#### Task 14.4: Implement Exponential Integrals
+**File**: `src/function/special/expint.ts`
+**Description**: Exponential integral functions
+**Interface**:
+```typescript
+expint(n: number, x: number): number  // E_n(x)
+ei(x: number): number  // Ei(x)
+li(x: number): number  // Logarithmic integral
+```
+
+#### Task 14.5: Implement Fresnel Integrals
+**File**: `src/function/special/fresnel.ts`
+**Description**: Fresnel S and C integrals
+**Interface**:
+```typescript
+fresnelS(x: number): number  // S(x) = ∫sin(πt²/2)dt
+fresnelC(x: number): number  // C(x) = ∫cos(πt²/2)dt
+fresnel(x: number): { S: number, C: number }
+```
+
+#### Task 14.6: Implement Sine and Cosine Integrals
+**File**: `src/function/special/trig_integrals.ts`
+**Description**: Si, Ci, Shi, Chi integrals
+**Interface**:
+```typescript
+si(x: number): number  // Sine integral
+ci(x: number): number  // Cosine integral
+shi(x: number): number  // Hyperbolic sine integral
+chi(x: number): number  // Hyperbolic cosine integral
+```
+
+#### Task 14.7: Implement Struve Functions
+**File**: `src/function/special/struve.ts`
+**Description**: Struve functions H_n(x) and L_n(x)
+**Interface**:
+```typescript
+struve(n: number, x: number): number  // H_n(x)
+struveL(n: number, x: number): number  // Modified L_n(x)
+```
+
+#### Task 14.8: Implement Polylogarithm
+**File**: `src/function/special/polylog.ts`
+**Description**: Polylogarithm function Li_s(z)
+**Interface**:
+```typescript
+polylog(s: number, z: number): number  // Li_s(z)
+```
+
+#### Task 14.9: Implement Dawson Function
+**File**: `src/function/special/dawson.ts`
+**Description**: Dawson's integral D(x)
+**Interface**:
+```typescript
+dawson(x: number): number  // D(x) = e^(-x²) ∫₀ˣ e^(t²) dt
+```
+
+#### Task 14.10: Implement Owen's T Function
+**File**: `src/function/special/owent.ts`
+**Description**: Owen's T function for bivariate normal
+**Interface**:
+```typescript
+owenT(h: number, a: number): number
+```
+
+---
+
+## Sprint 15: Symbolic Computation Enhancements
+**Priority**: MEDIUM
+**Estimated Complexity**: High
+**Dependencies**: Existing expression system
+
+### Tasks
+
+#### Task 15.1: Implement Symbolic Integration
+**File**: `src/function/algebra/integrate.ts`
+**Description**: Symbolic integration (antiderivatives)
+**Interface**:
+```typescript
+integrate(expr: Node | string, variable: string): Node
+```
+**Scope**: Polynomials, basic trig, exponentials, logarithms
+**Algorithm**: Pattern matching, Risch algorithm subset
+
+#### Task 15.2: Implement Limits
+**File**: `src/function/algebra/limit.ts`
+**Description**: Compute limits symbolically
+**Interface**:
+```typescript
+limit(expr: Node | string, variable: string, value: number | string, direction?: '+' | '-'): Node
+```
+**Handles**: L'Hôpital's rule, indeterminate forms
+
+#### Task 15.3: Implement Taylor/Maclaurin Series
+**File**: `src/function/algebra/taylor.ts`
+**Description**: Taylor series expansion
+**Interface**:
+```typescript
+taylor(expr: Node | string, variable: string, point: number, order: number): Node
+```
+**Output**: Polynomial approximation
+
+#### Task 15.4: Implement Symbolic Summation
+**File**: `src/function/algebra/symbolicSum.ts`
+**Description**: Simplify symbolic sums
+**Interface**:
+```typescript
+symbolicSum(expr: Node | string, variable: string, lower: number, upper: number | string): Node
+```
+**Handles**: Arithmetic/geometric series, common summation identities
+
+#### Task 15.5: Implement Equation Solver (solve)
+**File**: `src/function/algebra/solve.ts`
+**Description**: Solve equations symbolically
+**Interface**:
+```typescript
+solve(equation: Node | string, variable: string): Node[]  // All solutions
+solveSystem(equations: Array<Node | string>, variables: string[]): Object[]
+```
+**Scope**: Linear, quadratic, polynomial, some transcendental
+
+#### Task 15.6: Implement Polynomial Factorization
+**File**: `src/function/algebra/factor.ts`
+**Description**: Factor polynomials
+**Interface**:
+```typescript
+factor(expr: Node | string): Node
+factorList(expr: Node | string): Array<{ factor: Node, exponent: number }>
+```
+**Algorithm**: Integer factorization, rational root theorem
+
+#### Task 15.7: Implement Partial Fractions
+**File**: `src/function/algebra/partialFractions.ts`
+**Description**: Partial fraction decomposition
+**Interface**:
+```typescript
+partialFractions(expr: Node | string, variable: string): Node
+```
+**Use**: Integration of rational functions
+
+#### Task 15.8: Implement Expression Expansion
+**File**: `src/function/algebra/expand.ts`
+**Description**: Expand expressions (opposite of factor)
+**Interface**:
+```typescript
+expand(expr: Node | string): Node
+expandAll(expr: Node | string): Node  // Recursive expansion
+```
+
+---
+
+## Implementation Guidelines
+
+### Code Standards
+
+1. **TypeScript First**: All new code should be in TypeScript
+2. **Factory Pattern**: Use mathjs factory pattern for all functions
+3. **Typed Functions**: Support multiple numeric types (number, BigNumber, Complex, Matrix)
+4. **Pure Functions**: Implement core algorithms as pure functions in `src/plain/`
+5. **WASM**: Add WASM implementations for computationally intensive functions
+
+### File Structure Template
+
+```typescript
+// src/function/category/functionName.ts
+
+import { factory } from '../../utils/factory.js'
+import { functionNameNumber } from '../../plain/number/functionName.js'
+
+const name = 'functionName'
+const dependencies = ['typed', 'config', /* other deps */]
+
+export const createFunctionName = factory(name, dependencies, ({ typed, config }) => {
+  /**
+   * Description of the function.
+   *
+   * Syntax:
+   *    math.functionName(param1, param2)
+   *
+   * Examples:
+   *    math.functionName(1, 2)  // returns X
+   *
+   * @param {number | BigNumber | Complex | Matrix} param1  Description
+   * @param {number} param2  Description
+   * @return {number | BigNumber | Complex | Matrix} Description
+   */
+  return typed(name, {
+    'number': functionNameNumber,
+    'BigNumber': (x) => /* BigNumber implementation */,
+    'Complex': (x) => /* Complex implementation */,
+    'Array | Matrix': typed.referToSelf(self => (x) => deepMap(x, self, true))
+  })
+})
+```
+
+### Testing Template
+
+```javascript
+// test/unit-tests/function/category/functionName.test.js
+
+import assert from 'assert'
+import math from '../../../../src/defaultInstance.js'
+
+const { functionName, matrix, complex, bignumber } = math
+
+describe('functionName', function () {
+  it('should compute correct result for numbers', function () {
+    assert.strictEqual(functionName(1, 2), expectedResult)
+  })
+
+  it('should work with matrices', function () {
+    const result = functionName(matrix([[1, 2], [3, 4]]))
+    assert.deepStrictEqual(result.toArray(), [[...], [...]])
+  })
+
+  it('should throw for invalid input', function () {
+    assert.throws(function () { functionName('invalid') }, /TypeError/)
+  })
+})
+```
+
+### WASM Template
+
+```typescript
+// src-wasm/category/functionName.ts
+
+export function functionName(x: f64): f64 {
+  // High-performance implementation
+  return result
+}
+
+// For array operations
+export function functionNameArray(
+  input: Float64Array,
+  output: Float64Array,
+  length: i32
+): void {
+  for (let i: i32 = 0; i < length; i++) {
+    output[i] = functionName(input[i])
+  }
+}
+```
+
+---
+
+## Priority Matrix
+
+| Sprint | Functions Added | Cumulative | Scientific Computing % |
+|--------|----------------|------------|------------------------|
+| Current | 214 | 214 | 35% |
+| Sprint 1 | 8 | 222 | 40% |
+| Sprint 2 | 10 | 232 | 45% |
+| Sprint 3 | 10 | 242 | 50% |
+| Sprint 4 | 10 | 252 | 55% |
+| Sprint 5 | 8 | 260 | 60% |
+| Sprint 6 | 8 | 268 | 65% |
+| Sprint 7 | 8 | 276 | 70% |
+| Sprint 8 | 10 | 286 | 75% |
+| Sprint 9 | 10 | 296 | 80% |
+| Sprint 10 | 10 | 306 | 83% |
+| Sprint 11 | 8 | 314 | 86% |
+| Sprint 12 | 8 | 322 | 89% |
+| Sprint 13 | 8 | 330 | 92% |
+| Sprint 14 | 10 | 340 | 95% |
+| Sprint 15 | 8 | 348 | 97% |
+
+---
+
+## Success Criteria
+
+### Per-Sprint Completion Requirements
+
+1. **All tasks implemented** with TypeScript source files
+2. **Unit tests** achieving >90% code coverage
+3. **Documentation** in JSDoc format
+4. **Type definitions** updated in `types/index.d.ts`
+5. **Factory registration** in `src/factoriesAny.js`
+6. **Performance benchmarks** for WASM-accelerated functions
+
+### Overall Project Success
+
+1. **Scientific Computing Parity**: 95%+ feature coverage vs SciPy
+2. **Performance**: WASM acceleration for matrix operations and special functions
+3. **Type Safety**: Full TypeScript coverage
+4. **Test Coverage**: 90%+ overall
+5. **Documentation**: Complete API documentation for all new functions
+
+---
+
+## Appendix: Function Reference by Domain
+
+### A. Linear Algebra (Sprint 1, 12)
+- SVD, Cholesky, polar, rank, nullspace, orth, lstsq
+- BVP solvers, DAE solvers
+
+### B. Special Functions (Sprints 2, 8, 13, 14)
+- Bessel: j0, j1, jn, y0, y1, yn, i0, i1, in, k0, k1, kn, hankel1, hankel2
+- Airy: ai, aip, bi, bip
+- Elliptic: ellipk, ellipe, ellipkinc, ellipeinc, ellippi, jacobi sn/cn/dn
+- Hypergeometric: hyp1f1, hyp2f1, hypU, hyppfq
+- Orthogonal: legendre, hermite, laguerre, chebyshev, jacobi, gegenbauer
+- Other: digamma, polygamma, lambertw, expint, fresnel, struve, polylog
+
+### C. Probability & Statistics (Sprints 3, 4, 9)
+- Distributions: normal, t, chi2, F, beta, gamma, exponential, weibull, lognormal, poisson, binomial, etc.
+- Tests: ttest, chi2test, ftest, kstest, anova
+- Descriptive: skewness, kurtosis, cov, corrcoef, kde, bootstrap
+
+### D. Numerical Methods (Sprints 5, 6, 7)
+- Root finding: brentq, newton, fzero, fsolve
+- Optimization: goldenSection, fminbnd, nelderMead, bfgs
+- Integration: trapz, simps, quad, romberg, gausslegendre, dblquad
+- Interpolation: interp1, spline, pchip, interp2, polyfit, curvefit
+
+### E. Signal Processing (Sprints 10, 11)
+- Filters: butter, cheby1, cheby2, ellip, fir1, filter, filtfilt
+- Windows: hamming, hanning, blackman, kaiser, bartlett
+- Spectral: periodogram, pwelch, stft, conv, xcorr
+
+### F. Differential Equations (Sprint 12)
+- ODE: BDF, Adams-Bashforth, event detection
+- DAE, BVP, DDE solvers
+- PDE: pdepe, method of lines
+
+### G. Symbolic (Sprint 15)
+- integrate, limit, taylor, symbolicSum, solve, factor, partialFractions, expand
+
+---
+
+*Document Version: 1.0*
+*Created: 2025-11-28*
+*Target Completion: 15 sprints (150 days with 10 days/sprint)*