diff --git a/docs/src/index.md b/docs/src/index.md index 6236314..4267595 100644 --- a/docs/src/index.md +++ b/docs/src/index.md @@ -17,6 +17,17 @@ which defines the following classical orthogonal polynomials: 5. Laguerre: `L_n^{(α)}(x)` 6. Hermite: `H_n(x)` +These special polynomials have many applications and can be used as a basis for any function given their domain conditions are met, however these polynomials have some advantages due to their formulation: + +- Because of their relation to Laplace’s equation, **Legendre polynomials** can be useful as a basis for functions with spherical symmetry. +- **Chebyshev polynomials** are generally effective in reducing errors from numerical methods such as quadrature, interpolation, and approximation. +- Due to the flexibility of its parameters, **Jacobi polynomials** are capable of tailoring the behavior of an approximation around its endpoints, making these polynomials particularly useful in boundary value problems. +- **Ultraspherical polynomials** are advantageous in spectral methods for solving differential equations. +- **Laguerre polynomials** have a semi-infinite domain, therefore they are beneficial for problems involving exponential decay. +- Because of its weight function, **Hermite polynomials** can be useful in situations where functions display a Gaussian-like distribution. + +These are just a few applications of these polynomials. They have many more uses across mathematics, physics, and engineering. + ## Evaluation The simplest usage of this package is to evaluate classical