Special functions

2014-07-25 — 2016-12-21

Wherein a wunderkammer of useful curves is presented, and Bessel, Gamma, Erf and orthogonal polynomials are catalogued with applications to function approximation and probabilistic integrals.

physics

probability

statistics

A wunderkammer of useful curves.

Figure 1: Otto Lueger, 1904, Lexikon der gesamten Technik (dictionary of technology).

Chaitanya Rao, Some special functions and their applications: Bessel and related functions (Hankel, Airy), Hermite, Laguerre, Chebyshev, Legendre polynomials, various fun integrals such as Fresnel integrals, Erf and so on, as made popular by all of functional analysis and its application in function approximation.

1 Erf

See Gaussian distribution

2 Gamma function

A way to discover the Gamma function
John Cook’s Gamma function identities

3 Bessel functions

Bessel function

4 t-density

Shoichi Midorikawa’s Derivation of the t-Distribution
John Cook’s Student-t as a mixture of Normals