Special Functions: From the Factorial to Orthogonal Polynomials

The gamma function extends the factorial, defined only for whole numbers, to almost the whole complex plane while keeping the rule that its value at n plus one equals n times its value at n. Euler introduced it in the eighteenth century, and Legendre later supplied the standard capital gamma notation.

The beta function is closely related to the gamma function and to binomial coefficients, and is defined by a definite integral of two inputs. Euler studied it around 1730, and Jacques Binet gave it its name and its capital beta symbol.

Riemann extended the zeta function to complex variables in his 1859 paper and established its functional equation. Its Euler product over the primes ties it to the distribution of prime numbers, and the Riemann Hypothesis about its non-trivial zeros remains unsolved.

The error function is a nonelementary, sigmoid-shaped function that occurs often in probability, statistics and partial differential equations. Glaisher gave it its name in 1871 because of its connection with the theory of errors.

Bessel functions solve problems with circular or cylindrical symmetry, such as wave motion and heat conduction. They are named after Friedrich Bessel, who studied them systematically in 1824, and come in kinds that are either finite or infinite at the origin.

The Airy function is named after George Biddell Airy, who met it in his early study of optics and wrote about the intensity of light near a caustic in 1838. It is oscillatory for negative x and exponential for positive x, which makes its equation the simplest second order linear equation with a turning point.

Adrien-Marie Legendre introduced these polynomials in 1782 as coefficients in the expansion of the Newtonian potential. They form a complete system of orthogonal polynomials, which is why they recur in physical and numerical applications.

The Chebyshev polynomials come in two families, written T and U, that are tied to the cosine and sine functions. Their roots, called Chebyshev nodes, are used for polynomial interpolation that reduces Runge phenomenon, and Cornelius Lanczos showed their importance in approximation theory in 1952.