hermite

Hermite refers most often to Charles Hermite (1822–1901), a French mathematician whose work spanned analysis, number theory, and the theory of elliptic functions. Hermite made early foundational contributions to transcendence theory, proving in 1873 that the number e is transcendental; his methods influenced later results culminating in the Hermite–Lindemann theorem that e^α is transcendental for any nonzero algebraic α.

Several mathematical objects bear his name. Hermite polynomials H_n(x) form a family of orthogonal polynomials defined by the generating function exp(2xt - t^2). They satisfy the recurrence H_{n+1}(x) = 2x H_n(x) − 2n H_{n-1}(x) and form an orthogonal basis with weight e^{-x^2} on the real line, finding use in probability, physics (notably the quantum harmonic oscillator), and numerical analysis. The Hermite differential equation y'' − 2x y' + 2n y = 0 has polynomial solutions H_n.

Other concepts named after Hermite include Hermite interpolation, a form of polynomial interpolation that matches both function values and derivatives at specified nodes; Hermite normal form, a canonical upper-triangular form for integer matrices under unimodular transformations; and Hermite functions, the normalized eigenfunctions of the quantum harmonic oscillator given by e^{-x^2/2} H_n(x).