HermiteFunktion

Hermite function is a mathematical term. It refers to a class of orthogonal polynomials known as Hermite polynomials. These polynomials are solutions to the differential equation known as Hermite's equation. The equation is given by y'' - 2xy' + 2ny = 0, where n is a non-negative integer. The Hermite polynomials are denoted by H_n(x). The first few Hermite polynomials are H_0(x) = 1, H_1(x) = 2x, H_2(x) = 4x^2 - 2, and H_3(x) = 8x^3 - 12x. They have a recursive definition, which allows for the generation of higher-order polynomials from lower-order ones. Hermite polynomials are important in various areas of mathematics and physics. They appear in quantum mechanics, specifically in the study of the quantum harmonic oscillator, where they form the wave functions of the energy eigenstates. They are also used in probability theory and statistics, particularly in the context of the normal distribution. The orthogonality property of Hermite polynomials is crucial for their applications, meaning that the integral of the product of two different Hermite polynomials over their domain is zero. This property makes them suitable for use as basis functions in expansions and approximations.