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for some real constants a > 0, b, and c.

Gaussian functions with c² = 2 are eigenfunctions of the Fourier transform. This means that the Fourier transform of a Gaussian function is not only another Gaussian function but a scalar multiple of the function whose Fourier transform was taken.

Gaussian functions are among those functions that are "elementary" but lack "elementary antiderivatives", i.e., their antiderivatives are not among the functions ordinarily considered in first-year calculus courses. Nonetheless their definite integrals over the whole real line can be evaluated exactly:

This calculation can be performed by the residue theorem of complex analysis, but there is also a simple and instructive way to do the calculation. Call the value of this integral I. Then,

Note the renaming of the variable of integration from x to y (see dummy variable). We now change to plane polar coordinates

(The substitution u = r², du = 2r dr was used.)

1 Applications

The antiderivative of the Gaussian function is the error function.

Gaussian functions appear in many contexts in physics and mathematics, for example

• In statistics and probability theory, Gaussian functions appear as the density function of the normal distribution, which is a limiting probability distributionIn mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. In technical terms, a probability distribution is a probability measure whose domain is the Borel algebra of complicated sums, according to the central limit theoremCentral limit theorems are a set of weak-convergence results in probability theory. Intuitively, they all express the fact that any sum of many independent identically distributed random variables is approximately normally distributed. These results expla.
• A Gaussian function is the wave function of the ground stateIn physics, the ground state of a quantum mechanical system is its lowest- energy state. An excited state is any state with energy greater than the ground state. If more than one ground state exists, they are said to be degenerate''. Many systems have deg of the quantum harmonic oscillatorThe quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because, as in classical mechanics, a wide variety of physical situations can be re.
• Mathematically, the Gaussian functionA Gaussian function (named after Carl Friedrich Gauss, which rhymes with house is a function of the form: : for some real constants a > 0, b and c''. Gaussian functions with c''2 2 are eigenfunctions of the Fourier transform. This means that the Fourier t plays an important role in the definition of the Hermite polynomials.
• consequently, Gaussian functions (and functionalGenerally, functional refers to something with and able to fulfill its purpose or function. In medicine, the term functional is sometimes used to describe symptoms that have no organic basis, e. if they are a result of psychological or perceptual dysfuncts) are also associated with the vacuum state in quantum field theory.