Gaussian function
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A Gaussian function (named after Carl Friedrich Gauss) is a function of the form:
- <math>f(x) = a e^{-(x-b)^2/c^2}<math>
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 improper integrals over the whole real line can be evaluated exactly:
- <math>\int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt{\pi}.<math>
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,
- <math>I^2 = \int_{-\infty}^\infty e^{-x^2}\,dx \int_{-\infty}^\infty e^{-y^2}\,dy = \int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(x^2+y^2)}\,dx\,dy.<math>
Note the renaming of the variable of integration from x to y (see dummy variable). We now change to plane polar coordinates
- <math>I^2 = \int_0^{2\pi}\int_0^\infty e^{-r^2}r\,dr\,d\theta = 2\pi\int_0^\infty e^{-r^2}r\,dr=\pi\int_0^\infty e^{-u}\,du=\pi.<math>
(The substitution u = r2, du = 2r dr was used.)
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 distribution of complicated sums, according to the central limit theorem.
- A Gaussian function is the wave function of the ground state of the quantum harmonic oscillator.
- Mathematically, the Gaussian function plays an important role in the definition of the Hermite polynomials.
- consequently, Gaussian functions (and functionals) are also associated with the vacuum state in quantum field theory.