Gaussian integral

The Gaussian integral is the integral of a Gaussian function; the classic improper integral of e⁻ˣ² over the real line equals √π.

Polar coordinates: A coordinate system that locates points by distance and angle from an origin. The squared integral becomes radially symmetric, making polar coordinates the decisive substitution.
Gaussian function: A function shaped like a bell curve, commonly written as an exponential of a negative quadratic. The integral's integrand is the unnormalized Gaussian function e⁻ˣ².
Normal distribution: A continuous probability distribution whose density has a symmetric bell-shaped curve. Its density's normalization constant follows directly from the Gaussian integral.
Pierre-Simon Laplace: A French mathematician and astronomer who developed influential methods in probability and celestial mechanics. Laplace's approximation methods helped establish Gaussian integrals in probability calculations.
Laplace transform: An integral transform that maps a function of time to a function of a complex frequency variable. Unlike the fixed Gaussian area, it varies a kernel parameter to encode an entire function.
Fubini's theorem: A theorem allowing suitable multiple integrals to be evaluated as iterated integrals. It justifies interpreting the square of the one-dimensional integral as a two-dimensional integral.
Exponential function: The function whose rate of change is proportional to its value, commonly written eˣ. Exponential decay makes the integrand rapidly approach zero as |x| grows.
Gaussian distribution: A probability distribution with a density proportional to the exponential of a negative quadratic. The integral determines the constant that makes its density integrate to one.
Carl Friedrich Gauss: A German mathematician whose work shaped number theory, astronomy, geometry, and statistics. His use of the bell-shaped error law made Gaussian normalization central to statistics.
Fresnel integral: An oscillatory improper integral involving sine or cosine of a quadratic phase. Its quadratic exponent is imaginary rather than negative real, so cancellation replaces decay.

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