What is the Gaussian Integral?
The function e^(−x²) is the bell curve: it peaks at 1 when x = 0 and falls symmetrically to 0 in both directions. The area under it across the entire real line equals exactly √π ≈ 1.7724. This is remarkable: e and π, usually encountered in separate contexts, are united in the simplest integral of probability theory.
The integral of e^(−x²) over all x equals √π ≈ 1.7725. This is the Gaussian integral. Its square root divided by √(2π) gives the standard normal distribution curve.
The proof is one of mathematics' most elegant tricks. Let I = ∫e^(−x²)dx. Compute I² by writing it as a double integral over x and y, then switch to polar coordinates r, θ. The integrand becomes e^(−r²) and the area element becomes r·dr·dθ. The r makes the integral elementary: ∫₀^∞ re^(−r²)dr = 1/2. Multiplying by ∫₀^(2π) dθ = 2π gives I² = π, so I = √π.
The normal distribution, the central limit theorem, quantum wave functions (which use Gaussian wave packets), and Stirling's approximation for factorials all rest on this single integral. The value √π appears wherever e^(−x²) is integrated, which turns out to be almost everywhere in continuous probability.
The Gaussian integral: the integral from -infinity to +infinity of e^(-x^2) dx = sqrt(pi). The elegant proof squares the integral, converts to polar coordinates, and evaluates it exactly. This is the key calculation behind the normal distribution: the probability density (1/sqrt(2*pi))*e^(-x^2/2) integrates to 1. The Gaussian function appears in quantum mechanics, heat diffusion, Stirling's approximation, and the central limit theorem.
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