Introduction

The Gaussian integral, also known as the Euler-Poisson integral, is the integral of the Gaussian function $f(x) = e^{-x^2}$ over the entire real line. Named after the German mathematician Carl Friedrich Gauss, the integral is:

Although no elementary function exists for the error function, as can be proven by the Risch algorithm, the Gaussian integral can be solved analytically through the methods of multivariable calculus. That is, there is no elementary indefinite integral for

but the definite integral can be evaluated. The definite integral of an arbitrary Gaussian function is:

Computation by polar coordinates

A standard way to compute the Gaussian integral, the idea of which goes back to Poisson, is to make use of the property that:

Consider the function $e^{-(x^2+y^2)} = e^{-r^2}$ on the plane $\mathbb{R}^2$, and compute its integral two ways:

1. On the one hand, by Fubini’s theorem:

2. On the other hand, using polar coordinates $(r, \theta)$:

Comparing these two computations yields the result:

Applications

The Gaussian integral appears throughout mathematics and physics:

• Probability theory: The normalization constant for the normal distribution
• Quantum mechanics: Path integrals and wave function normalization
• Statistical mechanics: Partition functions
• Signal processing: Fourier transforms of Gaussian signals