---
title: "Gaussian Integral"
slug: gaussian-integral
type: evergreen
stime: 2025-01-15 @ 12:00AM
status: evergreen
certainty: certain
importance: 7
tags: [math, calculus, analysis]
---

## 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:

$$
\int_{-\infty}^{+\infty} e^{-x^2}\, dx = \sqrt{\pi}
$$

Although no elementary function exists for the [error function](error-function.md), 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

$$
\int e^{-x^2}\, dx
$$

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

$$
\int_{-\infty}^{+\infty} e^{-a(x+b)^2}\, dx = \sqrt{\frac{\pi}{a}}
$$

## 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:

$$
\left( \int_{-\infty}^{\infty} e^{-x^2}\, dx \right)^2 = \int_{-\infty}^{\infty} e^{-x^2}\, dx \int_{-\infty}^{\infty} e^{-y^2}\, dy = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2)}\, dx\, dy
$$

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](fubinis-theorem-order-swap.md):

$$
\int_{\mathbb{R}^2} e^{-(x^2+y^2)}\, dA = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2} e^{-y^2}\, dx\, dy = \left( \int_{-\infty}^{\infty} e^{-x^2}\, dx \right)^2
$$

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

$$
\int_{\mathbb{R}^2} e^{-r^2}\, dA = \int_0^{2\pi} \int_0^{\infty} e^{-r^2} r\, dr\, d\theta = 2\pi \int_0^{\infty} r e^{-r^2}\, dr = 2\pi \cdot \frac{1}{2} = \pi
$$

Comparing these two computations yields the result:

$$
\left( \int_{-\infty}^{\infty} e^{-x^2}\, dx \right)^2 = \pi \implies \int_{-\infty}^{\infty} e^{-x^2}\, dx = \sqrt{\pi}
$$

## 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

## Related

- [Notation of a Limit](notation-of-a-limit.md)
- [When a Limit Exists](when-a-limit-exists.md)
- [Function Notation](function-notation.md)
- [Methods Of Integration](methods-of-integration.md)
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- [Glasser's Master Theorem](glassers-master-theorem.md)
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- [Ramanujan's Master Theorem](ramanujans-master-theorem.md)
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- [Taylor / Power Series Expansion](taylor-power-series-expansion.md)
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