I created a tutorial for how to make websites last year, and I’m quickly checking that everything still works.

One of the most celebrated integrals in the history of science, especially due to its significance in physics and mathematical analysis, is the Gaussian integral. This integral is pivotal in probability theory, physics, and signal processing. The Gaussian integral, also known as the Euler-Poisson integral, directly leads to the normalization constant for the Gaussian function, which is foundational in the study of statistics and the normal distribution. Here’s the integral in \LaTeX format, including definitions of the variables involved:

\[\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}\]

This integral asserts that the area under the curve of the function \(e^{-x^2}\) from \(-\infty\) to \(\infty\) is equal to the square root of \(\pi\). This result is not only non-intuitive but also remarkably elegant, showcasing the deep interconnection between exponential functions and the geometry of circles (as \(\pi\) appears in the result).