The Dirac delta function, often denoted as $\delta(x)$, is a mathematical “function” or distribution that is not a function in the conventional sense but rather an idealised construct useful for modelling and analysis in various fields, especially physics and engineering. It is defined by the following properties: ## Sifting Property The Dirac delta function has the “sifting” property, which allows it to “pick out” the value of a function at a specific point. For any continuous function $f(x)$, $ \int_{-\infty}^{\infty} f(x) \delta(x - a) \, dx = f(a). $ This property makes the delta function extremely useful in applications involving sampling or impulses. ## Behaviour as a Distribution The Dirac delta is not a traditional function but rather a distribution. It is often visualised as a “spike” at $x = 0$ with an infinite height and zero width, such that its integral over the entire real line is equal to 1: $ \int_{-\infty}^{\infty} \delta(x) \, dx = 1. $ ## Application in Physical Systems In physics, $\delta(x)$ models point charges, masses, or other singularities. For example, in electromagnetism, it can represent a point charge density. ## Fourier Transform The Fourier transform of the Dirac delta function in one dimension is a constant. Specifically, $ \mathcal{F}\{\delta(x)\} = 1. $ In summary, the Dirac delta function serves as a tool to represent idealised point sources or instantaneous changes within integrals and differential equations, allowing us to model systems with sudden, localised effects effectively.