The _Dirac delta function_ may be defined in a [piecewise manner](analysis%20(index)#Piecewise%20functions) as $\delta(y-x)=\begin{cases} 0, & x\neq y\\ \infty, & x=y \end{cases}.$ which we may, for example, plot as, follows when $y=0:$  The arrow indicates a point at infinity. Although we refer to it as a "function," it is in fact a [distribution](Dirac%20delta%20function#^eba2ab), which needs to be defined in terms of what [it does to a function when we integrate over it.](Dirac%20delta%20function#^eba2ab) A _[Dirac delta function](Dirac%20delta%20function.md)_ may also be defined as the _kernel_ of a _[linear integral operator](Integral%20transform.md)_, meaning that it is defined through its effect on functions rather than as a function in and of itself. This linear integral operation is expressed as $F[f(x)]=\int_{-\infty}^{\infty} dx f(x) \delta(y-x) = f(y)$ where $f$ is a [[Smooth function]] defined as $f:\mathbb{R} \rightarrow \mathbb{R}$ that is continuous at $x=y$ and the Dirac delta is defined as $\delta: \mathbb{R}\rightarrow \mathbb{R},x\rightarrow \delta(y-x)$, i.e. as a map from a function space to a real number space. ^eba2ab Notice that when given [as an integrand](Dirac%20delta%20function#^eba2ab), the Dirac delta function allows us to 'pick' $f$ at $x=y$ or in other words, project to specific locations on a function's curve. It follows that we may then write $\int_{-\infty}^{\infty} dx \delta(y-x) = 1.$ This also defines a [completeness relation](Complete%20sequence.md) in many contexts. ^37bd77 # Properties ## Property 1 $\delta(y-x)=\delta(x-y)$ where, we sometimes see an alternative notation where we write $\delta_y(x)=\delta_x(y)$. This means we can exchange the variables in the defintion to write: $\int_{-\infty}^{\infty} dx f(y) \delta(y-x) = f(x)$ ## Property 2 $\delta(cx)=\frac{1}{|c|}\delta{(x)}$ ## Property 3 - Symmetry [Property 1](Dirac%20delta%20function.md#Property%201) implies $\delta(x)=\delta(-x)$. (i.e. $\delta(x)$ is symmetrical). ## Property 4 - Piecewise representation $\delta(g(x))=\begin{cases} 0, & g(x)\neq 0\\ \infty, & g(x)=0 \end{cases}$ ## Property 5 $f(x)\delta(x) = f(0)\delta(x)=0$ because the width of $\delta(x)$ is $0$. ## Property 6 If we can expand $g(x)$, such that $g(x)=g(x_0)+g'(x_0)(x-x_0)+...=g'(x_0)(x-x_0)+...$ and $x_0$ is a zero of $g(x_0)$ (by [Property 4 - Piecewise representation](Dirac%20delta%20function.md#Property%204%20-%20Piecewise%20representation)), we can substitute into [Property 2](Dirac%20delta%20function.md#Property%202) to obtain: $\int dx \delta(g(x))f(x)=\frac{f(x_0)}{\bigg|\frac{dg(x_0)}{dx}\bigg|}$ If $g(x)$ has multiple zeros at values $x_i$: $\int dx \delta(g(x))f(x)=\sum_i\frac{f(x_i)}{\bigg|\frac{dg(x_i)}{dx}\bigg|}$ ^e89ca3 And it follows that $\delta(g(x)) = \sum_i \Bigg|\frac{dg(x_i)}{dx}\Bigg|^{-1}\delta(x-x_i)$ ## Property 7 $\delta(x)$ is non-differentiable in the most elementary sense but it is defined in terms of a test function $f$ such that $\int_{-\infty}^{\infty} dx f(x) \delta'(y-x)=-\int_{-\infty}^{\infty} dx f'(x) \delta(y-x)= -f'(y)$ ([Proof](Dirac%20delta%20function.md#Proof%20of%20Property%207%20Dirac%2020delta%2020function%20md%20Property%20207)). ## Property 8 $\delta((x-x_1)(x-x_2)) = \frac{\delta(x-x_1)+\delta(x-x_2)}{|x_1-x_2|}$ ([Proof](Dirac%20delta%20function.md#Proof%20of%20Property%208%20Dirac%2020delta%2020function%20md%20Property%20208)) # Derivative of the Dirac delta function # Representations of 𝛿(y-x) We can define functions, $\delta_n(y-x)$, which converge to $\delta(y-x)$. This acts as a representation of $\delta(x)$. This is useful when we can't evaluate the integral, $\int_{-\infty}^{\infty} dx f(x) \delta(y-x) = f(y)$. In such a case we may use: .md#%5E337830) Note that in place of $n$ we sometimes write $1/\epsilon$ where in the limit $\epsilon \rightarrow 0.$ Thus $\delta_n(y-x)=\delta^{\epsilon}(y-x)$. ([... see more](Representation%20of%20𝛿(y-x).md)) # Relation of 𝛿(y-x) to the Heaviside function $\delta(x)$ is the derivative of the [[Heaviside function]]. # Multi-dimensional Dirac delta functions ([... see more](Multi-dimensional%20Dirac%20delta%20function.md)) --- # Proofs and Examples ## Proof of [Property 7](Dirac%20delta%20function.md#Property%207)  ## Proof of [Property 8](Dirac%20delta%20function.md#Property%208)     --- # Recommended reading The [Dirac delta function](Dirac%20delta%20function.md) is an important tool for many physics problems and thus it commonly introduced and discussed in mathematics textbooks aimed at physicists such as: * [Altland, A. von Delft, J., _Mathematics for Physicists_. Cambridge University Press. 2019.](Altland,%20A.%20von%20Delft,%20J.%20Mathematics%20for%20Physicists,%20Cambridge%20University%20Press,%202019.md) pgs. 272-276. This entry is largely based on this excerpt. Since the [Dirac delta function](Dirac%20delta%20function.md) features especially often in quantum mechanical calculations exercises based around it appear here: * [Schollwöck, U. Homework 1, _Quantum Mechanics 1_ (German) (2019/2020).](Schollwöck,%20U.%20Homework%201,%20Quantum%20Mechanics%201%20(German)%20(2019-2020).md) Many of the [properties](Dirac%20delta%20function.md#Properties) of the [Dirac delta function](Dirac%20delta%20function.md) given here and the proofs of some of those properties are solutions to this problem set. #MathematicalFoundations/Analysis/GeneralizedFunctions #MathematicalFoundations/Analysis/Functions