• Zero Everywhere Except at Zero:
  $$\delta (x)=0\text{for}x\mathrm{\ne }0$$

• Infinite at Zero: The Dirac delta function is infinitely large at
  $x$$=$$0$, but in a very special way.

• Unit Integral: The defining property of the Dirac delta function is that its integral over all space is equal to 1:
  $$\int_{-\infty}^{\infty} \delta(x) \, dx = 1$$
What makes the delta function so useful is that it "picks out" the value of a function at a specific point. For any continuous function
$f$$($$x$$)$, the delta function has the following property:


• Sifting Property: $$\int_{-\infty}^{\infty} f(x) \delta(x - a) \, dx = f(a)$$This property allows us to evaluate a function at a particular point $x = a$ by integrating it with the Dirac delta function. In essence, the delta function isolates or "samples" the value of $f(x)$ at the point $x = a$.

Properties of the Dirac Delta Function:

1. Scaling Property

2. Shifted Delta Function

• The Dirac delta function $\delta(x - a)$ is zero everywhere except at $x = a$, where it is "infinite" in such a way that its integral over the entire real line is 1.
• The function $f(x)$ is evaluated at $x = a$ because the delta function essentially "picks out" the value of $f(x)$ at that point.

3. Symmetry

4. Derivative of the Delta Function

The Dirac delta function has a well-defined derivative in the sense of distributions:

$$\frac{d}{dx} \delta(x)$$

This is not a "normal" derivative in the classical sense but is used in situations where we want to differentiate with respect to a parameter inside the delta function. For any smooth function
$f$$($$x$$)$, we have:

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\frac{d}{dx}\delta (x-a)dx=-f^{\mathrm{\prime }}(a)$$

The delta function's derivative is crucial for modeling impulse responses that involve sudden changes, such as forces or voltage spikes.

Derivation:

The Dirac delta function can be differentiated in the sense of distributions. The derivative of the delta function, δ′(x), satisfies the following property

$$\int_{-\infty}^{\infty} f(x) \delta'(x) \, dx = -f'(0)$$In general, the nth derivative of the delta function δ(n)(x) satisfies:

$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x){\delta }^{(n)}(x-a)dx=(-1{)}^n{f}^{(n)}(a)<br><br><br>$$

$$\int_{-\infty}^{\infty} f(x) \delta'(x) \, dx = -f'(0)$$

To derive this, consider an integration by parts:

$$\int_{-\infty}^{\infty} f(x) \delta'(x) \, dx = \left[ f(x) \delta(x) \right]_{-\infty}^{\infty} - \int_{-\infty}^{\infty} f'(x) \delta(x) \, dx$$
Since f(x) is zero at the boundaries, this simplifies to:

$$\int_{-\infty}^{\infty} f(x) \delta'(x) \, dx = -f'(0)$$

$$$$

5. Fourier Transform of the Delta Function

The Dirac delta function is also critical in Fourier analysis. Its Fourier transform is a constant:

$$\int_{-\infty}^{\infty} \delta(x) e^{-ikx} \, dx = 1$$

This is vital in signal processing, as it implies that the delta function is the neutral element under convolution, meaning it leaves other signals unchanged when convolved with them.

The Fourier transform of the Dirac delta function is a constant:

$$\mathcal{F}\{\delta (x)\}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta (x)e^{-ikx}dx=1$$This result shows that the delta function contains all frequencies equally and is neutral in the frequency domain.

Inverse Fourier Transform:

$$\delta (x)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{ikx}dk$$

This is a useful result in signal processing and quantum mechanics because it shows how the delta function is related to the harmonic components of a signal.

6. Convolution with the Delta Function

Convolution of any function f(x) with the delta function leaves the function unchanged:

$$(f\ast \delta )(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x^{\mathrm{\prime }})\delta (x-x^{\mathrm{\prime }})d{x}^{\mathrm{\prime }}=f(x)$$This is because the delta function "samples" the value of the function at point x.

7. Algebraic Properties

Multiplication by a Variable:

One of the essential algebraic properties of the Dirac delta function is that any term involving x⋅δ(x) vanishes:

$$x\delta (x)=0$$This is a direct consequence of the fact that δ(x) is zero everywhere except at x=0, so multiplying it by x eliminates any non-zero contribution.
More generally, for any positive integer n:
$$(x-a{)}^n\delta (x-a)=0$$
because δ(x−a)only has support at x=a, where $(x-a{)}^n=0$.

Uniqueness of Distributions:

If two distributions f(x) and g(x)satisfy x⋅f(x)=x⋅g(x), then they must differ only by a multiple of δ(x):

$<br>$

$$f(x)=g(x)+c\delta (x)$$where c is a constant. 

This is due to the special nature of δ(x) and its "point-like" behavior.

8. Compositions with Functions

A particularly useful property of the delta function is its behavior under composition with smooth functions. If g(x) is a continuously differentiable function, then the following change of variables formula holds:

$$\int_{-\infty}^{\infty} \delta(g(x)) f(g(x)) |g'(x)| \, dx = \int_{g(\mathbb{R})} \delta(u) f(u) \, du$$provided that g′(x)≠0 everywhere. This formula generalizes the delta function to more complex arguments and is essential when working with transformations of variables in integrals.

$$$$In the special case where g(x) has a root at x0​, we have:$$\delta (g(x))=\frac{\delta (x-x_0)}{\mathrm{\mid }{g}^{\mathrm{\prime }}(x_0)\mathrm{\mid }}$$This property is used to define the delta function in terms of the roots of smooth functions.

For example, if,

g(x)=x2−α2

then the roots are
$x$$=$$\pm$$\alpha$, and we get:

$$\delta (x^2-{\alpha }^2)=\frac{1}{2\mathrm{\mid }\alpha \mathrm{\mid }}[\delta (x+\alpha )+\delta (x-\alpha )]$$

$$<br>$$

Derivation:

$$<b><u><br></u></b>$$

If you have a function

$$g(x) = x^2 - \alpha^2$$

$$\delta(g(x))$$

$$g(x)$$

$$1.$$

Indefinite Integral of the Dirac Delta Function

The indefinite integral of the Dirac delta function is related to the Heaviside step function H(x):$$\int_{-\infty}^{x} \delta(t - a) \, dt = H(x - a)$$
Thus, integrating the delta function up to a certain point gives a step-like function that jumps from 0 to 1 at x=a.

Delta Function in Multiple Dimensions

The Dirac delta function generalizes to higher-dimensional spaces. In
$n$-dimensional space, the delta function is written as:

$$\delta^{(n)}(\mathbf{r}) = \delta(x_1) \delta(x_2) \cdots \delta(x_n)$$
This multi-dimensional delta function is used in problems involving point charges, point sources, or any phenomenon localized to a single point in n-dimensional space.

In particular, the delta function satisfies the following scaling property in n dimensions:$$\delta(\alpha \mathbf{r}) = \frac{1}{|\alpha|^n} \delta(\mathbf{r})$$This property is important for understanding how the delta function behaves under changes of scale in different dimensions.

Delta Function as a Surface Integral

In advanced applications, such as potential theory, the Dirac delta function is related to surface integrals over certain manifolds. For a smooth hypersurface S in n-dimensional space, the delta function can be associated with the surface integral:$${\delta }_S[g]={\int }_{S}g(\mathbf{s})d\sigma (\mathbf{s})$$where σ(s) is the measure on the surface S. This concept appears in problems involving layer potentials, surface charges, and other phenomena localized to surfaces rather than points.

The Dirac Delta Function in Higher Dimensions

In n-dimensional space, the Dirac delta function generalizes to:

$${\delta }^{(n)}(\mathbf{r})=\delta (x_1)\delta (x_2)\dots \delta (x_n)$$

In three dimensions, for example, the delta function is written as:

$$\delta^{(3)}(\mathbf{r}) = \delta(x) \delta(y) \delta(z)$$

It is used to describe point sources such as point charges in electromagnetism or point masses in gravitation. For example, the charge density of a point charge q at position r0​ is:

$$\rho(\mathbf{r}) = q \delta^{(3)}(\mathbf{r} - \mathbf{r}_0)$$

The Dirac Delta Function in 3-D Dimensional spherical coordinate

In three dimensions, the delta function is represented in spherical coordinates by:

The Functionality of the Dirac Delta in Physics

1. Modeling Point Charges in Electromagnetism

In electromagnetism, the delta function is used to describe point charges. For example, the charge density of a point charge q located at a position r0 can be written as:

$$\rho (\mathbf{r})=q\delta (\mathbf{r}-{\mathbf{r}}_0)$$

This allows for a precise mathematical description of a charge distribution that is concentrated at a single point, an idealization used frequently in theoretical physics.

2. Green’s Functions and Solutions to Differential Equations

The Dirac delta function is central to the construction of Green's functions, which are used to solve inhomogeneous differential equations. A Green's function G(x,x′) satisfies

$$L G(x, x') = \delta(x - x')$$where L is a differential operator. In this context, the delta function represents a localized source or disturbance, and the Green’s function provides the response of the system to that disturbance.

3. Quantum Mechanics and the Schrödinger Equation

In quantum mechanics, the Dirac delta function is used to describe potential wells or barriers that are infinitely narrow but still affect particle behavior. It also appears in the Born approximation for scattering and in the representation of states in terms of wave functions:

$$⟨x\mathrm{\mid }p⟩=\frac{1}{\sqrt{2\pi \mathrm{\hslash }}}{e}^{ipx\mathrm{/}\mathrm{\hslash }}$$Here, the delta function plays a role in normalizing the states and defining localized particles in space.

4. Impulse Response in Classical Mechanics

In classical mechanics, the delta function is used to model impulse forces. An impulse, that applies a large force over a very short time interval, can be represented by a delta function in the time domain:

$$F(t)=F_0\delta (t-t_0)$$

This expresses the idea that a sudden, instantaneous force is applied at a particular moment in time
$t_{}$${\mathrm{0 }}_{}$

5. Signal Processing and Systems Theory

The delta function is essential in signal processing as it represents the "impulse" input. The response of a system to a delta function input is called the impulse response, which can be used to understand the behavior of linear systems. Convolution with the delta function yields the original signal, demonstrating its role as a mathematical "identity."

Geometrical and Physical Interpretation

• Geometrically: The delta function is a spike at x=0, with an infinitely small width and an infinitely large height, constrained such that the area under the spike remains 1. It is a limiting case of functions that get narrower and taller, focusing all their "weight" on a single point.

• Physically: The Dirac delta function is used to model idealized point phenomena. For example, in physics, a point charge or point mass can be modeled using a delta function to represent a quantity (charge, mass, or force) that is concentrated at a single point in space.

Importance of the Dirac Delta Function in Physics

The Dirac delta function’s ability to represent localized phenomena makes it an indispensable tool in mathematical physics. Its unique properties allow physicists and engineers to model idealized systems where a physical quantity (such as charge, mass, or force) is concentrated at a single point. Furthermore, its use in solving differential equations makes it a fundamental element in nearly every area of physics:

• In quantum mechanics, it helps describe particle localization and state normalization.
• In electrodynamics, it defines the behavior of point charges and current distributions.
• In classical mechanics, it models instantaneous forces and momentary impacts.

In short, the Dirac delta function is more than just a mathematical curiosity—it's an essential tool for simplifying complex systems and providing precise solutions to a wide range of physical problems. Its versatility in applications, from quantum theory to signal processing, proves its importance in both theoretical and applied physics.

The Dirac Delta as an Idealized Spike

The Dirac delta function is a powerful mathematical tool that represents an idealized "spike" concentrated at a single point. It is not a function in the traditional sense, but rather a distribution that becomes infinitely narrow and tall in a limit process. Its ability to model point phenomena, such as point charges or instantaneous forces, makes it invaluable in physics and engineering. The delta function’s sifting property allows it to isolate the value of other functions at a given point, making it essential for solving differential equations, defining Green's functions, and analyzing signals in systems theory.

By thinking of the Dirac delta as the limit of a family of functions with vanishing width and increasing height, we can appreciate its role as a spike in mathematical and physical systems.

To understand the Dirac delta function δ(x) as a "spike," we can begin by thinking of it as an idealized limiting process. The Dirac delta function can be approximated by a family of functions that have certain characteristics, such as being nonzero over a narrow region but integrating to 1 over their domain. As the width of that region becomes infinitesimally small, the function approaches the Dirac delta function.

We’ll explore this concept by introducing a representation of δ(x) using a parameter ϵ that defines the width of the spike. As ϵ approaches 0, the function becomes more and more concentrated at a single point, while still having an integral of 1 over the entire real line.

Approximation to Dirac Delta as a Limit

1. Initial Approximation Using a Simple Function:

Consider a function δϵ(x) that is defined as:$${\delta }_{ϵ}(x)=\{\begin{array}{ll}\frac{1}{ϵ}& \text{if }\mathrm{\mid }x\mathrm{\mid }<\frac{ϵ}{2}\\ 0& \text{otherwise}\end{array}$$This is a rectangular-shaped function of height $\frac{1}{ϵ}$ and width $ϵ$. Its total integral over the real line is 1, since:$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\delta }_{ϵ}(x)dx={\int }_{-ϵ\mathrm{/}2}^{ϵ\mathrm{/}2}\frac{1}{ϵ}dx=1$$So, as long as ϵ\epsilon is finite, this function represents a "bump" that is localized around x=0 with area 1.

2. Taking the Limit
$\mathrm{<br>}$$ϵ\to 0$:

Now, as we make ϵ smaller and smaller, the function
${\delta }_{}$${ϵ}_{}$$($$x$$)$ becomes narrower and taller, while still maintaining its total integral as 1. In the limit as
$ϵ$$\to$$0$, the function
${\delta }_{}$${ϵ}_{}$$($$x$$)$ becomes infinitely tall at
$x$$=$$0$, but is zero everywhere else. Formally, this limiting process defines the Dirac delta function:

$$\underset{ϵ\to 0}{\lim }{\delta }_{ϵ}(x)=\delta (x)$$The result is a spike centered at x=0 that is zero everywhere except at that point, and yet its integral over all space is still 1. This is the key idea behind the Dirac delta function: it is an "infinitely thin" and "infinitely tall" spike with a total area of 1.

 

3. Mathematical Representation:

The Dirac delta function can thus be thought of as:

$$\delta (x)=\underset{ϵ\to 0}{\lim }{\delta }_{ϵ}(x)$$Where δϵ(x) is any appropriate sequence of functions that satisfies the following two key properties:

• It integrates to 1 over all space: $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\delta }_{ϵ}(x)dx=1$$
  
• It becomes concentrated at $x=0$ as $ϵ\to 0$:

  $${\delta }_{ϵ}(x)\to 0\text{for}x\mathrm{\ne }0$$

4. Why is it a Spike?:

As ϵ→0, the function δϵ(x) becomes infinitely narrow and approaches zero everywhere except at x=0, where it becomes infinitely tall. In other words, the "height" of the function increases as its "width" decreases, but its total integral always remains 1. The result is a spike that effectively selects out the value of a function at the point where the spike occurs, which leads to the sifting property:$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\delta (x-a)dx=f(a)$$This property captures the essence of the delta function's ability to "pick out" the value of a function at a particular point, much like how a spike isolates a single point in space or time.

Other Representations of the Dirac Delta Function

There are various other ways to approximate the Dirac delta function using different families of functions. Here are two examples:

1. Gaussian Approximation:
   $$\delta_\epsilon(x) = \frac{1}{\sqrt{\pi\epsilon}} e^{-x^2 / \epsilon}$$
   As
   $ϵ$$\to$$0$, this Gaussian function becomes narrower and taller, and in the limit, it approaches the Dirac delta function.
2. Lorentzian Approximation:
   $$\delta_\epsilon(x) = \frac{\epsilon}{\pi(x^2 + \epsilon^2)}$$This Lorentzian (Cauchy) distribution also becomes a spike at x=0 as ϵ→0.

demonstration of delta spike from three special functions

SUMMARISING

Delta Function as a Limit of Functions

The Dirac delta function can be constructed as the limit of a family of functions that become more "concentrated" around x=0 while preserving the integral of 1. These functions include:

(a) Gaussian Approximation:

The Gaussian function:$$\delta_\epsilon(x) = \frac{1}{\sqrt{\pi \epsilon}} e^{-x^2 / \epsilon}$$
As ϵ→0, this Gaussian becomes infinitely narrow and tall, and approaches the Dirac delta function:

$$\lim_{\epsilon \to 0} \delta_\epsilon(x) = \delta(x)$$

The total integral remains 1, even in the limit:$$\int_{-\infty}^{\infty} \delta_\epsilon(x) \, dx = 1$$

(b) Lorentzian Approximation:

The Lorentzian (Cauchy) function:$$\delta_\epsilon(x) = \frac{\epsilon}{\pi(x^2 + \epsilon^2)}$$​
As ϵ→0, this function also converges to the Dirac delta function:

$$\underset{ϵ\to 0}{\lim }{\delta }_{ϵ}(x)=\delta (x)$$

(c) Rectangular Approximation:

The rectangular function:$$\delta_\epsilon(x) = \begin{cases} \frac{1}{\epsilon} & \text{if } |x| < \frac{\epsilon}{2} \\ 0 & \text{otherwise} \end{cases}$$As ϵ→0, this function becomes infinitely tall and narrow, and approaches the Dirac delta function.

Fourier Series Representation of Dirac Comb:

Relationship between Dirac Delta and Kronecker Delta:

• Dirac Delta (Continuous): The Dirac delta applies to continuous spaces, and it "picks out" values of a continuous variable:
  $$\int_{-\infty}^{\infty} f(x) \delta(x - a) dx = f(a)$$
  
• Kronecker Delta (Discrete): The Kronecker delta applies to discrete spaces, and it "picks out" specific values in summations:
  $$\sum _{i}f(i){\delta }_{ij}=f(j)$$

Conclusion