Delta Function and Distributions: Foundations for Generalized Function Theory

Table of Contents

1. Introduction
2. Motivation: Beyond Classical Functions
3. The Dirac Delta Function
4. Properties of the Delta Function
5. Representations and Approximations
6. Delta Function in Multiple Dimensions
7. Derivatives of the Delta Function
8. Heaviside Step Function and Relation to Delta
9. Distributions and Test Functions
10. Action of Distributions on Test Functions
11. Schwartz Distributions and Linear Functionals
12. Fourier Transform of the Delta Function
13. Delta Function in Physics
14. Applications in Quantum Mechanics and Electrodynamics
15. Conclusion

1. Introduction

The Dirac delta function and the theory of distributions extend classical calculus to include objects like impulses, point charges, and discontinuities. While not functions in the traditional sense, they are indispensable in physics and engineering for modeling idealized phenomena.

2. Motivation: Beyond Classical Functions

In many physical situations, we want to describe:

• A point mass or point charge
• An instantaneous force
• A perfect impulse at a single point in time or space

Classical functions cannot handle these cases. Distributions (or generalized functions) allow us to rigorously define and manipulate such concepts.

3. The Dirac Delta Function

The Dirac delta function \( \delta(x) \) is defined via its action:

\[
\int_{-\infty}^{\infty} \delta(x) f(x) dx = f(0)
\]

More generally:

\[
\int_{-\infty}^{\infty} \delta(x – a) f(x) dx = f(a)
\]

It “picks out” the value of the function at the point \( x = a \).

4. Properties of the Delta Function

• Localization: \( \delta(x – a) = 0 \) for \( x \ne a \)
• Sifting property:
  \[
  \int_{-\infty}^{\infty} \delta(x – a) f(x) dx = f(a)
  \]
• Evenness: \( \delta(-x) = \delta(x) \)
• Scaling:
  \[
  \delta(ax) = \frac{1}{|a|} \delta(x)
  \]
• Change of variable:
  \[
  \delta(g(x)) = \sum_i \frac{\delta(x – x_i)}{|g'(x_i)|}
  \]
  where \( x_i \) are the simple roots of \( g(x) \)

5. Representations and Approximations

The delta function can be approximated by a sequence of functions:

• Gaussian:
  \[
  \delta_\epsilon(x) = \frac{1}{\epsilon \sqrt{\pi}} e^{-x^2/\epsilon^2}
  \]
• Lorentzian:
  \[
  \delta_\epsilon(x) = \frac{1}{\pi} \frac{\epsilon}{x^2 + \epsilon^2}
  \]
• Rectangular:
  \[
  \delta_\epsilon(x) =
  \begin{cases}
  \frac{1}{2\epsilon}, & |x| < \epsilon \
  0, & \text{otherwise}
  \end{cases}
  \]

As \( \epsilon \to 0 \), all tend weakly to \( \delta(x) \).

6. Delta Function in Multiple Dimensions

In \( \mathbb{R}^n \):

\[
\int_{\mathbb{R}^n} \delta(\vec{x} – \vec{a}) f(\vec{x}) \, d^n x = f(\vec{a})
\]

Used in electrostatics, quantum field theory, and Green’s functions.

7. Derivatives of the Delta Function

Distributions can be differentiated:

\[
\int \delta'(x – a) f(x) dx = -f'(a)
\]

This extends to higher derivatives and plays a role in solving differential equations with singular sources.

8. Heaviside Step Function and Relation to Delta

The Heaviside function \( \theta(x) \) is defined as:

\[
\theta(x) =
\begin{cases}
0, & x < 0 \
1, & x \ge 0
\end{cases}
\]

Its derivative (in the distributional sense) is the delta function:

\[
\frac{d}{dx} \theta(x) = \delta(x)
\]

9. Distributions and Test Functions

A distribution is a linear functional \( T \) acting on a test function \( \phi(x) \in \mathcal{D}(\mathbb{R}) \), the space of smooth functions with compact support:

\[
T(\phi) = \int_{-\infty}^{\infty} t(x) \phi(x) dx
\]

Where \( t(x) \) may be singular.

10. Action of Distributions on Test Functions

All calculations are done in terms of pairing:

\[
\langle \delta(x – a), \phi(x) \rangle = \phi(a)
\]

\[
\langle \delta^{(n)}(x – a), \phi(x) \rangle = (-1)^n \phi^{(n)}(a)
\]

This avoids direct evaluation of singularities.

11. Schwartz Distributions and Linear Functionals

In the Schwartz distribution framework:

• Distributions generalize functions
• Differentiation becomes continuous
• Singular behaviors like jumps and spikes are manageable

This is used in quantum field theory and generalized Green’s functions.

12. Fourier Transform of the Delta Function

\[
\mathcal{F}\{\delta(x)\} = 1
\]

\[
\mathcal{F}\{1\} = 2\pi \delta(\omega)
\]

These identities are fundamental in signal analysis and quantum mechanics.

13. Delta Function in Physics

• Point charge: \( \rho(\vec{r}) = q \delta(\vec{r} – \vec{r}_0) \)
• Initial conditions: \( f(t) = \delta(t – t_0) \)
• Green’s functions: \( L G(x, x’) = \delta(x – x’) \)
• Scattering theory: transition amplitudes involve delta functions

14. Applications in Quantum Mechanics and Electrodynamics

• Wavefunctions: normalized via delta
  \[
  \langle x | x’ \rangle = \delta(x – x’)
  \]
• Commutation relations: involve delta
• Green’s functions: fundamental solutions to field equations

15. Conclusion

The Dirac delta function and the theory of distributions revolutionized the mathematical handling of singularities and idealizations in physics. From modeling point charges to solving field equations, these generalized objects provide clarity, precision, and powerful tools in theoretical and applied sciences.

Understanding them is essential for any serious student of physics or applied mathematics.