# Dirac Delta Function: Definition & Mathematics

• Vahsek
In summary, the conversation discusses the decision to shift focus from self-studying physics to rigorous mathematics. The topic of the Dirac delta function is brought up and it is questioned if it is rigorously defined in any area of mathematics and if so, which topic it falls under. The answer is that it is defined in distributions, which requires knowledge of functional analysis.
Vahsek
It's been quite some time now since I decided to stop self-studying physics and to pay more attention to the math behind. I'm working towards gaining an understanding of 100% rigorous mathematics for now.

One thing that has always bothered me is the Dirac delta function. What I want to know is the following: is the dirac delta function rigorously defined in any area of mathematics or is it just some kind of "fuzzy" mathematics to help with computations? Secondly, if it is rigorously defined somewhere, in which topic of mathematics is it? (real analysis, complex analysis, functional analysis... ?)

See distributions for the rigorous treatment. It requires knowledge of functional analysis.

1 person

## 1. What is the Dirac Delta Function?

The Dirac Delta Function, also known as the impulse function, is a mathematical concept that represents a spike or point of infinite height and zero width at a specific point on a graph. It is often used to model a point source of energy or mass in physics and engineering applications.

## 2. What is the mathematical definition of the Dirac Delta Function?

The mathematical definition of the Dirac Delta Function is denoted as δ(x) and is defined as follows: δ(x) = 0 for x ≠ 0 and ∫δ(x)dx = 1. Essentially, the Dirac Delta Function is zero everywhere except at x = 0, where it has an infinitely high and narrow peak.

## 3. How is the Dirac Delta Function used in calculus?

In calculus, the Dirac Delta Function is used as a tool for solving differential equations and evaluating integrals involving discontinuous functions. It is also commonly used in Fourier analysis to represent periodic functions as a sum of sinusoidal functions.

## 4. What are some real-world applications of the Dirac Delta Function?

The Dirac Delta Function has many practical applications in various fields of science and engineering. It is used to model point sources of energy or mass, such as in nuclear physics and electricity and magnetism. It is also used in signal processing, control theory, and image processing.

## 5. Can the Dirac Delta Function be graphed?

Technically, the Dirac Delta Function cannot be graphed since it is not a traditional function with a defined value at each point. However, it is often represented graphically as a tall, narrow spike at x = 0 to illustrate its properties and use in mathematical equations.

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