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Dirac delta function

Dirac delta function is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Dirac delta function rather than just read about it. In short: In mathematical analysis, the Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, where it is infinite, and whose integral over the entire real line is equal to one. Thus it can be represented heuristically as δ ( x ) = { 0 , x ≠ 0 ∞ , x = 0 {\displaystyle \delta…

Dirac delta function — main illustration
Dirac delta function — illustration

Key takeaways

  • Dirac delta function belongs to mathematics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Dirac delta function to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Dirac delta function from memory before moving on to harder problems.

Reference excerpt

In mathematical analysis, the Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, where it is infinite, and whose integral over the entire real line is equal to one. Thus it can be represented heuristically as δ ( x ) = { 0 , x ≠ 0 ∞ , x = 0 {\displaystyle \delta (x)={\begin{cases}0,&x\neq 0\\{\infty },&x=0\end{cases}}} such that ∫ − ∞ ∞ δ ( x ) d x = 1. {\displaystyle \int _{-\infty }^{\infty }\delta (x)\,dx=1.}

Since no function has this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions. The delta function is named after physicist Paul Dirac, and has been applied routinely in physics and engineering to model point masses and concentrated loads. It is called the delta function because it is a continuous analogue of the Kronecker delta function. The mathematical rigor of the delta function was disputed until Laurent Schwartz developed the theory of distributions, where it is defined as a linear form acting on functions.

Motivation and overview The graph of the Dirac delta is usually thought of as following the whole x {\displaystyle x} -axis and the positive y {\displaystyle y} -axis. The Dirac delta is used to model a tall narrow spike function (an impulse), and other similar abstractions such as a point charge or point mass. For example, to calculate the dynamics of a billiard ball being struck, one can approximate the force of the impact by a Dirac delta. In doing so, one can simplify the equations and calculate the motion of the ball by only considering the total impulse of the collision. In applied mathematics, the delta function is often manipulated as a kind of limit (a weak limit) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero. (However, even in some applications, highly oscillatory functions are used as approximations to the delta function, see below.) The Dirac delta, given the desired properties outlined above, cannot be a function with domain and range in real numbers. For example, the objects f ( x ) = δ ( x ) {\displaystyle f(x)=\delta (x)} and g ( x ) = 0 {\displaystyle g(x)=0} are equal everywhere except at x = 0 {\displaystyle x=0} yet have integrals that are different. According to Lebesgue integration theory, if f {\displaystyle f} and g {\displaystyle g} are functions such that f = g {\displaystyle f=g} almost everywhere, then f {\displaystyle f} is integrable if and only if g {\displaystyle g} is integrable and the integrals of f {\displaystyle f} and g {\displaystyle g} are identical. A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right uses measure theory or the theory of distributions.

History As part of his development of quantum mechanics, Paul Dirac introduced the δ {\displaystyle \delta } -function in a 1927 paper, subsequently popularized in his 1930 book The Principles of Quantum Mechanics. He called it the "delta function" since he used it as a continuum analog of the discrete Kronecker delta. However, it had been used by multiple mathematical scientists in the nineteenth century. Dirac biographer Graham Farmelo surmised that Oliver Heaviside was likely a direct influence on Dirac, given Dirac's background in engineering. Indeed, Heaviside introduced the δ {\displaystyle \delta } -function in his work on electromagnetism and electrical engineering. In a 1963 interview, Dirac stated, "All electrical engineers are familiar with the idea of a pulse, and the δ {\displaystyle \delta } -function is just a way of expressing a pulse mathematically." Mathematicians refer to the same concept as a generalized function or distribution rather than a function in the ordinary sense. The earliest known use of the δ {\displaystyle \delta } -function is in the works of Jean-Baptiste Joseph Fourier. Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur (1822) in the form:

… excerpt ends here. Continue reading the full article.

Illustrations

Dirac delta function: Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.
Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.
Dirac delta function: The Dirac delta as the limit as a → 0 {\displaystyle a\to 0} (in the sense of distributions) of the sequence of zero-centered normal distributions δ a ( x ) = 1 | a | π e − ( x / a ) 2 {\displaystyle \delta _{a}(x)={\frac {1}{\left|a\right|{\sqrt {\pi }}}}e^{-(x/a)^{2}}}
The Dirac delta as the limit as a → 0 {\displaystyle a\to 0} (in the sense of distributions) of the sequence of zero-centered normal distributions δ a ( x ) = 1 | a | π e − ( x / a ) 2 {\displaystyle \delta _{a}(x)={\frac {1}{\left|a\right|{\sqrt {\pi }}}}e^{-(x/a)^{2}}}
Dirac delta function: A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of T
A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of T

Worked examples

Example 1 — a first encounter with Dirac delta function

Start with the simplest possible case. Write down what Dirac delta function claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Dirac delta function before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Dirac delta function ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Dirac delta function

In research
Dirac delta function appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Dirac delta function in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Dirac delta function is common in secondary-school and first-year university syllabi. It links to neighbouring topics Digital signal processing, Fourier analysis, Generalized functions, so understanding it makes those chapters shorter.
In everyday life
Look for Dirac delta function outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Dirac delta function in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Dirac delta function means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Dirac delta function out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Dirac delta function in simple terms?

In mathematical analysis, the Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, where it is infinite, and whose integral over the entire real…

Why does Dirac delta function matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Dirac delta function?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Dirac delta function.

Tags

  • Digital signal processing
  • Fourier analysis
  • Generalized functions
  • Measure theory
  • Paul Dirac
  • Schwartz distributions

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