The Dirac delta function question

In summary, the conversation discusses the Dirac delta function and its properties as a distribution or generalized function. The function is defined as a "gadget" that modifies how integrals work and is often approximated by a sequence of functions. It is commonly used in mathematical analysis and physics.
  • #1
MathematicalPhysicist
Gold Member
4,699
371
in the attatch file there is the dd function.
what i want to know is: when x doesn't equal 0 the function equals 0 and the inegral is the integral of the number 0 which is any constant therefore i think the integral should be equal 0.
can someone show me how this integral equals 1?


for your convinience here is the website that the gif was taken from:
http://www.engr.unl.edu/~glibrary/home/DefineG/node6.html [Broken]


another thing that i don't understand is how can you restrict an integral with b and a when it equals a constant for example int(b-a):0dx. the anti derevative of 0 is some C now how can you put into a number numbers that propety is only for variables.
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
The dirac delta function isn't really a function! Basically, the dirac delta function is really a gadget that modifies how the integral works rather than being something you integrate.

This gadget can be approximated by functions, though. For example, we can define the class of functions:

[tex]
\delta_n(x) = \frac{n}{\sqrt{\pi}} e^{-(nx)^2}
[/tex]

And then we have, under reasonable circumstances:

[tex]
\int_a^b f(x) \delta(x) \, dx
= \lim_{n \rightarrow \infty} \int_a^b f(x) \delta_n(x) \, dx
[/tex]


If you plot a few of the functions [itex]\delta_n(x)[/itex], you'll notice that for [itex]x \neq 0[/itex] these functions converge to [itex]0[/itex], and for [itex]x = 0[/itex] these functions diverge to [itex]+\infty[/itex]. Also, the area under each of these curves is [itex]1[/itex]. This is why you think intuitively of the dirac delta "function" as being infinite at [itex]x = 0[/itex] and [itex]0[/itex] everywhere else in such a way that its integral is [itex]1[/itex] iff the region contains [itex]0[/itex].
 
Last edited:
  • #3
To back up Hurkyl's answer, the delta "function" is defined as the "function" such that the integral of δ(x) over any set not containing 0 is 0 and the integral of δ(x) over any set containing 0 is 1.

Of course, there is no such function. It is, more correctly, a "distribution" or "generalized function". One can also define it as the "operator" on functions that, to any function f(x), assigns the value f(0). This is true because the integral of f(x)δ(x) over all real number is f(0).

If you don't like exponentials, instead of the approximation Hurkyl gave, you can use "dn(x)= 0 for x< -1/n,
n for -1/n<= x<= 1/n
0 for x> 1/n.
dn has the property that the integral from -1/n to 1/n is 1. The "limit" as n goes to infintiy is &delta;(x). I put limit in quotes because, of course, that doesn't actually converge but you can do things like find the Fourier transform of the delta function by finding the Fourier transform of each dn and then take the limit of that.

Hurkyl's sequence consists of differentiable functions, mine, only continuous functions. They would both be referred to as "delta sequences".
 
  • #4
Minor correction: HallsofIvy's sequence should be either

[tex]
d_n(x) = \left\{
\begin{array}{ll}
0, & x < -\frac{1}{2n} \\
n, & -\frac{1}{2n} \leq x \leq \frac{1}{2n} \\
0, & \frac{1}{2n} < x
\end{array}
\right
[/tex]

or

[tex]
d_n(x) = \left\{
\begin{array}{ll}
0, & x < -\frac{1}{n} \\
n (1 - |x|), & -\frac{1}{n} \leq x \leq \frac{1}{n} \\
0, & \frac{1}{n} < x
\end{array}
\right
[/tex]

(I'm not sure which one he intended)
 
Last edited:
  • #5
The first was what I intended and you are right- I forgot the "1/2" needed since the rectangle extends a distance 1/n on both sides (and it's not continuous). Thanks.
 
  • #6
The dirac delta function isn't really a function!
Well, yes it is. It is not a real-function of a real variable, but it is certainly a continuous linear scalar function on some appropriate space of test functions (usually either the Schwarz space of smooth functions of rapid decrease, or the space D of smooth functions with compact support). It is defined by [tex]\delta(f) \equiv f(0)[/tex]. Although it cannot be represented as integration against some locally integrable function:
[tex]T_g(f) = \int g(x)f(x)[/tex]
it is the limit of such function(als) in the topology of distributions, hence often the suggestive if slightly misleading notation:
[tex]\delta(f)\equiv f(0)= \int \delta(x)f(x)[/tex]
 
Last edited:

1. What is the Dirac delta function?

The Dirac delta function, also known as the impulse function, is a mathematical function that is zero everywhere except at one point, where it is infinite. It is commonly used in mathematics and physics to model a point source or an instantaneous change in a system.

2. How is the Dirac delta function represented mathematically?

The Dirac delta function is represented by the symbol δ(x) or δx. It is defined as δ(x) = 0 for all x ≠ 0, and δ(x) = ∞ for x = 0. It is also known as the limit of a sequence of functions that approach zero everywhere except at the origin.

3. What are the properties of the Dirac delta function?

The Dirac delta function has several important properties, including the sifting property, scaling property, and derivative property. The sifting property states that the integral of the function over any interval that contains the origin is equal to 1. The scaling property states that the function can be scaled by a constant factor without affecting its integral. The derivative property states that the derivative of the function is zero everywhere except at the origin.

4. How is the Dirac delta function used in applications?

The Dirac delta function has many applications in mathematics and physics. It is commonly used in differential equations to model point sources or impulses, such as in the study of electrical circuits. It is also used in signal processing to represent a signal that is concentrated at a single point. In quantum mechanics, it is used to represent the position of an electron in an atom.

5. What are the limitations of the Dirac delta function?

While the Dirac delta function is a useful mathematical tool, it has some limitations. It is not a true function in the traditional sense, as it is not defined at the point where it is infinite. It also violates some mathematical rules, such as the Leibniz integral rule. Additionally, it can be difficult to work with in some applications, as it can lead to non-smooth solutions.

Similar threads

  • Calculus
Replies
25
Views
907
Replies
2
Views
845
Replies
32
Views
3K
  • Calculus
Replies
8
Views
2K
Replies
18
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
Replies
8
Views
8K
Replies
4
Views
978
Replies
2
Views
10K
Back
Top