The dirac delta function

In summary, the conversation discusses the difference between writing δ(x-x') and δ(x') to indicate that the function is infinite at x=x' and 0 everywhere else. The former notation is more commonly used and makes more sense when considering the equality involving integration. However, it is also possible to define a "delta function" for each real number x, where x indicates where the function has its peak. It is important to note that δ(x') is not a function, but a continuous linear functional or distribution.
  • #1
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What's the reason that you write δ(x-x') rather than just δ(x') both indicating that the function is infinite at x=x' and 0 everywhere else? For me that notation just confuses me, and in my opinion the other notation is easier.
 
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  • #2
aaaa202 said:
What's the reason that you write δ(x-x') rather than just δ(x') both indicating that the function is infinite at x=x' and 0 everywhere else? For me that notation just confuses me, and in my opinion the other notation is easier.

I don't understand exactly what you want to do, but it doesn't seem to make sense. Consider the equality $$\int f(x')\delta(x'-x)dx'=f(x).$$ Are you suggesting that we write the left-hand side as
$$\int f(x')\delta(x')dx'$$ or as $$\int f(x')\delta(x)dx'\ ?$$ The former is independent of x, so it can't be equal to f(x). The latter at least contains an x, but since the "δ(something)" doesn't contain an x', it's not involved in the integration, so we should be able to take it outside the integral.

It would however make sense to define one "delta function" ##\delta_x## for each real number x: ##\delta_x(y)=\delta(y-x)##. Now x indicates where the "function" has its peak.
 
  • #3
[itex]\delta(x')[/itex] is not a function, it is a number. Specifically, it is 0 if x' is not 0, undefined if x= 0.
 
  • #4
No, it is a continuous linear functional, or distribution. Physicists just call it a function but it is not a number.
 
  • #5
You really need to think before you speak. [itex]\delta(x)[/itex] and [itex]\delta(x- x')[/itex], for fixed x', are "functionals" or "generalized functions". [itex]\delta(x')[/itex] is a number, just as [itex](x- x')^2[/itex], for fixed x', is a function of x, but [itex]x'^2[/itex] is a number.
 
  • #6
I didn't say for fixed x prime. The delta distribution is defined in terms of the independent variable. Look at the second integral in Fredrik's post above. Clearly well defined and equal to f(0). I can call my variables anything that I want and it doesn't change the meaning of the distribution. I still think that calling it a number is rather loose.
 

What is the Dirac delta function?

The Dirac delta function, also known as the delta function or the impulse function, is a mathematical function that is used to model an infinitesimally narrow and infinitely tall spike at a specific point on a function's graph.

What is the purpose of the Dirac delta function?

The Dirac delta function is commonly used in mathematics and physics to represent point masses or point charges, as well as to describe idealized point sources in fields such as fluid mechanics and electromagnetism.

What is the mathematical representation of the Dirac delta function?

The Dirac delta function is often denoted as δ(x) and is defined as zero for all values of x except for x=0, where it is infinite. This can be written mathematically as: δ(x) = 0 for x ≠ 0 and ∫δ(x)dx = 1 for x = 0.

How is the Dirac delta function different from a regular function?

The Dirac delta function is not a regular function in the traditional sense, as it does not have a well-defined value for most inputs. Instead, it is considered a distribution or generalized function that is used to represent certain mathematical concepts and physical phenomena.

What are some real-world applications of the Dirac delta function?

The Dirac delta function has a wide range of applications in various fields such as signal processing, quantum mechanics, and engineering. It is commonly used to model signals, impulses, and point sources in these fields, as well as in the solution of differential equations and Fourier analysis.

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