- #1
gabriellelee
- 21
- 1
- Homework Statement
- What is the integral of the following equation?
- Relevant Equations
- See below for the equation.
Integral Involving the Dirac Delta Function
- Thread starter gabriellelee
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SUMMARY
The integral involving the Dirac delta function is defined mathematically as $$\int_{-\infty}^{+\infty}\delta(x) f(x) dx = f(0)$$, indicating that the integral evaluates to the value of the function at zero. The Dirac delta function is not a conventional function but a distribution, which complicates its integration. It is essential to understand the context of test functions, particularly within the framework of functional analysis, where the space of test functions can vary based on the problem at hand. The discussion emphasizes the importance of using appropriate spaces, such as Schwartz space or compact support functions, to properly define the Dirac delta function.
PREREQUISITES- Understanding of Dirac delta function and its properties
- Familiarity with functional analysis concepts
- Knowledge of test functions and Schwartz space
- Basic principles of distributions in mathematical physics
- Study the properties of the Dirac delta function in detail
- Learn about Schwartz space and its applications in functional analysis
- Explore the concept of rigged Hilbert spaces in quantum mechanics
- Investigate the role of test functions in the theory of distributions
Mathematicians, physicists, and students in advanced calculus or functional analysis courses who are looking to deepen their understanding of distributions and the Dirac delta function.
- #2
Abhishek11235
- 174
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What are your limits of integral?
- #3
gabriellelee
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-∞ to +∞Abhishek11235 said:
- #4
Abhishek11235
- 174
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So what do you think?gabriellelee said:
- #5
gabriellelee
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1?Abhishek11235 said:
- #6
Abhishek11235
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Yesgabriellelee said:
- #7
gabriellelee
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I know intuitively that the integral is 1, can you explain it to me mathematically?Abhishek11235 said:
- #8
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By definition:gabriellelee said:
$$\int_{-\infty}^{+\infty}\delta(x) f(x) dx = f(0)$$
- #9
wrobel
Science Advisor
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Actually it is not an integral it just a symbol. As well as ##\delta-##function is not a function in usual sense. These things are clarified in functional analysis
- #10
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Hm, is this an exercise in a physics course or a trick question in a mathematics course? In the latter case, I think, the answer is that the integral is undefined, because the exponential function is not a proper test function in the domain (of, e.g., Schwartz functions), the Dirac distribution is defined on.
- #11
wrobel
Science Advisor
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There is no sense to speak about integrals here. It is nothing more than symbol in this context.
A space of test functions is a conditional question. In different problems this space is introduced differently this depends of problem we consider . Loosely speaking, the smaller space of test functions the bigger space of generalized functions.
For example, consider a subspace of ##\mathcal D'(\mathbb{R})## that consists of generalized functions with compact support. Such generalized functions are naturally defined on ##C^\infty(\mathbb{R})##.
A space of test functions is a conditional question. In different problems this space is introduced differently this depends of problem we consider . Loosely speaking, the smaller space of test functions the bigger space of generalized functions.
For example, consider a subspace of ##\mathcal D'(\mathbb{R})## that consists of generalized functions with compact support. Such generalized functions are naturally defined on ##C^\infty(\mathbb{R})##.
Last edited:
- #12
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I'm only a bit familiar with the formalism, but I don't think that ##\exp(\mathrm{i} x)## is a proper test function in any case. I'm a bit unsure, whether the Dirac ##\delta## function (which has compact support, namely only 1 point) is defined on the entire ##C^{\infty}(\mathbb{R})##. Isn't it usually either ##C_0^{\infty}(\mathbb{R})## or the "Schwartz space of quickly falling functions"?
- #13
wrobel
Science Advisor
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What is the problem to define the Dirac δ function on ##C^\infty(\mathbb{R})##?
Just put
$$(\delta, \varphi):=\varphi(0),\quad \varphi\in C^\infty(\mathbb{R})$$
This is a continuous linear functional with respect the standard topology in ##C^\infty(\mathbb{R})##
Just put
$$(\delta, \varphi):=\varphi(0),\quad \varphi\in C^\infty(\mathbb{R})$$
This is a continuous linear functional with respect the standard topology in ##C^\infty(\mathbb{R})##
- #14
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I don't know. Maybe you can do that, but in quantum mechanics you often have confused students, because they think ##\exp(\mathrm{i} p x)## is a "eigenstate of the momentum operator", but it is not square integrable nor in the domain of the momentum operator as an essentially self-adjoint operator. Here you need the rigged-Hilbert space description!