A question on the Dirac delta distribution

In summary, the Dirac delta distribution is a mathematical function that is defined as zero everywhere except at a single point, where it is infinite. It is used to represent point charges or impulses in systems, and has different properties and behaviors compared to regular functions. Its integral is equal to 1 and it is used in practical applications such as signal processing and quantum mechanics, but it also has limitations such as being a distribution and not a true function, having infinite magnitude at the defined point, and not being defined for all points on the real line.
  • #1
user1139
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Homework Statement
Please refer below
Relevant Equations
Please refer below
Is it correct to say that

$$\int e^{-i(k+k’)x}\,\mathrm{d}x$$

is proportional to ##\delta(k+k’)##?
 
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  • #2
In the distributional sense, yes.

Consider a test function ##\hat \phi(k)## being the Fourier transform of ##\phi(x)##. It holds that
$$
\int \hat\phi(k) \int e^{-i(k+k')x} dx\, dk = \int e^{-ik'x} \int \hat\phi(k) e^{-ikx} dk\, dx
= \int e^{-ik'x} \phi(-x) dx = \int e^{ik'x} \phi(x) dx = 2\pi \hat \phi(-k') = \int 2\pi \delta(k+k') \hat\phi(k) dk.
$$
 

Related to A question on the Dirac delta distribution

1. What is the Dirac delta distribution?

The Dirac delta distribution, also known as the Dirac delta function, is a mathematical concept used to represent a point mass or impulse at a specific location. It is often used in physics and engineering to model point sources of energy or mass.

2. How is the Dirac delta distribution different from a regular function?

The Dirac delta distribution is not a traditional function in the mathematical sense, as it cannot be evaluated at a specific point. Instead, it is defined as a distribution or generalized function that has certain properties, such as a value of zero everywhere except at the point of interest where it has a value of infinity.

3. What are some applications of the Dirac delta distribution?

The Dirac delta distribution has many applications in physics and engineering, such as modeling point sources of force or charge, analyzing signal processing systems, and solving differential equations. It is also used in quantum mechanics to represent the position of a particle.

4. Can the Dirac delta distribution be graphed?

Technically, the Dirac delta distribution cannot be graphed as it is not a traditional function. However, it is often represented as a spike or impulse at the specific location of interest on a graph, with a height of infinity and a width of zero.

5. What is the relationship between the Dirac delta distribution and the Kronecker delta?

The Dirac delta distribution and the Kronecker delta are closely related but have different domains. The Dirac delta distribution is a continuous function defined over the real numbers, while the Kronecker delta is a discrete function defined over the integers. They both have similar properties, such as being zero everywhere except at a specific point, and are often used interchangeably in different contexts.

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