Understanding the Delta Function: Exploring the Role of k in Equations

In summary, the conversation discussed the reasons for the use of both positive and negative values for k in an equation, and how it relates to the domain of the function. It also touched on the Dirac distribution and its applications in Fourier's integral and the fundamental theorem of algebra. The conversation also mentioned Cauchy's and L Schwartz's theories of distributions and Dirac's naming of the delta function.
  • #1
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in this equation why does k take on both positive and negative values? isn't k a fixed constant that can only be positive or negative
 

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  • #2
mathnerd15 said:
in this equation why does k take on both positive and negative values? isn't k a fixed constant that can only be positive or negative

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  • #3
oh you have to click on the attachment and then the picture to see the equation. so the domain of the function is kx so if k ranges from -infinity to infinity then you need a +- before the integral?
 
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  • #4
It's just an example for pretty unclear notation. I guess the formula is an attempt to prove the equation
[tex]\delta(k x)=\frac{1}{|k|} \delta(k x).[/tex]
Of course you have to assume that [itex]k \neq 0[/itex]. Otherwise the equation doesn't make any sense to begin with.

Then you have to just do the substitution [itex]y=k x[/itex] in the integral with the distribution times an arbitrary test function to prove this formula. For [itex]k>0[/itex] you find
[tex]\int_{\mathbb{R}} \mathrm{d} x f(x) \delta(k x)=\int_{\mathbb{R}} \mathrm{d}y \frac{1}{k} f(y/k) \delta(y)=\frac{1}{k} f(0).[/tex]
For [itex]k<0[/itex] you have
[tex]\int_{\mathbb{R}} \mathrm{d} x f(x) \delta(k x)=\int_{\mathbb{R}} \mathrm{d} y \left (-\frac{1}{k} \right ) f(y/k) \delta (y)=-\frac{1}{k} f(0).[/tex]
On the other hand the distribution
[itex]\frac{1}{|k|} \delta(x)[/itex]
has the same outcome under an integral with an arbitrary test function, which proves the above statement about the Dirac distribution.

You can generalize this for arbitrary function [itex]y(x)[/itex] which have only single-order zeros, i.e., [itex]y(x_k)=0[/itex] but [itex]y'(x_k) \neq 0[/itex] for [itex]k \in \{1,\ldots,n\}[/itex]. Then you can prove in pretty much the same way as the above example
[tex]\delta[y(x)]=\sum_{k=1}^{n} \frac{1}{|y'(x_k)|} \delta(x-x_k).[/tex]
 
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  • #5
[tex]f(x)=\frac{1}{2\pi}\int_{R}e^{ipx}\left ( \int_{R}f(\alpha)d\alpha \right )dp=\frac{1}{2\pi}\int_{R}\left (\int_{R}e^{ipx}e^{-ip\alpha}\right)f(\alpha)d\alpha= \int_{R}\delta (x-\alpha)f(\alpha)d\alpha.[/tex] where,[itex] \delta(x-a)=\int_{R}e^{ip(x-\alpha)}\left dp. [/tex]as you know Cauchy expressed Fourier's integral as exponentials and the delta distribution can be expressed in this way (he also pointed out that the integrals are non-commutative in some circumstances). In modern times there is L Schwartz's theory of distributions. Dirac called it the delta function because he used it as a continuous analogue of the discrete Kronecker delta.
Here is a vast generalization of the fundamental theorem of algebra
 

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  • #6
[tex](x+y)^{N+1}=_{0}^{N+1}\textrm{C}x^{N+1}+\sum_{k=1}^{N}(_{k}^{N}\textrm{C}+_{k-1}^{N}\textrm{C})x^{N+1-k}y^{k}+_{N+1}^{N+1}\textrm{C}y^{N+1}=_{0}^{N+1}\textrm{C}x^{N+1}+\sum_{k=1}^{N}_{k}^{N+1}\textrm{C}x^{(N+1)-k}y^{k}+_{N+1}^{N+1}\textrm{C}y^{N+1}=\sum_{k=0}^{N+1}_{k}^{N+1}{C}x^{(N+1)-k}y^{k}[/tex]
 
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What is a delta function?

A delta function is a mathematical function that is used to represent the concentration of a point mass or a point charge at a specific location. It is also known as the Dirac delta function, named after the physicist Paul Dirac.

What is the purpose of a delta function?

The purpose of a delta function is to simplify mathematical calculations by representing a point mass or charge as a single point. It is also used in physics and engineering to model physical phenomena, such as point forces or point charges.

How is a delta function different from other functions?

A delta function is different from other functions because it is defined as infinite at a specific point and zero everywhere else. It also has a value of zero when integrated over any interval that does not include the point of singularity.

What are the properties of a delta function?

The properties of a delta function include being infinitely tall at the point of singularity, having an area of one when integrated over its entire domain, and being zero everywhere else. It also satisfies the sifting property, which states that when a function is multiplied by a delta function, the result is the value of the original function at the point of singularity.

Where is a delta function used?

A delta function is used in various fields of science and engineering, such as electromagnetics, signal processing, and quantum mechanics. It is also used in probability and statistics to represent point probabilities, and in computer science to model point processes.

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