Proper proof of a delta function

In summary: This is the definition of the Schwartz product. Note that in this case we have ##f(0) = 0##. The proof is easy. Let ##\varphi## be a test function, then we have\int dt t \delta(t) \varphi(t) = \int dt t \delta(t) \varphi(t) = \int dt \delta(t) t \varphi(t) = \int dt \delta(t) \varphi(t) = \varphi(0) \int dt \delta(t) = 0where we used integration by parts.
  • #1
tolove
164
1
Prove:
tδ(t) = 0

The answer our TA has given isn't doing it for me:
[itex]\int dt \delta(t)f(t) = (0)f(0) = 0[/itex]

I'm wanting to write:
[itex] t \frac{d}{dt}\int \delta(t) dt = t \frac{d}{dt}(1) = t * 0 = 0 [/itex]

Am I right here? This doesn't make use of a test function. I'm very sloppy with proofs!

Thanks for your time!
 
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  • #2
do you mean ##\delta (t)## is the dirac delta function?

also, you are incorrect in stating this:

tolove said:
[itex] t \frac{d}{dt}\int \delta(t) dt = t \frac{d}{dt}(1) [/itex]
note that ##\frac{d}{dt}\int \delta (t) dt= const. ##, thus and arbitrary ##\delta (t)## would work in the above "proof" (assuming it fits the criteria for the fundamental theorem). first, you misuse the fundamental theorem (no bounds). secondly, you assume a function of ##t## is not in fact a function of ##t##. and lastly, if this question is a homework question, i think it belongs in that section.
 
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  • #3
Both ##t\delta(t)## and ##0## are distributions here. So you have to prove an equality of distributions. By definition, two distributions ##F## and ##G## are equal if

[tex]\int \varphi(t)F(t)dt = \int \varphi(t)G(t)dt[/tex]

for all test functions ##\varphi##. So this is what you need to check.
 
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  • #4
tolove said:
Prove:
tδ(t) = 0

The answer our TA has given isn't doing it for me:
[itex]\int dt \delta(t)f(t) = (0)f(0) = 0[/itex]

What don't you like about it? It's exactly the Schwartz product, that is if ##f## is a smooth function and ##\mu## a distribution and ##g## a test function then
##\langle f \, \mu , g \rangle := \langle \mu , fg \rangle##.
 

FAQ: Proper proof of a delta function

1. What is a delta function?

A delta function is a mathematical representation of a point mass, named after the Greek letter delta (Δ). It is defined as an infinitely narrow and infinitely tall function with an area of 1 under its curve. It is often used in physics and engineering to model point sources, such as point charges or point masses.

2. How is a delta function different from a regular function?

A regular function has a well-defined value at every point, whereas a delta function only has a value of 0 everywhere except at the point it represents, where its value is infinite. Additionally, the integral of a delta function over its entire domain is equal to 1, while the integral of a regular function may be any finite value.

3. What is the proper proof of a delta function?

The proper proof of a delta function involves showing that it satisfies the defining properties of a delta function, namely that it is infinitely narrow, infinitely tall, and has an area of 1 under its curve. This can be done through various mathematical techniques, such as using limits, Fourier transforms, or distributions.

4. Why is the delta function useful in scientific research?

The delta function is useful in scientific research because it allows for the modeling and analysis of point sources, which are common in many areas of science and engineering. It also has many mathematical properties that make it a powerful tool for solving certain types of problems, such as those involving differential equations or convolutions.

5. Can the delta function be used in real-world applications?

Yes, the delta function has many real-world applications, particularly in physics and engineering. It is commonly used to model point sources in electromagnetic or gravitational fields, and also has applications in signal processing, image analysis, and probability theory. However, it should be noted that the delta function is an idealized mathematical concept and cannot be physically realized in the strict sense.

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