# Integration of dirac delta composed of function of integration variable

• kmdouglass
In summary, the conversation is about a particular equation (Eq. 66) in a paper by Chandrasekhar. The original poster is having trouble understanding the steps to progress through the equation and is seeking help from others. Two different results have been obtained, with one of them having inconsistent units. The original poster also notes that there may be typos in the paper.
kmdouglass
Hi all,
I'm working through Chandrasekhar's http://prola.aps.org/abstract/RMP/v15/i1/p1_1" and can not understand the steps to progress through Eq. (66) in Chapter 1. The integral is:

$$\prod^{N}_{j=1} \frac{1}{l^{3}_{j}|\rho|}\int^{\infty}_{0} sin(|\rho|r_{j})r_{j}\delta (r^{2}_{j}-l^{2}_{j})dr_{j} = \prod^{N}_{j=1} \frac{sin(|\rho|l_{j})}{|\rho|l_{j}}$$

Could anyone show the steps on how this result was obtained? I am aware of how to simplify a dirac delta that is composed of a function, but it does not lead me to the above result. Thanks.

-kmd

Last edited by a moderator:
Weird, I didn't get that one either. I got

$$\prod^{N}_{j=1} \frac{sin(|\rho|l_{j})}{2|\rho|l_{j}^3}$$

phsopher said:
Weird, I didn't get that one either. I got

$$\prod^{N}_{j=1} \frac{sin(|\rho|l_{j})}{2|\rho|l_{j}^3}$$
That seems more reasonable. In the equation posted by the OP, the units are inconsistent between the two sides, so it can't be right.

Yes, you are right about the units. And someone else aside from myself got phsopher's result as well.

A few equations back, the author defines the probability distribution that he is using, and if I integrate over all angles and radial distances, I don't get unity. I think there are significant typos in this section. Thanks for the help.

kmd

## 1. What is the Dirac delta function?

The Dirac delta function is a mathematical construct used to represent a point mass or impulse at a specific point in a system. It is defined as zero everywhere except at the point of interest, where it is infinitely large, and has a total integral of one.

## 2. How is the Dirac delta function integrated with a function of the integration variable?

The integration of the Dirac delta function with a function of the integration variable is done using the sifting property of the delta function. This property states that the integral of the product of the delta function and any continuous function is equal to the value of the continuous function at the point where the delta function is located.

## 3. What is the significance of integrating a Dirac delta function composed of a function of the integration variable?

The integration of a Dirac delta function composed of a function of the integration variable allows us to evaluate the value of the function at a specific point without having to explicitly evaluate the function at that point. This is useful in many applications, such as signal processing and probability theory.

## 4. Can the Dirac delta function be used in higher dimensions?

Yes, the Dirac delta function can be extended to higher dimensions. In one dimension, it is represented as a spike, while in two dimensions it is represented as a surface delta, and in three dimensions it is represented as a volume delta. The properties and rules for integrating the delta function remain the same in higher dimensions.

## 5. Are there any limitations to using the Dirac delta function?

While the Dirac delta function is a useful mathematical tool, it does have limitations. It can only be used to represent point masses or impulses at specific points, and cannot be used to model continuous distributions. Additionally, it is not a function in the traditional sense, as it is not defined at the point where it is infinitely large. Care must be taken when using the delta function to avoid mathematical inconsistencies.

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