# Proving that the Dirac Delta is the limit of Gaussians

• xWaffle
In summary, to prove the given statement, one can split the integral into three parts and use the same method as for φ = constant. This involves transforming the integral into a double integral and then integrating by parts. The limit of the integral with no 1/λ term can be proved to approach 0, while the remaining term will give the desired result of φ(0).
xWaffle

## Homework Statement

I need to prove for arbitrary functions φ(x) that:

$$\lim_{\lambda \to 0} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi} \lambda} exp\left( \frac{-x^{2}}{2 \lambda^{2}} \right) \varphi(x) dx = \varphi(0),$$

which, in the sense of distributions is basically the delta function,

$$\frac{1}{\sqrt{2 \pi} \lambda} exp\left( \frac{-x^{2}}{2 \lambda^{2}} \right) = \delta (x)$$

(see above)

## The Attempt at a Solution

I can easily explain graphically what happens as λ→0, i.e., as λ approaches zero, the width of the distribution gets thinner and thinner, and the height of the peak gets higher and higher; the limit, of course, being an infinitely high and infinitesimally thin spike (which we know as the Dirac delta function).

Now, to prove this mathematically, I've seen some weird and completely different approaches being taken here, in random lecture notes I find around the web, and on stackoverflow. Everyone's approach is different but none really hit home with me.

After discussing with my professor a bit, he hinted I should try to split the integral up (I'm assuming into a part from -∞ to some arbitrary constant -c, -c to c, and c to +∞. Nothing I can find online or in my textbook resembles this, which makes me think we're supposed to be really stretching our brains with this one.

So, if I split the integral up,

$$\int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi} \lambda} exp\left( \frac{-x^{2}}{2 \lambda^{2}} \right) \varphi(x) dx = \int_{-\infty}^{-c} \frac{1}{\sqrt{2 \pi} \lambda} exp\left( \frac{-x^{2}}{2 \lambda^{2}} \right) \varphi(x) dx + \int_{-c}^{c} \frac{1}{\sqrt{2 \pi} \lambda} exp\left( \frac{-x^{2}}{2 \lambda^{2}} \right) \varphi(x) dx + \int_{c}^{\infty} \frac{1}{\sqrt{2 \pi} \lambda} exp\left( \frac{-x^{2}}{2 \lambda^{2}} \right) \varphi(x) dx$$

Now what..? How does this help me? I realize c is free to be whatever we want it to be, but how can that put us onto a path to proving:

$$\lim_{\lambda \to 0} \int_{-\infty}^{\infty} \frac{1}{\sqrt{2 \pi} \lambda} exp\left( \frac{-x^{2}}{2 \lambda^{2}} \right) \varphi(x) dx = \varphi(0)?$$

hi xWaffle!

have you tried the same method as for φ = constant?

ie make it the square-root of the double integral ∫∫ (1/2πλ2) e-r22 φ(rcosθ)φ(rsinθ) rdrdθ,

integrate wrt θ, then integrate by parts:

i think the [] part is [φ(0)]2, and the ∫ part has no 1/λ and so hopefully can be proved to –> 0

## 1. What is the Dirac Delta function and why is it important?

The Dirac Delta function, denoted by δ, is a mathematical concept used to represent a point mass or impulse at a specific location. It is important because it allows us to mathematically model and analyze systems with point-like behavior, such as point charges in electrostatics or point masses in celestial mechanics.

## 2. How is the Dirac Delta function related to Gaussians?

The Dirac Delta function can be seen as the limit of a sequence of Gaussian functions with increasing standard deviation. As the standard deviation approaches 0, the Gaussian function becomes more concentrated at the origin, resulting in the Dirac Delta function.

## 3. Can the Dirac Delta function be graphically represented?

No, the Dirac Delta function cannot be graphically represented in the traditional sense as it is not a continuous function. However, it can be represented as a spike or impulse at the origin, with an infinite height and zero width.

## 4. How is the Dirac Delta function used in physics and engineering?

The Dirac Delta function is used in many areas of physics and engineering, such as signal processing, quantum mechanics, and fluid dynamics. It is often used to model point-like sources of energy or mass, and to simplify mathematical calculations in these fields.

## 5. Is there a rigorous mathematical proof for the Dirac Delta function as the limit of Gaussians?

Yes, there are several mathematical proofs that show the convergence of Gaussians to the Dirac Delta function as the standard deviation approaches 0. These proofs involve concepts from calculus, such as the notion of a limit and the use of the concept of a generalized function.

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