Proving that the Dirac Delta is the limit of Gaussians

[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,

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

Homework Equations

(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:

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

 

[tiny-tim]

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^-r2/λ2 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