# Why is $e^{-x^2}$ such a big deal?

It seems like integrating it is a big deal and everything, but I don't understand why. My teacher is making it seem really important in my calculus class for seemingly no reason.

Help would be appreciated.

• It has no elementary anti-derivative, yet has a very simple expression for its integral along the positive real axis. Oct 26 '14 at 1:54
• This question is probably overly broad, precisely because the Gaussian function is such a big deal. Oct 26 '14 at 2:01
• The normal distribution in probability is one major example of where this function arises. Oct 26 '14 at 2:50
• I don't have any background in probability/statistics past what I learned about permutations and combinations in my eighth grade algebra class. That might be why I don't understand the importance yet; I really never knew that it went that far. I am determined to do some further research to figure this stuff out; what do you guys recommend to get me up to speed? Oct 26 '14 at 2:52

The importance of the integral isn't limited to the fact that there is no closed form anti-derivative. The integral is important, in my opinion, because it makes many calculations possible which would not be possible otherwise. These calculations provide us with insights into the nature of certain mathematical problems which is a hard thing to come by.

Here is a list of important instances of the Gaussian that I can think of off the top of my head,

• The sequence of functions $\frac{n}{\sqrt{\pi}} exp{-x^2/n^2}$ defines the generalized function known as the "Dirac Delta Function". This is so important I can't even get into it here, look on wikipedia to learn more.

• Gaussian integration can be used to prove the Fourier inversion theorem for a certain class of functions.

• The Fourier transform of a Gaussian can actually be computed (because we know the integral) which allows us to construct a Gaussian wave packet in quantum mechanics. Wave packets are an invaluable tool for gaining intuition about a problem.

• In probability theory the central limit theorem tells us that the sum of a large number of random outcomes has a Gaussian probability distribution regardless of the nature of the original distribution. Since averages and variances require integration of the probability distribution we couldn't solve many important problems in probability without being able to evaluate this integral.

You may have heard the phrase "When you have a hammer every problem looks like a nail". In this analogy the Guassian integral is a really good hammer and it allows us to turn normally intractable problems into nice easy nails. Hope this helps.

• I'm skeptical as to whether the fact about the Dirac Delta function is significant - for any positive, symmetrical $f$ with area $1$ below, the limit $\lim_{n\rightarrow\infty}nf(\frac{x}n)$ is the Dirac Delta function. There's nothing special about $e^{-x^2}$ here, really. Oct 26 '14 at 2:48
• True, the second sentence was meant to refer to the importance of the Delta Function rather than the particular sequence. Bad writing on my part. I do maintain that many proofs of the Delta's properties are more easily executed using a sequence of Guassians than many other representations. Oct 26 '14 at 3:14

Are you familiar at all with probability and statistics? The famous "bell curve" of the normal distribution is given by $ke^{-x^2}$, for an appropriate constant $k$, or some translate of that curve, so integrating this function is essential to many calculations in statistics.

• It's "tricky" to do its definite integral on $\mathbb{R},$ but it's a different story for the antiderivative, which is not an elementary function. Oct 26 '14 at 1:57