## Tuesday, April 10, 2012

### Integrating exponentials

Often in physics, and sometimes in life, you come across the need to integrate an exponential of the form

$\large A = \int_{-\infty}^{\infty}e^{-a x^2} dx$

Allow me to show you how to handle this using simple dimensional analysis rather than calculus and memorization. Dimensional analysis can get you out of a bind when working on a plane (sans wireless), in an oral exam or even during Q&A after your colloquium!

First, note that the units of A must be the same as the units of x since exponentials are dimensionless and dx has units of x. Further, examination of the quantity in the exponent reveals that a must have units of 1/x^2, since the argument of an exponential must be dimensionless, too. Thus, the integral must have units of x and involve a, like so:

$\large A = \int_{-\infty}^{\infty}e^{-a x^2} dx \propto \frac{1}{\sqrt{a}}$

This is most of the way there. It turns out that there's a missing factor of the square-root of pi:

$\large A = \int_{-\infty}^{\infty}e^{-a x^2} dx = \sqrt{\frac{\pi}{a}}$

But I think it's pretty cool that you can get to within a factor of root-pi (1.77) without any calculus! I can pretty easily remember the pi part after I get the dimensions correct. Even if I forget, being within a factor of two is good enough for astronomy in most applications.

You might notice that this is the form of the Gaussian function, centered on x=0 with

$\large a = \frac{1}{2\sigma^2}$

Once normalized, the Gaussian function becomes the normal distribution so frequently used in data analysis (and CA1). Note the distinction between a Gaussian function and a normal distribution. The difference is important, but frequently ignored in the scientific literature. For example, a Gaussian has three free parameters. A normal distribution has only two. And only one of these is a proper probability distribution function (pdf).

For more "Street Fighting Mathematics" like this, check out this book.