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

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:

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

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

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.

No comments:

Post a Comment