The Gaussian integral

One of my favorite “mathematical tricks” is given in Appendix K of the 4th edition of Intermediate Physics for Medicine and Biology. The goal is to calculate the integral of the Gaussian function, e^-x2, or bell shaped curve. (This is often called the Gaussian Integral). The indefinite integral cannot be expressed in terms of elementary functions (in fact, “error functions” are defined as the integral of the Gaussian), but the definite integral integrated over the entire x axis (from –∞ to ∞) is amazingly simple: the square root of π. Here is how Russ Hobbie and I describe how to derive this result:

Integrals involving e^-ax2 appear in the Gaussian distribution. The integral

can also be written with y as the dummy variable:

There can be multiplied together to get

A point in the xy plane can also be specified by the polar coordinates r and θ (Fig. K.1). The element of area dxdy is replaced by the element rdrdθ:

To continue, make the substitution u = ar², so that du = 2ardr. Then

The desired integral is, therefore,

Of course, if you let a =1, you get the simple result I mentioned earlier. Isn’t this a cool calculation?

To learn more, click here. For those of you who prefer video, click here.

The integral and function are, of course, named after the German mathematician Johann Karl Friedrich Gauss (1777-1855). Asimov’s Biographical Encyclopedia of Science and Technology (2nd Revised Edition) says

“Gauss, the son of a gardener and a servant girl, had no relative of more than normal intelligence apparently, but he was an infant prodigy in mathematics who remained a prodigy all his life. He was capable of great feats of memory and of mental calculation. There are those with this ability who are only average or below-average mentality, but Gauss was clearly a genius. At the age of three, he was already correcting his father’s sums, and all his life he kept all sorts of numerical records, even useless ones such as the length of lives of famous men, in days. He was virtually mad over numbers.

Some people consider him to have been one of the three great mathematicians of all time, the others being Archimedes and Newton.”