*e*, 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

^{-x2}*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 involvingOf course, if you leteappear in the Gaussian distribution. The integral^{-ax2}

can also be written withyas the dummy variable:

There can be multiplied together to get

A point in thexyplane can also be specified by the polar coordinatesrandθ(Fig. K.1). The element of areadxdyis replaced by the elementrdrdθ:

To continue, make the substitutionu = ar, so that^{2}du = 2ardr. Then

The desired integral is, therefore,

*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.”

## No comments:

## Post a Comment