Gaussian Function

The area under the curve is:

 

 The square of I0(a) can be written in the form

Now convert to plane polar coordinates, letting r2 = x2 + y2 and dxdy = rdrdq. The appropriate limits of integration over all space in spherical polar coordinates are 0 £ r < ¥ and 0 £ q £ p/2 so

which is elementary and gives

so that

 

Note that the Gaussian function

is centered at zero. In general, we can express a Gaussian centered at any position x0 as

For example, if we let a = 0.0001 we can plot two Gaussian using the above functional form.

Notice that if we integrate either one of these the magnitude of the integral is the same. This is evident from inspection since the integral is the area under the curve and the areas are both the same. To see this mathematically, consider the integral

We make the substitution u = x – x0, then du = dx. In that case

Which is identical to the integral we started with above.

Note that often Gaussian are written in the form

Integration leads to an area of

In this case s is called the standard deviation. We can also relate s to the full-width-at-half-maximum. This is done by recognizing that the magnitude of the function is 1 at its peak. Therefore, we can solve for x when

so

and the fwhm is .