# 4.1: Error Function

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- 7228

Before we start this chapter, let’s just make sure that we are familiar with the error function erf a. We may need it during this chapter.

Here is a graph of the gaussian function

\[ y = \frac{1}{ \sqrt{ \pi}} e^{-x^2}.\]

I have chosen the coefficient \(1/ \sqrt{\pi}\) so that the area under the curve, from − ∞ to + ∞ is 1. The maximum value, which occurs at x = 0, is \(1/ \sqrt{ \pi} = 0.5642\), and it is easy to show that the half width at half the maximum is \( \sqrt{ \ln 2} = 0.8326\). Also of some interest (though not particularly in this chapter) is the square root of the second moment of area around the y-axis. In a mechanical context this would be called the **radius of gyration**. In a statistical context it would be called the standard deviation. Either way, its value is \(1/ \sqrt{2} = 0.7071\). We shall meet the gaussian function again in Chapter 6.

In the present chapter we shall need to make use of the **error function** erf *a*. This is the area under the gaussian curve from *x* = *-a* to *x *= +*a*:

\[ \text{erf} a = \frac{1}{ \sqrt{ \pi}} \int_{- a}^{+ a} e^{-x^2} dx.\]

The area outside the limits x = ±a, which is the area under the two “tails” of the gaussian function, is sometimes called the *complementary error function*:

\[ \text{erfc} a = 1 - \text{erf} a\]

It will be clear that erf a goes from 0 to 1 as a goes from 0 to infinity. Note also that

erfc (one standard deviation) = 0.3173

erfc (two standard deviations) = 0.0455.

Here are graphs of erf *a* (continuous line) and erfc *a* (dashed line) versus *a*.