**Gaussian, Error and Complementary Error function**

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Table of Contents -
Glossary -
Active Figures -
Equation Sheet -
Study Aids -
1
2
3
4
5
6
7
8
9

In this section:
- The Gaussian function
- The Error function
- The Complementary Error function

The Gaussian function (also refered to as bell-shaped or "bell" curve)
is of the following form:
(x19)

where s is refered to as the spread
or standard deviation and *A* is a constant. The function can be normalized so that
the integral from minus infinity to plus infinity equals one yielding
the normalized Gaussian:
(x18)

by using the following definite integral:
(x17)

The gaussian function goes to zero at plus and minus infinity while
all the derivatives of any order evaluated at *x* = 0 are zero.
The error function equals twice the integral of a normalized gaussian
function between 0 and *x*/sÖ2:
(x10)

The relation between the normalized gaussion distribution and the error
function equals:
(x20)

A series approximation for small value of *x* of this function
is given by:
(x11)

while an approximate expression for large values of *x* can be obtained
from:
(x12)

The complementary error function equals one minus the error function yielding:
(x13)

which, combined with the series expansion of the error function listed
above, provides
approximate expressions for small and large values of *x*:
(x14)

(x15)

## General

The gaussian function, error function and complementary error
function are frequently used in probability theory since the
normalized gaussian curve represents the probability distribution
with standard deviation s relative to the average
of a random distribution.
The error function represents the probability that the parameter of
interest is within a range between
-*x*/sÖ2
and *x*/sÖ2, while the complementary
error function provides the probability that the parameter is
outside that range.
All three functions are shown in the figure below:

gaussian.xls - gausslin.gif
*Normalized gaussian with
s = 1 (black curve), error function
(red curve) and complementary error function (green curve)*

gaussian.xls - gausslog.gif
*Gaussian (upper) and the complementary error
function on a semi-logarithmic scale. The constant A
is chosen to equal one
and s = 1*

© Bart J. Van Zeghbroeck, 1998