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Error Function erf(x) & Complimentary Error Function erfc(x)

Note: The error function and complimentary function used in communications are not exactly the same as the ones typically used in statistics. The relationship between the two is given at the bottom of the page.

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations.

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics, materials science, and partial differential equations. - Wikipedia


The Gaussian probability density function with mean = 0 and variance =1 is

Probability density function equation

The error function erf(x) is defined as:

Error function equation for communications


Note that erf(0) = 0.5, and that erf(∞)=1.


The complimentary error function erfc(x) is defined as:

Complimentary error function equation for communications


The following graph illustrates the region of the normal curve that is being integrated.

Complimentary error function graph - RF Cafe



For large values of x (>3), the complimentary error function can be approximated by:

Approximation complimentary error function equation for communications


The error in the approximation is about -2% for x=3, and -1% for x=4, and gets progressively better with larger values of x.


Approximations

RF Cafe visitor Ilya L. provided an approximation for the error function and complimentary error function that was published by Sergei Winitzki titled, "A handy approximation for the error function and its inverse." February 6, 2008.

Here are the main results:

Error function approximation:
      RF Cafe - Error function approximation, where  RF Cafe - Error function approximation constant "a"


Complimentary error function:
      RF Cafe - Complimentary error function approximation


NOTE: I used to have an alternative approximation formula for the complimentary error function for large values of x, but decided to remove it since the source for it is not generally available to the public. It can be found as equation #13, on page 641, of IEEE Transactions on Communications volume COM-27, No. 3, dated March 1979. A subscription to the IEEE service is required to access the article.


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