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

Note: The error function and complementary 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." - Wikipedia

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

   Probability density function equation - RF Cafe

The error function erf(x) is defined as:

   Error function equation for communications - RF Cafe

 

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

 

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

   Complimentary error function equation for communications - RF Cafe

 

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 complementary error function can be approximated by:

   Approximation complimentary error function equation for communications - RF Cafe

 

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 complementary error function that was published by Sergei Winitzki titled, "A handy approximation for the error function and its inverse," February 6, 2008 (Google Drive file - slow to load).

Here are the main results:

Error function approximation:

 

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

 

Complementary error function:

      Complimentary error function approximation

 

 

NOTE: I used to have an alternative approximation formula for the complementary 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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    BSEE - KB3UON

RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The World Wide Web (Internet) was largely an unknown entity at the time and bandwidth was a scarce commodity. Dial-up modems blazed along at 14.4 kbps while tying up your telephone line, and a nice lady's voice announced "You've Got Mail" when a new message arrived...

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