Appendix E. erf( )
Appendix E erf()
NOTE: In the mathematical literature you will see many tabulations of the function erf ( ), sometimes with definitions different from what is presented here. Although the definitions may be transformed from one to another fairly easily, it is sometimes confusing when you need one function and have only a table for another. Table E.1 shows how to convert tabulated data (or software functions) provided under an alternate definition into my format. The functions defined here as erf ( ) and erfc ( ) may be computed from any of the other forms:
Equation E.1
Equation E.2
Equation E.3
Equation E.4
Equation E.5
Equation E.6
Equation E.7
Equation E.8
Table E.1. Alternate Definitions for the Error Function
|
Function name |
Definition |
Range for a > 0 |
Comment |
|---|---|---|---|
|
My error function [136] , [137] erf ( a ) |
|
[0,1] |
I like the minimal simplicity of this definition. |
|
My complementary error function erfc ( a ) |
|
[0,1] |
Used in probability analysis and communication theory to emphasize connection with Gaussian probability density function. |
|
Error function [138] (alt. definition) erf 2 ( a ) |
|
[ ½,1] |
Used in probability analysis and communication theory to emphasize connection with Gaussian probability density function. |
|
Yet another error function [139] erf 3 ( a ) |
|
[0, ½] |
Variant; same as ( erf 2 ( a ) “ ½). |
|
Complementary error function [140] , [141] Q ( a ) |
|
[ ½,0] |
Variant; same as (1 “ erf 2 ( a )). |
|
MathCad built-in function Pnorm ( a , m , s ) |
|
[ ½,1] |
For m = 0 and s = 1, same as erf 2 ( x ); NOTE: In versions of MathCad earlier than 2001 do not use the built-in function erf ( ), as it is a totally unrelated function. Use Pnorm instead. |
[136] John M. Wozencraft, Irwin Mark Jacobs, Principles of Communication Engineering , John Wiley & Sons, 1965, ISBN 0 471 96240 6
[137] John A. Aseltine, Transform Method in Linear System Analysis , McGraw-Hill, 1958, U.S. Lib. Congress cat. no. 58-8038
[138] Harry Van Trees, Detection, Estimation, and Modulation Theory: Part I , John Wiley and Sons, 1968 ISBN 471 89955 0
[139] Athanasios Papoulis, Probability, Random Variables, and Stochastic Processes , McGraw-Hill, 1965, ISBN 07-048448-1
[140] John M. Wozencraft, Irwin Mark Jacobs, Principles of Communication Engineering , John Wiley & Sons, 1965, ISBN 0 471 96240 6
[141] Bernard Sklar, Digital Communications , Prentice Hall, 1988, ISBN 0-13-211939-0