signum function
The signum function is the function
The following properties hold:
-
1.
For all ,
-
2.
For all ,
-
3.
For all , .
Here, we should point out that the signum function is often defined simply as for and for . Thus, at , it is left undefined. See for example [1]. In applications such as the Laplace transform this definition is adequate, since the value of a function at a single point does not change the analysis. One could then, in fact, set to any value. However, setting is motivated by the above relations. On a related note, we can extend the definition to the extended real numbers by defining and .
A related function is the Heaviside step function defined as
Again, this function is sometimes left undefined at . The motivation for setting is that for all , we then have the relations
This first relation is clear. For the second, we have
Example Let be real numbers, and let be the piecewise defined function
Using the Heaviside step function, we can write
(1) |
almost everywhere. Indeed, if we calculate using equation 1 we obtain for , for , and . Therefore, equation 1 holds at all points except and .
1 Signum function for complex arguments
For a complex number , the signum function is defined as [2]
In other words, if is non-zero, then is the projection of onto the unit circle . Clearly, the complex signum function reduces to the real signum function for real arguments. For all , we have
where is the complex conjugate of .
References
- 1 E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 1993, 7th ed.
- 2 G. Bachman, L. Narici, Functional analysis, Academic Press, 1966.
Title | signum function |
Canonical name | SignumFunction |
Date of creation | 2013-03-22 13:36:41 |
Last modified on | 2013-03-22 13:36:41 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 11 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 30-00 |
Classification | msc 26A06 |
Related topic | ModulusOfComplexNumber |
Related topic | HeavisideStepFunction |
Related topic | PlusSign |
Related topic | SineIntegralInInfinity |
Related topic | ListOfImproperIntegrals |
Defines | Heavyside step function |
Defines | step function |