The following properties hold:
For all ,
For all ,
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 . 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
Using the Heaviside step function, we can write
1 Signum function for complex arguments
- 1 E. Kreyszig, Advanced Engineering Mathematics, John Wiley & Sons, 1993, 7th ed.
- 2 G. Bachman, L. Narici, Functional analysis, Academic Press, 1966.
|Date of creation||2013-03-22 13:36:41|
|Last modified on||2013-03-22 13:36:41|
|Last modified by||yark (2760)|
|Defines||Heavyside step function|