Sign
The sign of a real number, also called sgn or signum, is 
for a negative number (i.e., one with a minus sign "
"), 0 for the number zero, or 
for a positive number (i.e., one with a plus sign "
"). In other words, for real 
,
![sgn(x)=<span style=]() {-1 for x<0; 0 for x=0; 1 for x>0. " src="http://mathworld.wolfram.com/images/equations/Sign/NumberedEquation1.gif" style="height:62px; width:151px" /> |
(1)
|
For real 
, this can be written
 |
(2)
|
and satisfies
 |
(3)
|
for real 
can also be defined as
 |
(4)
|
where 
is the Heaviside step function.
The sign function is implemented in the Wolfram Language for real 
as Sign[x]. For nonzero complex numbers, Sign[z] returns 
, where 
is the complex modulus of 
.
can also be interpreted as an unspecified point on the unit circle in the complex plane (Rich and Jeffrey 1996).
REFERENCES:
Bracewell, R. "The Sign Function, 
." In The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 61-62, 1999.
Rich, A. and Jeffrey, D. "Function Evaluation on Branch Cuts." SIGSAM Bull., No. 116, 25-27, June 1996.
