signum function

The signum function is the functionMathworldPlanetmath sgn:

sgn(x) = {-1whenx<0,0whenx=0,1whenx>0.

The following properties hold:

  1. 1.

    For all x, sgn(-x)=-sgn(x).

  2. 2.

    For all x, |x|=sgn(x)x.

  3. 3.

    For all x0, ddx|x|=sgn(x).

Here, we should point out that the signum function is often defined simply as 1 for x>0 and -1 for x<0. Thus, at x=0, 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 sgn(0) to any value. However, setting sgn(0)=0 is motivated by the above relations. On a related note, we can extend the definition to the extended real numbers ¯={,-} by defining sgn()=1 and sgn(-)=-1.

A related function is the Heaviside step function defined as

H(x) = {0whenx<0,1/2whenx=0,1whenx>0.

Again, this function is sometimes left undefined at x=0. The motivation for setting H(0)=1/2 is that for all x, we then have the relations

H(x) = 12(sgn(x)+1),
H(-x) = 1-H(x).

This first relation is clear. For the second, we have

1-H(x) = 1-12(sgn(x)+1)
= 12(1-sgn(x))
= 12(1+sgn(-x))
= H(-x).

Example Let a<b be real numbers, and let f: be the piecewise defined function

f(x) = {4whenx(a,b),0otherwise.

Using the Heaviside step function, we can write

f(x) = 4(H(x-a)-H(x-b)) (1)

almost everywhere. Indeed, if we calculate f using equation 1 we obtain f(x)=4 for x(a,b), f(x)=0 for x[a,b], and f(a)=f(b)=2. Therefore, equation 1 holds at all points except a and b.

1 Signum function for complex arguments

For a complex numberMathworldPlanetmathPlanetmath z, the signum function is defined as [2]

sgn(z) = {0whenz=0,z/|z|whenz0.

In other words, if z is non-zero, then sgnz is the projection of z onto the unit circleMathworldPlanetmath {z|z|=1}. Clearly, the complex signum function reduces to the real signum function for real arguments. For all z, we have


where z¯ is the complex conjugateDlmfMathworldPlanetmath of z.


  • 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