Sign function

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In mathematics and especially in computer science, the sign function is a logical function which extracts the sign of a real number. To avoid confusion with the sine function, this function is often called the signum function (after the Latin form of "sign"). The sign function is often represented as sgn and can be defined thus:

Any real number can be expressed as the product of its absolute value and its sign function:

From equation (1) it follows that whenever x is not equal to 0 we have

The signum function is the derivative of the absolute value function (up to the indeterminacy at zero):

The signum function is differentiable with derivative 0 everywhere except at 0. It is not differentiable at 0 in the ordinary sense, but under the generalised notion of differentiation in distribution theory, the derivative of the signum function is two times the Dirac delta function,

The signum function is related to the Heaviside step function H_1/2(x) thus

where the 1/2 subscript of the step function means that H_1/2(0) = 1/2. The signum can also be written using the Iverson bracket notation:

For k > 0, a smooth approximation of the step function is

See Heaviside step function.

[edit] Complex Signum

The signum function can be generalized to complex numbers as

for any z ∈ except z=0. The signum of a given complex number, z is the point on the unit circle of the complex plane that is nearest to z.

Because zero has an equal distance to all points on the unit circle, it is generally given the signum 0.

[edit] See also

• Negative and non-negative numbers
• Absolute value
• Rectangular function