Sign function
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. The sign function is often represented as sgn and can be defined thus:
or using the Iverson bracket notation:
Any real number can be expressed as the product of its
absolute value and its sign function:
From equation it follows that whenever
x is not equal to 0 we have
The signum function is the
derivative of the absolute value function :
Encyclopedia
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. The sign function is often represented as
sgn and can be defined thus:
or using the Iverson bracket notation:
Any real number can be expressed as the product of its
absolute value and its sign function:
From equation it follows that whenever
x is not equal to 0 we have
The signum function is the
derivative of the absolute value function :
The signum function is differentiable with derivative 0 everywhere except at 0. It is not differentiable at 0 in the ordinary sense, but under a somewhat generalised notion of differentiation , we can say that the derivative of the signum function is two times the
Dirac delta function,
The signum function is related to the
Heaviside step function H1/2 thus
where the 1/2 subscript of the step function means that
H1/2 = 1/2
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.
See also
