In mathematics, the sign function or signum function (from signum, Latin for "sign") is a function that has the value , or according to whether the sign of a given real number is positive or negative, or the given number is itself zero. In mathematical notation.

Definition

is a piecewise function which is defined as follows:

The law of trichotomy states that every real number must be positive, negative or zero. The signum function denotes which unique category a number falls into by mapping it to one of the values , or which can then be used in mathematical expressions or further calculations.

Basic properties

Any real number can be expressed as the product of its absolute value

Similarly, for anyFrom the following:

:

Hence also:

Some algebraic identities

The signum can also be written using the Iverson bracket

The signum can also be written using the floor is accepted to be equal to 1

Properties in mathematical analysis

Discontinuity at zero

Although the sign function takes the value is negative, the ringed point . Instead, the value jumps abruptly to the solid point at is either positive or negative.

These observations are confirmed by any of the various equivalent formal definitions of continuity in mathematical analysis can be approximated arbitrarily closely by the sequence becomes sufficiently large. In the notation of mathematical limits The arrow symbol can be read to mean approaches, or tends to, and it applies to the sequence as a whole.

. This counterexample at zero that is visible in the plot.

Despite the sign function having a very simple form, the step change at zero causes difficulties for traditional calculus techniques, which are quite stringent in their requirements. Continuity is a frequent constraint. One solution can be to approximate the sign function by a smooth continuous function; others might involve less stringent approaches that build on classical methods to accommodate larger classes of function.

Smooth approximations and limits

is the hyperbolic tangent is the inverse tangent).

See .

Differentiation

is differentiable Its derivative

This follows from the differentiability of any constant function acts as a constant function when it is restricted to the negative open region where it equals where the corresponding constant is . Although these are two different constant functions, their derivative is equal to zero in each case.

, because there is a discontinuity there.

Because its derivative is 0, the n-th derivative of a function multiplied by the signum function simplifies to:

in the ordinary sense, under the generalized notion of differentiation in distribution theory, the derivative of the signum function is two times the Dirac delta function. This can be demonstrated using the identity is the Heaviside step function formalism. Using this identity, it is easy to derive the distributional derivative:

Integration

The signum function has a definite integral between any pair of finite values and , even when the interval of integration includes zero. The resulting integral for and

In fact, the signum function is the derivative of the absolute value function, except where there is an abrupt change in gradient

whose derivative is the constant value there.

Because the absolute value is a convex function, there is at least one subderivative at every point, including at the origin. Everywhere except zero, the resulting subdifferential. A subderivative value , which might be thought of informally as "filling in" the graph of the sign function with a vertical line through the origin, making it continuous as a two dimensional curve.

In integration theory, the signum function is a weak derivative of the absolute value function. Weak derivatives are equivalent if they are equal almost everywhere, making them impervious to isolated anomalies at a single point. This includes the change in gradient of the absolute value function at zero, which prohibits there being a classical derivative.

are constants of integration can take on any value, most commonly 0. When considering the Dirac delta function. This can be proven using integration by parts is the Dirac delta function. Integrating, the following expression can be obtained:

is the Heaviside step function has to mach in both terms; otherwise, the derivative of the signum function is 0, so it can be multiplied by any constant in the second term.

Fourier transform

The Fourier transform of the signum function is means taking the Cauchy principal value.

Generalizations

Complex signum

The signum function can be generalized to complex numbers is the point on the unit circle of the complex plane is the complex argument function.

,.

Polar decomposition of matrices

Thanks to the Polar decomposition is a unitary matrix. This leads to each invertible matrix.

. In this sense, polar decomposition generalizes to matrices the signum-modulus decomposition of complex numbers.

Signum as a generalized function

, it is possible to define a generalized function. This generalized signum allows construction of the algebra of generalized functions, but the price of such generalization is the loss of commutativity. In particular, the generalized signum anticommutes with the Dirac delta function.)

See also

• Absolute value

• Heaviside step function

• Negative number

• Rectangular function

• Sigmoid function (Hard sigmoid)

• Step function (Piecewise constant function)

• Three-way comparison

• Zero crossing

• Polar decomposition

• Parity of a permutation

Notes