In mathematics, the sign function or signum function (from signum, Latin for "sign") is an odd mathematical function that extracts the sign of a real number. In mathematical expressions the sign function is often represented as sgn. To avoid confusion with the sine function, this function is usually called the signum function.
Any real number can be expressed as the product of its absolute value and its sign function:
It follows that whenever x is not equal to 0 we have
Similarly, for any real number x,
We can also ascertain that:
The signum function is the derivative of the absolute value function, up to (but not including) the indeterminacy at zero. More formally, in integration theory it is a weak derivative, and in convex function theory the subdifferential of the absolute value at 0 is the interval [−1, 1], "filling in" the sign function (the subdifferential of the absolute value is not single-valued at 0). Note, the resultant power of x is 0, similar to the ordinary derivative of x. The numbers cancel and all we are left with is the sign of x.
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, which can be demonstrated using the identity
where H(x) is the Heaviside step function using the standard H(0) = 1/2 formalism. Using this identity, it is easy to derive the distributional derivative:
where p. v. means Cauchy principal value.
The signum can also be written using the Iverson bracket notation:
For k ≫ 1, a smooth approximation of the sign function is
Another approximation is
which gets sharper as ε → 0; note that this is the derivative of √. This is inspired from the fact that the above is exactly equal for all nonzero x if ε = 0, and has the advantage of simple generalization to higher-dimensional analogues of the sign function (for example, the partial derivatives of √).
The signum function can be generalized to complex numbers as:
where arg is the complex argument function.
For reasons of symmetry, and to keep this a proper generalization of the signum function on the reals, also in the complex domain one usually defines, for z = 0:
Another generalization of the sign function for real and complex expressions is csgn, which is defined as:
where Re(z) is the real part of z and Im(z) is the imaginary part of z.
We then have (for z ≠ 0):
At real values of x, it is possible to define a generalized function–version of the signum function, ε(x) such that ε(x)2 = 1 everywhere, including at the point x = 0, unlike sgn, for which (sgn 0)2 = 0. 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
in addition, ε(x) cannot be evaluated at x = 0; and the special name, ε is necessary to distinguish it from the function sgn. (ε(0) is not defined, but sgn 0 = 0.)