Sign function

 ( Special Functions ) 

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.

For example:

It follows that whenever $x$ is not equal to 0 we have $$\backslash sgn\; x\; =\; \backslash frac\{x\}$$

Similarly, for any real number $x$, $$|x|\; =\; x\backslash sgn\; x\backslash ,.$$ We can also be certain that: $$\backslash sgn\; (xy)=(\backslash sgn\; x)(\backslash sgn\; y)\backslash ,,$$ and so $$\backslash sgn\; (x^n)=(\backslash sgn\; x)^n\backslash ,.$$

The signum can also be written using the floor and the absolute value functions: $$\backslash sgn\; x\; =\; \backslash Biggl\backslash lfloor\; \backslash frac\{x\}$$

      &= \lim_{n\to\infty}\frac{2}{\pi}\operatorname{arctan}(nx)\\
      &= \lim_{n\to\infty}\tanh(nx)\\
      &= \lim_{\varepsilon\to 0} \frac{x}{\sqrt{x^2 + \varepsilon^2}}.
     

We can understand this as before by considering the definition of the absolute value $|x|$ on the separate regions $x>0$ and For example, the absolute value function is identical to $x$ in the region $x>0,$ whose derivative is the constant value , which equals the value of $\backslash sgn\; x$ 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 consists of a single value, equal to the value of the sign function. In contrast, there are many subderivatives at zero, with just one of them taking the value $\backslash sgn(0)\; =\; 0$. A subderivative value occurs here because the absolute value function is at a minimum. The full family of valid subderivatives at zero constitutes the subdifferential interval $-1,1$, 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.

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 $z=0$: $$\backslash sgn(0+0i)=0$$

Another generalization of the sign function for real and complex expressions is $\backslash text\{csgn\}$,Maple V documentation. May 21, 1998 which is defined as: $$$$

\operatorname{csgn} z= \begin{cases}
1 & \text{if } \mathrm{Re}(z) > 0, \\
-1 & \text{if } \mathrm{Re}(z) < 0, \\
\sgn \mathrm{Im}(z) & \text{if } \mathrm{Re}(z) = 0
     

We then have (for $z\backslash ne\; 0$): $$\backslash operatorname\{csgn\}\; z\; =\; \backslash frac\{z\}\{\backslash sqrt\{z^2\}\}\; =\; \backslash frac\{\backslash sqrt\{z^2\}\}\{z\}.$$

In the special case where $\backslash mathbb\; K=\backslash mathbb\; R,\backslash \; n=2,$ and the (invertible) matrix , which identifies with the (nonzero) complex number $a+\backslash mathrm\; i\; b=c$, then the signum matrices satisfy and identify with the complex signum of $c$, $\backslash sgn\; c\; =\; c/|c|$. In this sense, polar decomposition generalizes to matrices the signum-modulus decomposition of complex numbers.

• 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