Sign function

The sign function or signum function (from Latin signum 'sign' ) is a function in mathematics that assigns its sign to a real or complex number .

Sign function on the real numbers

definition

Sign function graph

The real sign function maps the set of real numbers into the set and is usually defined as follows: ${\displaystyle \{-1,0,1\}}$

${\displaystyle \operatorname {sgn}(x):={\begin{cases}+1&\;{\text{falls}}\quad x>0\\\;\;\,0&\;{\text{falls}}\quad x=0\\-1&\;{\text{falls}}\quad x<0\\\end{cases}}}$

She assigns the positive numbers the value +1, the negative numbers the value −1 and the 0 the value 0.

In computing technology applications, the special position of 0 is sometimes dispensed with; this is then assigned to the positive, negative or both number ranges. This allows the sign of a number to be encoded in a single bit . Since the zero is a zero set below the Lebesgue measure , this is not important for all practical applications.

In the event that is set, there is the following relationship to the Heaviside function : ${\displaystyle \operatorname {sgn}(0)=1}$ ${\displaystyle \Theta (x)}$

${\displaystyle \operatorname {sgn}(x)=2\Theta (x)-1}$

Calculation rules

By differentiating between cases it is easy to prove:

• For everyone with amount applies . ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle |x|}$ ${\displaystyle x=|x|\cdot \operatorname {sgn}(x)\;{\text{sowie}}\;x\cdot \operatorname {sgn}(x)=|x|}$

• The signum function is an odd function :

${\displaystyle \operatorname {sgn}(-x)=-\operatorname {sgn}(x)}$for everyone . ${\displaystyle x\in \mathbb {R} }$

• If is a constant and an odd function , then ${\displaystyle k\in \mathbb {R} }$ ${\displaystyle f}$

${\displaystyle f(k\cdot x)\quad =f(\operatorname {sgn}(k)\cdot |k|\cdot x)\quad =\operatorname {sgn}(k)\cdot f(|k|\cdot x)}$

• For the transition to the reciprocal number is compatible with the signum function and does not change its value: ${\displaystyle x\neq 0}$

${\displaystyle \operatorname {sgn}(x^{-1})=(\operatorname {sgn}(x))^{-1}=\operatorname {sgn}(x)}$for everyone . ${\displaystyle 0\neq x\in \mathbb {R} }$

• The signum function is compatible with multiplication:

${\displaystyle \operatorname {sgn}(x)\cdot \operatorname {sgn}(y)=\operatorname {sgn}(x\cdot y)}$for everyone . ${\displaystyle x,y\in \mathbb {R} }$

• The signum function is idempotent :

${\displaystyle \operatorname {sgn}(\operatorname {sgn}(x))=\operatorname {sgn}(x)}$for everyone . ${\displaystyle x\in \mathbb {R} }$

From the last two calculation rules mentioned, it follows, for example, that the signum function in an argument composed of any number of factors can be replaced by a factor without changing the function value: ${\displaystyle x_{j}}$ ${\displaystyle \operatorname {sgn}(x_{j})}$

${\displaystyle \operatorname {sgn} {\bigg (}x_{j}\cdot \prod _{i}x_{i}{\bigg )}=\operatorname {sgn} {\bigg (}\operatorname {sgn}(x_{j})\cdot \prod _{i}x_{i}{\bigg )}}$for any . ${\displaystyle x_{i},x_{j}\in \mathbb {R} }$

Derivative and integral

The sign function is not continuous at the 0 position.

The sign function is not continuous at this point and therefore not classically differentiable there. For all other places the sign function can be differentiated with . The sign function also has no weak derivative . However, it is differentiable in the sense of distributions , and its derivative is , where denotes the delta distribution . ${\displaystyle x=0}$ ${\displaystyle x\neq 0}$ ${\displaystyle \operatorname {sgn} ^{\prime }(x)=0}$ ${\displaystyle 2\delta }$ ${\displaystyle \delta }$

Furthermore applies to everyone ${\displaystyle x\in \mathbb {R} }$

${\displaystyle |x|=\int _{0}^{x}\operatorname {sgn}(t)\,dt\,.}$

The sign function is also the weak derivative of the absolute value function .

Sign function on the complex numbers

definition

Signum of four complex numbers

Compared to the sign function of real numbers, the following extension to complex numbers is rarely considered:

${\displaystyle \operatorname {sgn}(z):={\begin{cases}{\frac {z}{|z|}}&\;{\text{falls}}\quad z\neq 0\\0&\;{\text{falls}}\quad z=0\\\end{cases}}}$

The result of this function is on the unit circle and has the same argument as the output value, in particular applies ${\displaystyle z\neq 0}$

${\displaystyle \operatorname {sgn} \left(r\mathrm {e} ^{\mathrm {i} \varphi }\right)=\mathrm {e} ^{\mathrm {i} \varphi },\qquad \mathrm {falls} \ r>0.}$

Example: (red in the picture) ${\displaystyle z_{1}=2+2\mathrm {i} }$

${\displaystyle \operatorname {sgn} (z_{1})=\operatorname {sgn} (2+2\mathrm {i} )={\frac {2+2\mathrm {i} }{\left|2+2\mathrm {i} \right|}}={\frac {2+2\mathrm {i} }{2{\sqrt {2}}}}={\frac {1+\mathrm {i} }{\sqrt {2}}}={\frac {\sqrt {2}}{2}}+{\frac {\sqrt {2}}{2}}{\mathrm {i} }.}$

Calculation rules

The following calculation rules apply to the complex sign function:

For all complex numbers and the following applies: ${\displaystyle z}$ ${\displaystyle w}$

• ${\displaystyle z=|z|\cdot \operatorname {sgn} z}$for all where denotes the amount of ; ${\displaystyle z,}$ ${\displaystyle |z|}$ ${\displaystyle z}$
• ${\displaystyle \operatorname {sgn} ({\bar {z}})={\overline {\operatorname {sgn} (z)}}}$where the slash denotes the complex conjugation ;
• ${\displaystyle \operatorname {sgn}(z\cdot w)=\operatorname {sgn} z\cdot \operatorname {sgn} w}$, in particular
  • ${\displaystyle \operatorname {sgn}(\lambda \cdot z)=\operatorname {sgn} z}$for positive real ones , ${\displaystyle \lambda }$
  • ${\displaystyle \operatorname {sgn}(\lambda \cdot z)=-\operatorname {sgn} z}$for negative reals , ${\displaystyle \lambda }$
  • ${\displaystyle \operatorname {sgn} (-z)=-\operatorname {sgn} (z)}$;
• ${\displaystyle \operatorname {sgn} (|z|)=|\operatorname {sgn} (z)|}$.
• If is, also applies ${\displaystyle z\neq 0}$

${\displaystyle \operatorname {sgn} (z^{-1})=\operatorname {sgn} (z)^{-1}={\overline {\operatorname {sgn} (z)}}}$.

literature

• Königsberger: Analysis 1 . 6th edition. Springer, Berlin 2003, ISBN 3-540-40371-X , p. 101 .
• Hildebrandt: Analysis 1 . 2nd Edition. Springer, Berlin 2005, ISBN 3-540-25368-8 , pp. 133 .

Web links

Wikibooks: Algorithm Collection: Number Theory: Signum  - Learning and Teaching Materials

• Eric W. Weisstein : Sign . In: MathWorld (English).
• yark, matte, Cam McLeman: Signum function . In: PlanetMath . (English)