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Sign function

The sign function
 sgn(x)
, also called the signum function, of a real number
 x
is defined as
${\begin{array}{l}\displaystyle {\operatorname {sgn}(x):={\begin{cases}-1&{\text{if }}x<0,\\0&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}}\end{array}}$ This may be compactly written as

${\begin{array}{l}\displaystyle {\operatorname {sgn}(x):=[x\geq 0]-[x\leq 0],\quad x\in \mathbb {R} ,}\end{array}}$ where
 [·]
is the Iverson bracket.

"Sign" of complex number

The "sign" of a complex number
 z = a + b i = r e i θ = | z | e i arg(z)
, where
 | z |
is the absolute value of a complex number and
 arg(z) = θ
is the argument of a nonzero complex number (the argument of 0 being undefined), would generalize to
$\operatorname {sgn}(z):={\begin{cases}0&{\text{if }}z=0,\\e^{i\arg(z)}=e^{i\theta }&{\text{if }}z\neq 0.\end{cases}}$ The "sign" of a nonzero complex number would thus give the complex number on the unit circle of the complex plane which has the same argument as
 z
.

If we consider the real numbers as a subset of the complex numbers, then

$\operatorname {sgn}(x):={\begin{cases}e^{i\pi }&{\text{if }}x<0,\\0&{\text{if }}x=0,\\e^{i0}&{\text{if }}x>0.\end{cases}}$ 