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Sign function

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The sign function
sgn(x)
, also called the signum function, of a real number
x
is defined as

This may be compactly written as

where
[·]
is the Iverson bracket.

"Sign" of complex number

The "sign" of a complex number
z = a + bi = rei θ =
| z |
ei 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
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

See also