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Sign function

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

\sgn(x) :=
\begin{cases}
  -1 & \text{if } x < 0, \\
   0 & \text{if } x = 0, \\
   1 & \text{if } x > 0. 
\end{cases}

This may be compactly written as

\sgn(x) := [x \ge 0] - [x \le 0],\quad x \in \R, \,

where \scriptstyle [\cdot] \, is the Iverson bracket.

"Sign" of complex number

The "sign" of a complex number \scriptstyle z \,=\, a + bi \,=\, r e^{i \theta} \,=\, |z| e^{i \arg(z)} \,, where \scriptstyle |z| \, is the absolute value of a complex number and \scriptstyle \arg(z) \,=\, \theta \, is the argument of a nonzero complex number (the argument of 0 being undefined), would generalize to

\sgn(z) :=
\begin{cases}
  0 & \text{if } z = 0, \\
  e^{i \arg(z)} = e^{i \theta} & \text{if } z \ne 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 \scriptstyle z \,.

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

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

See also

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