Editing Sign function (section)

=== Complex signum ===
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The signum function can be generalized to [[complex numbers]] as:
<math display="block">\sgn z = \frac{z}{|z|} </math>
for any complex number <math>z</math> except <math>z=0</math>. The signum of a given complex number <math>z</math> is the [[point (geometry)|point]] on the [[unit circle]] of the [[complex plane]] that is nearest to <math>z</math>.  Then, for <math>z\ne 0</math>,
<math display="block">\sgn z = e^{i\arg z}\,,</math>
where <math>\arg</math> is the [[Argument (complex analysis)|complex argument function]].

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 <math>z=0</math>: 
<math display="block">\sgn(0+0i)=0</math>

Another generalization of the sign function for real and complex expressions is <math>\text{csgn}</math>,<ref>Maple V documentation. May 21, 1998</ref> which is defined as:
<math display="block">
 \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
\end{cases}
</math>
where <math>\text{Re}(z)</math> is the real part of <math>z</math> and <math>\text{Im}(z)</math> is the imaginary part of <math>z</math>.

We then have (for <math>z\ne 0</math>):
<math display="block">\operatorname{csgn} z = \frac{z}{\sqrt{z^2}} = \frac{\sqrt{z^2}}{z}. </math>