information:highlighting, structural:expansion, information:classification
Complex signum

The signum function can be generalized to complex numbers as: sgn z = z | z | {\displaystyle \operatorname {sgn} z={\frac {z}{|z|}}} for any complex number z {\displaystyle z} except z = 0 {\displaystyle z=0}. The signum of a given complex number z {\displaystyle z} is the point on the unit circle of the complex plane that is nearest to z {\displaystyle z}. Then, for z ≠ 0 {\displaystyle z\neq 0}, sgn z = e i arg z, {\displaystyle \operatorname {sgn} z=e^{i\arg z}\,,} where arg {\displaystyle \arg } is the 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 z = 0 {\displaystyle z=0}: sgn (0 + 0 i) = 0 {\displaystyle \operatorname {sgn}(0+0i)=0}

Another generalization of the sign function for real and complex expressions is csgn {\displaystyle {\text{csgn}}}, which is defined as: csgn z = { 1 if R e (z) > 0, − 1 if R e (z) < 0, sgn I m (z) if R e (z) = 0 {\displaystyle \operatorname {csgn} z={\begin{cases}1&{\text{if }}\mathrm {Re} (z)>0,\\-1&{\text{if }}\mathrm {Re} (z)<0,\\\operatorname {sgn} \mathrm {Im} (z)&{\text{if }}\mathrm {Re} (z)=0\end{cases}}} where Re (z) {\displaystyle {\text{Re}}(z)} is the real part of z {\displaystyle z} and Im (z) {\displaystyle {\text{Im}}(z)} is the imaginary part of z {\displaystyle z}.

We then have (for z ≠ 0 {\displaystyle z\neq 0}): csgn z = z z 2 = z 2 z. {\displaystyle \operatorname {csgn} z={\frac {z}{\sqrt {z^{2}}}}={\frac {\sqrt {z^{2}}}{z}}.}