Sign Function - Complex Signum

Complex Signum

The signum function can be generalized to complex numbers as

for any z ∈ except z = 0. The signum of a given complex number z is the point on the unit circle of the complex plane that is nearest to z. Then, for z ≠ 0,

where 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:

Another generalization of the sign function for real and complex expressions is csgn, which is defined as:

$\operatorname{csgn}(z)= \begin{cases} 1 & \text{if } \Re(z) > 0, \\ -1 & \text{if } \Re(z) < 0, \\ \sgn(\Im(z)) & \text{if } \Re(z) = 0 \end{cases}$

where is the real part of z, is the imaginary part of z.

We then have (except for z = 0):

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