The csgn function is used to determine in which half-plane ("left" or "right") the complex-valued expression or number x lies. With the exception described in the next bullet point, it is defined by

For the case of a complex number in which the real component is one of the floating point values -0. or +0., csgn returns the sign of the real part.  For more information, see Numeric Computation in Maple.

The value of csgn(0) is controlled by the environment variable _Envsignum0.  The 3-argument calling sequence csgn(0, x, y) sets _Envsignum0 = y for the duration of the call to csgn. See signum for further information.

The decision of whether or not to perform many of the automatic symmetry transformations in maple is based on the value of csgn. For example, if csgn(x) = -1, the transformation $\sin \left(x\right)\to -\sin \left(-x\right)$ is done.

csgn uses signum to determine the signs of $\mathrm{\Re }\left(x\right)$ and $\mathrm{\Im }\left(x\right)$.

The derivative of csgn is denoted by csgn(1, x).  This is 0 for all non-purely-imaginary numbers, and is undefined otherwise.

For mathematical consistency, the value of csgn(0), as determined either by the value of _Envsignum0 or by the third argument to csgn, should be either 0 (the default) or one of 1, -1, or undefined.

$\mathrm{csgn}\left(1-\frac{2}{3}\mathrm{I}\right)$

$1$

$\mathrm{csgn}\left(-1-\frac{2}{3}\mathrm{I}\right)$

$\mathrm{-1}$

$\mathrm{csgn}\left(-1+\frac{2}{3}\mathrm{I}\right)$

$\mathrm{-1}$

$\mathrm{csgn}\left(1+\frac{2}{3}\mathrm{I}\right)$

$1$

$\mathrm{csgn}\left(-\frac{2}{3}\mathrm{\pi }\mathrm{I}\right)$

$\mathrm{-1}$

$\mathrm{csgn}\left(\exp \left(\frac{2\mathrm{\pi }}{3}\mathrm{I}\right)\right)$

$\mathrm{-1}$

$\mathrm{csgn}\left(\mathrm{\pi }\right)$

$1$

$\mathrm{diff}\left(\mathrm{csgn}\left(x\right)\,x\right)$

$\mathrm{csgn}\left(1\,x\right)$

$\mathrm{diff}\left(\mathrm{csgn}\left(x\right)\,x\,x\right)$

$\mathrm{csgn}\left(1\,x\right)$

$\mathrm{csgn}\left(1\,-3+\mathrm{I}\right)$

$0$

$\mathrm{csgn}\left(0\right)$

$0$

$\mathrm{csgn}\left(0\,0\,-1\right)$

$\mathrm{-1}$

$\mathrm{csgn}\left(0+\mathrm{I}\right)$

$1$

$\mathrm{csgn}\left(-0.+1.\mathrm{I}\right)$

$\mathrm{-1}$

$\mathrm{csgn}\left(0-\mathrm{I}\right)$

$\mathrm{-1}$

$\mathrm{csgn}\left(0.-1.\mathrm{I}\right)$

$1$