Sign of the imaginary part of a complex number
This functionality does not run in MATLAB.
signIm(z) represents the sign of Im(z).
signIm(z) indicates whether the complex number z lies in the upper or in the lower half plane: signIm(z) yields 1 if Im(z) > 0, or if z is real and z < 0. At the origin: signIm(0)=0. For all other numerical arguments, - 1 is returned. Thus, signIm(z)=sign(Im(z)) if z is not on the real axis.
If the position of the argument in the complex plane cannot be determined, then a symbolic call is returned. If appropriate, the reflection rule signIm(-x) = -signIm(x) is used.
The following relation holds for arbitrary complex z and p:
Further, for arbitrary complex z:
For numerical values, the position in the complex plane can always be determined:
signIm(2 + I), signIm(- 4 - I*PI), signIm(0.3), signIm(-2/7), signIm(-sqrt(2) + 3*I*PI)
Symbolic arguments without properties lead to symbolic calls:
signIm(x), signIm(x - I*sqrt(2))
Properties set via assume are taken into account:
assume(x, Type::Real): signIm(x - I*sqrt(2))
assume(x > 0): signIm(x)
assume(x < 0): signIm(x)
assume(x = 0): signIm(x)
signIm is a constant function, apart from the jump discontinuities along the real axis. These discontinuities are ignored by diff:
Also series treats signIm as a constant function:
series(signIm(z/(1 - z)), z = 0)
An arithmetical expression representing a complex number