signIm

Sign of the imaginary part of a complex number

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

signIm(z)

Description

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 functions diff and series treat signIm as a constant function. Cf. Example 2.

The following relation holds for arbitrary complex z and p:

.

Further, for arbitrary complex z:

and

.

Environment Interactions

Properties of identifiers set via assume are taken into account.

Examples

Example 1

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)

unassume(x):

Example 2

signIm is a constant function, apart from the jump discontinuities along the real axis. These discontinuities are ignored by diff:

diff(signIm(z), z)

Also series treats signIm as a constant function:

series(signIm(z/(1 - z)), z = 0)

Parameters

z

An arithmetical expression representing a complex number

Return Values

Either , 0, or a symbolic call of type "signIm".

Overloaded By

z

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