Sign of the imaginary part of a complex number
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signIm(z
)
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
.
Properties of
identifiers set via assume
are
taken into account.
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):
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)

An arithmetical expression representing a complex number 
Either
, 0,
or a symbolic call of type "signIm"
.
z