signIm
-- the sign of the
imaginary part of a complex numbersignIm
(z)
represents the sign of
Im(z)
.
signIm(z)
z |
- | an arithmetical expression representing a complex number |
either +/- 1, 0, or a symbolic call.
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.diff
and series
treat
signIm
as a constant function. Cf. example 2.(-z)^p=z^p*(-1)^(-p*signIm(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)
1, -1, -1, 1, 1
Symbolic arguments without properties lead to unevaluated calls:
>> signIm(x), signIm(x - I*sqrt(2))
1/2 signIm(x), signIm(x - I 2 )
Properties set via assume
are taken into account:
>> assume(x, Type::Real): signIm(x - I*sqrt(2))
-1
>> assume(x > 0): signIm(x)
-1
>> assume(x < 0): signIm(x)
1
>> assume(x = 0): signIm(x)
0
>> 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)
0
Also series
treats signIm
as
a constant function:
>> series(signIm(z/(1 - z)), z = 0)
/ z \ 6 signIm| ------- | + O(z ) \ - z + 1 /