signIm -- the sign of the
imaginary part of a complex number
IntroductionsignIm(z) represents the sign of
Im(z).
Call(s)signIm(z)
Parametersz |
- | an arithmetical expression representing a complex number |
Returnseither +/- 1, 0, or a symbolic call.
z
DetailssignIm(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)).
Example
1For 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):
Example
2signIm 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 /