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5. Complex Math.
Complex numbers are much more than a simple extension of the real numbers into two
dimensions. The Complex Plane is a mathematical domain with well-defined, own properties and
singularities, and it isn’t in the scope of this manual to treat all its fundamental properties. On
occasions there will be a short discussion for a few functions (notably the logarithms!), and some
analogies will be made to their geometric equivalences, but it is assumed throughout this manual
that the user has a good understanding of complex numbers and their properties.
5.1. Arithmetic and Simple Math.
Table-5.1:- Arithmetic functions.
Index
1
2
3
4
5a
5b
6
7
8
9
10
11
Function
Z+
ZZ*
Z/
ZINV
1/Z
ZDBL
ZHALF
ZRND
ZINT
ZFRC
ZPIX
Formula
Z=w+z
Z=w-z
Z=w*z
Z=w/z
Z=1/z
Z=1/r e^(-iArg)
z=2*z
z= z/2
Z=rounded(z)
Z=Int(z)
Z=Frc(z)
Z=zπ
Description
Complex addition
Complex subtraction
Complex multiplication
Complex division
Complex inversion, direct formula
Complex inversion, uses TOPOL
Doubles the complex number
Halves the complex number
Rounds Z to display settings precision
Takes integer part for Re(z) and Im(z)
Takes fractional part for Re(z) and Im(z)
Simple multiplication by pi
Here’s a description of the individual functions within this group.
Z+
ZZ*
Z/
Complex addition
Complex subtraction
Complex multiplication
Complex division
Z=w+z
Z=w-z
Z=w*z
Z=w/z
Does
Does
Does
Does
LastZ
LastZ
LastZ
LastZ
Complex arithmetic using the RPN scheme, with the first number stored in the W stack level and
the second in the Z stack level. The result is stored in the Z level, the complex stack drops
(duplicating U into V), and the previous contents of Z is saved in the LastZ register.
ZINV
1/Z
Direct Complex inversion
Uses POLAR conversion
Z=1/z
Z=1/r e^(-iArg)
Does LastZ
Does LastZ
Calculates the reciprocal of the complex number stored in Z. The result is saved in Z and the
original argument saved in the LastZ register. Of these two the direct method is faster and of
comparable accuracy – thus it’s the preferred one, as well as the one used as subroutine for other
functions.
This function would be equivalent to a particular case of Z/, where w=1+0j, and not using the
stack level W. Note however that Z/ implementation is not based on the ZINV algorithm [that is,
making use of the fact that : w/z = w * (1/z)], but based directly on the real and imaginary parts
of both arguments.
41Z User Manual Page 21