On Thursday 19 May 2005 09:39, Luke Palmer wrote:
On 5/19/05, Edward Cherlin [EMAIL PROTECTED] wrote:
It turns out that the domain and range and the location of
the cut lines have to be worked out separately for different
functions. Mathematical practice is not entirely consistent
in making these decisions, but in programming, there seems
to be widespread agreement that the shared definitions used
in the APL, Common LISP, and Ada standards are the best
available.
Do we want to get into all of this in Perl6?
I'm not really sure I know what you mean by do we want to get
into all of this?. If we're going to have a Complex class,
we have to. But getting into it might involve saying that
APL, CL, and Ada are the best, so we use those. This is the
kind of problem where, if someone wants to get more precise,
they turn to CPAN.
Luke
Math::Complex - complex numbers and associated mathematical
functions
http://cpan.uwinnipeg.ca/htdocs/perl/Math/Complex.html
lists the complex functions, but with no information given
there on domain, range (principal values), and branch cuts.
The APL standard costs $350 from ANSI or 260 Swiss Francs from
ISO, but the Common Lisp Hyperspec is available online for free.
http://www.lispworks.com/documentation/HyperSpec/
as is the Ada Reference Manual. The CL and Ada definitions are
incomplete. I'll have to find a copy of the APL standard.
log(z)
CL Hyperspec: The branch cut for the logarithm function of one
argument (natural logarithm) lies along the negative real axis,
continuous with quadrant II. The domain excludes the origin.
Thus the range would be defined as -pi Im(log(z)) = pi
sin(z)
CL Hyperspec: Not defined for complex arguments.
Ada RF says:
The functions have their usual mathematical meanings.
However, the arbitrariness inherent in the placement of branch
cuts, across which some of the complex elementary functions
exhibit discontinuities, is eliminated by the following
conventions:
(13)
* The imaginary component of the result of the Sqrt and
Log functions is discontinuous as the parameter X crosses the
negative real axis.
(14)
* The result of the exponentiation operator when the left
operand is of complex type is discontinuous as that operand
crosses the negative real axis.
(15)
* The real (resp., imaginary) component of the result of
the Arcsin and Arccos (resp., Arctanh) functions is
discontinuous as the parameter X crosses the real axis to the
left of -1.0 or the right of 1.0.
(16)
* The real (resp., imaginary) component of the result of
the Arctan (resp., Arcsinh) function is discontinuous as the
parameter X crosses the imaginary axis below -i or above i.
(17)
* The real component of the result of the Arccot function
is discontinuous as the parameter X crosses the imaginary axis
between -i and i.
(18)
* The imaginary component of the Arccosh function is
discontinuous as the parameter X crosses the real axis to the
left of 1.0.
(19)
* The imaginary component of the result of the Arccoth
function is discontinuous as the parameter X crosses the real
axis between -1.0 and 1.0.
(20)
The computed results of the mathematically multivalued
functions are rendered single-valued by the following
conventions, which are meant to imply the principal branch:
(21)
* The real component of the result of the Sqrt and
Arccosh functions is nonnegative.
(22)
* The same convention applies to the imaginary component
of the result of the Log function as applies to the result of
the natural-cycle version of the Argument function of
Numerics.Generic_Complex_Types (see G.1.1).
(23)
* The range of the real (resp., imaginary) component of
the result of the Arcsin and Arctan (resp., Arcsinh and Arctanh)
functions is approximately -Pi/2.0 to Pi/2.0.
(24)
* The real (resp., imaginary) component of the result of
the Arccos and Arccot (resp., Arccoth) functions ranges from 0.0
to approximately Pi.
(25)
* The range of the imaginary component of the result of
the Arccosh function is approximately -Pi to Pi.
--
Edward Cherlin
Generalist activist--Linux, languages, literacy and more
A knot! Oh, do let me help to undo it!
--Alice in Wonderland
http://cherlin.blogspot.com
