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
