[Cfp-interest 2734] Definition of the 'carg' function

[Damian McGuckin]

Given a complex number

 	z = x + i y

the definitions used in the Digital Library of Mathematical Functions (DLMF)
from the National Institute of Standards and Technology (NIST) say that

 	r = sqrt ( x * x + y * y )

and

 	theta = arctan ( y / x )

are called the modulus and phase. Reviewing a wide variety of course notes 
from university level courses, they all seem to concur with what NIST says 
although the modulus is often called 'magnitude' and similarly, the phase 
is often called 'argument', as the draft C23 standard mentions

But this raises a minor definitional error with 7.3.9.1 'carg' which uses

 	phase angle

As the above mentioned NIST reference attests, the concept should more
correctly be called just 'phase' even if phase is an angle.

So, the standard beyond C23 could read (to be more precise):

   Description
   The carg functions compute the phase (also called argument) of z, an angle
   expressed in radians, with a branch cut along the negative real axis.

   Returns
   The carg functions return the value of the phase in the interval [-pi,+pi]

or, swapping the words 'argument' and 'phase'

   Description
   The carg functions compute the argument (also called phase) of z, an angle
   expressed in radians, with a branch cut along the negative real axis.

   Returns
   The carg functions return the value of the argument in the interval [-pi,+pi]

The slightly more radical change reflects the defintion that was used in the
original work on complex numbers by Argand from the early 19th century:

   Argand, R. Essai sur une maniere de representer les quantities imaginaires
   dans les constructions geometriques. Paris: Albert Blanchard, 1971. Reprint
   of the 2nd ed., published by G. J. Hoel in 1874. First edition published
   Paris, 1806.

Once either change occurs, the index should put parentheses around the 
word angle

   phase (angle), 195 (or whatever 7.3.9.1 becomes)

Comments?

Thanks - Damian