CVS difference for ais/ai-00185.txt

Differences between 1.1 and version 1.2
Log of other versions for file ais/ai-00185.txt

--- ais/ai-00185.txt	1998/09/30 00:17:34	1.1
+++ ais/ai-00185.txt	1999/09/23 18:39:39	1.2
@@ -1,26 +1,98 @@
-!standard G.1.2    (15)                               97-03-19  AI95-00185/00
+!standard G.1.2    (15)                               99-09-18  AI95-00185/01
 !class binding interpretation 97-03-19
+!status work item 99-09-18
 !status received 97-03-19
 !priority Medium
 !difficulty Hard
 !subject Branch cuts of inverse trigonometric and hyperbolic functions
 
-!summary 97-03-19
+!summary
 
+Replace G.1.2(15-17) by:
 
-!question 97-03-19
+The imaginary component of the result of the Arcsin, Arccos and Arctanh
+functions is discontinuous as the parameter X crosses the real axis to the left
+of -1.0 or the right of 1.0.
 
+The real component of the result of the Arctan and Arcsinh functions is
+discontinuous as the parameter X crosses the imaginary axis below -i or above i.
 
-!recommendation 97-03-19
+The real component of the result of the Arccot function is discontinuous as
+the parameter X crosses the imaginary axis below -i or above i.
 
+!question
 
-!wording 97-03-19
+The definition of the branch cuts in RM95 G.1.2(15-17) seem contradictory with
+other rules regarding these functions, and inconsistent with common mathematical
+practice.
 
+!recommendation
 
-!discussion 97-03-19
+(See summary.)
 
+!wording
 
-!appendix 97-03-19
+(See summary.)
+
+!discussion
+
+G.1.2(17) defines the branch cut of Arccot as follows:
+
+"The real component of the result of the Arccot function is discontinuous as
+the parameter X crosses the imaginary axis between -i and i."
+
+G.1.2(24) defines the principal value of Arccot as follows:
+
+"The real component of the result of the Arccot function ranges from 0.0 to
+approximately Pi."
+
+These two paragraphs contradict each other.  Consider what happens when X is
+real and close to 0.0.  Mathematically, the Arccot of 0.0 is any odd multiple of
+Pi/2.0.  Because G.1.2(17) requires a discontinuity at 0.0, Arccot (-0.0) and
+Arccot (+0.0) must be two different odd multiples of Pi/2.0.  But G.1.2(24)
+constrains the range of Arccot so that the only acceptable multiple of Pi/2.0
+is Pi/2.0.
+
+We resolve the contradiction by following G.1.2(24), because this paragraph is
+consistent with the definition of Arccot for a real argument, which states that
+the Arccot function "ranges from 0.0 to approximately Pi."  (RM95 A.5.1(14-15))
+
+Now consider the rules related to Arcsin:
+
+"The real component of the result of the Arcsin function is discontinuous as the
+parameter X crosses the real axis to the left of -1.0 or the right of 1.0."
+(RM95 G.1.2(15))
+
+and:
+
+"The range of the real component of the result of the Arcsin function is
+approximately -Pi/2.0 to Pi/2.0."  (RM95 G.1.2(23))
+
+Remember that Arcsin is mathematically multivalued, so that, if Y is one
+possible result of Arcsin (X), then Pi - Y and Y + 2.0 * Pi are also possible
+results of Arcsin (X).
+
+Consider what happens when X crosses the real axis to the right of 1.0.  Let X =
+A + I * B a complex number where A > 0.0 and B is small compared to A (so that
+we can use first order approximation).  A first order approximation of Arcsin
+(X) is:
+
+Y = Pi / 2.0 + B / Sqrt (A**2 - 1.0) - I * Log (A + Sqrt (A**2 - 1.0))
+
+When B > 0.0, the real part of Y is slightly above Pi / 2.0.  In order to keep
+the real part of Arcsin (X) in the range -Pi / 2.0 .. Pi / 2.0, we have to use
+Y when B < 0.0 and Pi - Y when B > 0.0.  This cause the imaginary part to
+become discontinuous.  This illustrates that for this RM95 G.1.2(23) requires
+that the imaginary part, not the real part, be discontinuous when X crosses the
+real axis to the right of 1.0.
+
+A similar analysis could be performed for X to the left of -1.0 and for Arccos
+and Arcsinh.
+
+The rules given in the !summary correspond to the common mathematical
+definitions of these functions.
+
+!appendix
 
 !section G.1.2(15)
 !subject Branch cuts of inverse trigonometric and hyperbolic functions

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