!standard G.1.2 (15) 97-03-19 AI95-00185/00
!class binding interpretation 97-03-19
!status received 97-03-19
!priority Medium
!difficulty Hard
!subject Branch cuts of inverse trigonometric and hyperbolic functions
!summary 97-03-19
!question 97-03-19
!recommendation 97-03-19
!wording 97-03-19
!discussion 97-03-19
!appendix 97-03-19
!section G.1.2(15)
!subject Branch cuts of inverse trigonometric and hyperbolic functions
!reference RM95 G.1.2(15)
!reference RM95 G.1.2(16)
!reference RM95 G.1.2(17)
!reference RM95 G.1.2(24)
!from Pascal Leroy 97-03-10
!reference 97-15727.f Pascal Leroy 97-3-10>>
!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 seem to 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. So Arccot cannot be discontinuous at 0.0 after all...
Also, the paragraphs G.1.2(15) and G.1.2(16) define branch cuts as follows:
"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
The real (resp. imaginary) component of the result of the Arctan (resp
Arcsinh) functions is discontinuous as the parameter X crosses the imaginary
axis below -i or above i."
These rules are puzzling, because the natural mathematical definition of
Arcsin and Arccos is such that the real part is continuous; it is the
imaginary part which has branch cuts. Similarly, the natural mathematical
definition of Arcsinh is such that the imaginary part is continuous; it is the
real part which has branch cuts.
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