!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
Replace G.1.2(15-17) by:
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.
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
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
(See summary.)
!wording
(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
!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.
****************************************************************