!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

!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

!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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