Francois Maltey <[EMAIL PROTECTED]> writes: > Hi, > >> Somebody should check (re-check ?) branch cuts and singularities with this > >> patch. > >> > > > > (for convenience, I include the patch below) > > > > I guess the right place to check branch cuts and singularities is in > > src/input/elemnum.input. I looked at the common lisp hyperspec, the > > definition > > of branchcuts is quite clear there, so I suggest that we take it as > > definition, > > unless you have a better reference. > > > > I don't understand. > > Do you change asin z == atan (z/(1-z^2)^(1/2)) > to asin z == -i log(iz+(1-z^2)^(1/2)) ?
No, that's not possible for the definition over the reals. However we want that the branch cuts and singularities for the complex versions are as in the HyperSpec, and that the real version agrees with the complex version whenever the result of the complex version is real. > > so we need to test: > > > > real < -1 error > > > +1 error > > = -1 -%pi/2 > > = +1 -%pi/2 > Yes this test is necessary for the atan definition, with the denominator > (1-z^2)^(1/2). Exactly. > > complex x + 0*%i, x < -1 close to x + 0.1*%i > > complex x + 0*%i, x > 1 close to x - 0.1*%i > > > > Is this correct? > What definition do you call ? > with the log definition I see no sigularity arroud x+/-0%i for x >=1. Oh yes, there is: the sqrt has a branch cut. > x^2-1 is (c+) +/- 0%i : a positive constant (c+), and imaginary part is > arround 0. > sqrt(1-x^2) is arround (c+)*%i (you mean sqrt(c)*%i, I believe) or sqrt(c)*(-%i)... > If asin is defined from the complex atan, what definition do you use for > atan? I suppose you start from : > http://www.lispworks.com/documentation/HyperSpec/Body/f_asin_.htm > > i (log (1-iz)-log(1+iz)) / 2 = -i log ((1+iz)/(1+z^2)^(1/2)) I didn't work it out yet. One has to do a case by case analysis, depending on the quadrant of z. I'm not sure whether we can rely on the underlying common lisp to follow the standard, however. Maybe you could work out that one? Martin --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To post to this group, send email to [email protected] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/fricas-devel?hl=en -~----------~----~----~----~------~----~------~--~---
