On 21 November 2014 at 20:18, Ondřej Čertík <ondrej.cer...@gmail.com> wrote: > > I am still confused about one thing: is this issue is already > present in FriCAS before your changes? Because you can > already use conjugate, sin, +, *, ..., even without defining the > derivative for abs(x). I fail to see how defining the abs(x).diff(x) > in the way you did it can introduce issues that weren't present > in the first place. >
FriCAS currently does not implement a symbolic 'conjugate' operator. The issue concerns whether adding 'conjugate' is a good idea and only secondly how to differentiate it. > ----- > > I have finished the writeup, it starts here (you might want to refresh > your browser to see the latest changes): > > http://www.theoretical-physics.net/dev/math/complex.html#complex-conjugate > > and it was implemented with these two PRs: > > https://github.com/certik/theoretical-physics/pull/39 > https://github.com/certik/theoretical-physics/pull/40 > Thanks. > I must say one thing that I like about the "theta" is that it tells > you immediately if the function is analytic or not (if theta is > present it is not, if it is not present, then the expression does not > depend on theta, and thus is analytic). For example, for log(z), > the theta cancels, and so the result 1/z is analytic. > Still looks ugly to me. > I found a bug in these results from FriCAS: > >> (4) -> D(abs(f(x)),x) >> >> , _ _ , >> f(x)f (x) + f(x)f (x) >> >> (4) --------------------- >> 2abs(f(x)) >> Type: >> Expression(Integer) >> (5) -> D(abs(log(x)),x) >> >> _ _ >> xlog(x) + x log(x) >> (5) ------------------ >> _ >> 2xxabs(log(x)) >> Type: >> Expression(Integer) > > The bar must be over the whole f(x) as well as log(x), because > conjugate(log(x)) is only equal log(conjugate(x)) if x is not > negative real number. In FriCAS with my patch functions defined by f := operator 'f are currently assume to be holomorphic and log is holomorphic by definition so conjugate(log(x)) = log(conjugate(x)) Perhaps you are considering the wrong branch. > See the example here: > http://www.theoretical-physics.net/dev/math/complex.html#id1 where I > have it explicitly worked out. You can also check that easily in > Python: > > In [1]: from cmath import log > > In [2]: x = -1+1j > > In [3]: log(x).conjugate() > Out[3]: (0.34657359027997264-2.356194490192345j) > > In [4]: log(x.conjugate()) > Out[4]: (0.34657359027997264-2.356194490192345j) > > In [5]: x = -1 > > In [6]: log(x).conjugate() > Out[6]: -3.141592653589793j > > In [7]: log(x.conjugate()) > Out[7]: 3.141592653589793j > > In [8]: log(x.conjugate()) - 2*pi*1j > Out[8]: -3.141592653589793j > > > Where [3] and [4] are equal, but [6] and [7] are not (you need to > subtract 2*pi*i from [7], as in [8], in order to recover [6], > consistent with the formula in the writeup). > Complex 'log' is a multi-valued like 'sqrt' so you need to consider more than one branch. Bill. -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
