I'm really naive on this one: the problem I'm trying to solve is to write a recurrence for the Legendre Q(n,x) polynomials / Q(n,m,x) functions. Numeric results can be easily conjugated but a symbolic expression with conjugated log functions is tedious to use and, as it seems, impossible to dfifferentiate.
See also http://trac.sagemath.org/ticket/16813 On Sat, Sep 13, 2014 at 3:44 PM, maldun <[email protected]> wrote: > Hi! > > Be careful! conjugate(·) is not complex differentiable! The Example you gave > in your link had not conjugate(log(x)) as function but > conjugate(log(conjugate(x)). Which exists. > You can alos show it easily by using the facts, if log(·) is differentialbe > in an neighbourhood of x, it has a series expansion, and since conjugate is > a field isomorphism and continuous we in fact have > conjugate(log(conjugate(x)) = log(x). > > Complex conjugation is a very special case. In advanced complex analysis it > has even it's own calculus based on the so called wirtinger operators [1] > (maybe you are already familiar with that) > In addition there is also the very nice theorem f is holomorphic ⇔ > ∂ｆ/∂conjugate(z) = 0 > > And that's also the reason you have a problem since D[0](conjugate) is > simply not defined. I would'nt say this is a wrong behaviour since > conjugation isn't analytic (∂conjugate(z)/∂conjugate(z) = 1 ≠ 0) > > Hope this info is helpful. > > [1] http://en.wikipedia.org/wiki/Wirtinger_derivatives#Formal_definition > > > On Saturday, September 13, 2014 11:38:07 AM UTC+2, Ralf Stephan wrote: >> >> From searching the net (1), I gather that >> log(x).conjugate(x).diff(x) >> >> should yield >> (log(x)/x).conjugate() >> >> but Sage cannot evaluate such differentiated conjugates of functions: >> sage: ex=log(x).conjugate() >> sage: ex=log(x).conjugate(); ex >> conjugate(log(x)) >> sage: ex.diff(x) >> D[0](conjugate)(log(x))/x >> sage: ex.diff(x).subs(x=-1/2) >> -2*D[0](conjugate)(I*pi + log(1/2)) >> sage: ex.diff(x).subs(x=-1/2).n() >> >> --------------------------------------------------------------------------- >> TypeError: cannot evaluate symbolic expression numerically >> >> which should just be >> sage: conjugate(log(x)/x).subs(x=-1/2).n() >> 1.38629436111989 + 6.28318530717959*I >> if I understand it correctly. Am I wrong? >> >> (1) https://answers.yahoo.com/question/index?qid=20120210002819AAtlqub >> >> PS: Note also that Wolfram gives >> d/dx(log(x)^conjugate) = (Conjugate'(log(x)))/x >> >> when asked for the derivative of conjugate of log(x). >> >> >> Regards, > > -- > You received this message because you are subscribed to a topic in the > Google Groups "sage-support" group. > To unsubscribe from this topic, visit > https://groups.google.com/d/topic/sage-support/bEMPMEYeZKU/unsubscribe. > To unsubscribe from this group and all its topics, send an email to > [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/sage-support. > For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups "sage-support" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/sage-support. For more options, visit https://groups.google.com/d/optout.
