derivative of conjugate of function

[Ralf Stephan]

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,

[maldun]

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

[Ralf Stephan]

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

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