On 23 July 2014 07:45, Waldek Hebisch <[email protected]> wrote:
> Bill Page wrote:
>>
>> On 19 July 2014 13:53, Waldek Hebisch <[email protected]> wrote:
>> > Bill Page wrote:
>> >> ...
>> If the integrator is intended to work with expressions representing
>> complex numbers then I would think that it should be possible to
>> symbolically represent conjugate. Then of course we could write
>>
>> abs(x) = sqrt(x*conjugate(x))
>>
>> However the issue of branches and branch-cuts remains.
>
> I should write "functions which are differentiable in complex domain"
> instead of "complex functions". 'conjugate' is not differentiable
> as a function of complex variable (of course it is smooth as a
> function of two real variables).
>
I was a bit surprised to discover that Maple does differentiate
conjugate. It gives
diff(conjugate(f(x)),x) =
diff(f(x),x) * ( conjugate(f(x))/f(x) + 2*abs'(f(x))/signum(f(x)) )
This is apparently a consequence of a careful choice of definition of
abs and signum over the complex domain but I am not yet aware of any
specific application of this definition.
In any case here we are back to signum again ...
> We can do _some_ comutations with conjugate by using different
> definition, namely:
>
> conjugate(f)(z) = conjugate(f(conjugate(z)))
>
> Note that 'conjugate' on the LHS is operation on functions
> which preserves set of holomorphic functions.
>
Yes. It would be interesting to be able to detect in general which
functions are holomorphic.
> More generally, 'conjugate' is a "real" operation. Basically
> any real operation brings us into scope of Richardson
> theorem and equality of functions became undecidable.
> This is why we can not handle such operation in full
> generality (of course we may seek algorithms for decidable
> subclasses).
>
Yes.
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