Mirror symmetry and realpart/imagpart of functions

On Sun, Feb 1, 2009 at 1:26 PM, Dieter Kaiser <drdieterkaiser at web.de> wrote:
> ...If the function has mirror symmetry this simplifies to:
>
> realpart(f(x+%i*y)) = 1/2*   (f(x-%i) + f(x+%i*y))
> imagpart(f(x+%i*y)) = 1/2*%i*(f(x-%i) - f(x+%i*y))
>
> This is implemented directly into the routine risplit for the Erf function.
>
> It is possible to support this simplification in general for function which are
> declared to have mirror symmetry (or partially)....

These are nice transformations, which have the good property of
eliminating realpart/imagpart/abs/carg nouns, which means further
simplifications are possible.  However, I am not sure that a user
would always prefer to see

         sqrt(gamma(1-%i))*sqrt(gamma(%i+1))

than

         abs(gamma(1+%i))

I do not actually have an opinion on this, but I think the question is
worth thinking about....

              -s