This is somewhat of a follow-up question to ‘Elliptic integrals with parameter outside 0<m<1’.
I have an equation that I'm attempting to simplify that has terms that look something like this:
$ \Re\bigl( -i (c-1) \mathrm K(c) -2i \mathrm E(c) \bigr) $
where $c>1$. I.e., they're being evaluated outside their standard parameter range of $0\le m\le 1$. I know the equation has been derived with Mathematica, and I'm suspicious that the $\Re(i\mathrm K(c))$ terms, etc., are more complex than they need to be.
But I've taken a look at Legendre's Relation which seems to be related to splitting up these integrals into real and imaginary parts, and it doesn't seem to click—I suspect that the expression above is about as simple as I can get it (i.e., even if I manage to replace the integrals by alternate versions that have only real components, the resulting expression will be more complex than that above). Is there anything I'm missing?