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 clickâI 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?
Best Answer
Well, for the simplified case that you now presented in your current version of the question (again, as I said, the complete elliptic integral of the third kind requires a different sort of analysis), the identities in formula 19.7.3 in the DLMF can be rewritten in parameter form as
$$K(m)=\frac1{\sqrt{m}}\left(K\left(\frac1{m}\right)-i K\left(1-\frac1{m}\right)\right)$$
and
$$E(m)=\sqrt{m}\left(E\left(\frac1{m}\right)-\left(1-\frac1{m}\right)K\left(\frac1{m}\right)+i\left(E\left(1-\frac1{m}\right)-\frac1{m} K\left(1-\frac1{m}\right)\right)\right)$$
If we apply both to your expression above, it can be simplified to
$$2\sqrt{c}E\left(1-\frac1{c}\right)-\frac{1+c}{\sqrt{c}}K\left(1-\frac1{c}\right)$$