Dear Reader:


Let $K(k)$ and $E(k)$ be elliptic integrals of respectively the first and second kind, where $k$ is the elliptic modulus and $k'=\sqrt{1-k^2}$ is the complementary elliptic modulus. 


I happened to encounter the following numerical "fact" (when solving an engineering problem regarding energy conversion):


When I chose a $k$ such that $K(k) \[ K(k')-E(k') \] = \pi/2$, then seemingly $K(k)/K(k')$ is quite close to $\pi/4$, if not exactly. I wonder whether there is an expansion like $K(k)/K(k')=\pi/4+(\text{small terms})$ for this particular $k$. I am just curious. Does someone know? 


Thank you!


Best regards,


Hiroshi Okamoto



approximate equation involving elliptic integrals
Given the Legendre relation, your question is equally about K - E. This is a difference of hypergeometric function values (see <http://en.wikipedia.org/wiki/Elliptic_integral> for all of this). You seem to be setting a condition on *k* that also is simpler when read out of the Legendre relation, on E and K'. I would think the truth would come out of the power series in *k*, though I haven't looked at details.


