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