Complete Elliptic Integral of the First Kind Identity

Question

Is there an identity for $\frac{K'(k)}{K(k)}=?$ where $K(k)=\int_0^{\frac{\pi}{2}}\frac{1}{\sqrt{1-k^2\sin^2(x)}}dx=\int_0^1\frac{1}{\sqrt{(1-t^2)(1-k^2t^2)}}dt$ is the Complete Elliptic Integral of the First Kind and $K'(k)=K(\sqrt{1-k^2})$ ? The closest I got was showing that $$1.):\frac{K'(k)}{K(\frac{1-k}{1+k})}=\frac{2}{1+k}$$ and from the book "Pi and the AGM" i got an identity showing $$2.):\frac{K'(k)}{K(k)}=2\frac{K'(\frac{2\sqrt{k}}{1+k})}{K(\frac{2\sqrt{k}}{1+k})}$$ however this identity isn't particularly useful for me. Is there a identity that is similar to the identity I derived for $1.)$

Answer 1

The function $\lim_{x\to0}$ $\frac{2\ln(\frac{4}{x})}{\pi}$ = $\frac{K'(x)}{K(x)}$

Answer 2

The OEIS sequence A227503 may be what you are looking for. By definition the Jacobi nome q is $\, q = \exp(-\pi K'(k)/K(k))\,$ which implies $\,K'(k)/K(k) = -\log{(q)}/\pi.\,$ This can be expanded as a power series in $\,k.\,$ The sequence A227503 is closely related to that series expansion. $$ \frac{K'(k)}{K(k)} = -\frac1{\pi}\left(\log(x)+8\left(x + 13\, x^2/2 + 184\, x^3/3+\dots\right)\right) $$ where $\, x := k^2/16.\,$ and the coefficients are from OEIS A227503.

There are several other methods depending on your needs and you can choose among them.Read the DLMF Chapter 19 on Elliptic integrals for much more details. In particular, equations 19.5.5 and 19.5.6.

P.S. In your question you mention an identity from "Pi and the AGM".

Note that in the context of $\ q = \exp(-\pi K'(k)/K(k))\ $ that the map $\ k \to 2\sqrt{k}/(1+k)\ $ corresponds to $\ q \to \sqrt{q}\ $ and its inverse mapping $\ k \to (1-k')/(1+k')\ $ corresponds to $\ q \to q^2.\ $ These are known as ascending and descending transformations.

Answer 3

I don't know if you're asking for something like this: http://mathworld.wolfram.com/ModularEquation.html Or this: https://en.m.wikipedia.org/wiki/Elliptic_integral (look under complete elliptic integral of the first kind under "relation to Jacobi theta function") The relation you derived can be used in the identity for the theta function to find the value of it when the imaginary part is equal to 2, and using some relationships between the other theta functions you can evaluate values of all the theta functions for all powers of 2, which allows you to calculate values of the j invariant for all powers of 2 as well (just thought that was interesting)