I am looking into elliptic integrals and I am slightly confused about something. The material I'm using for reference is mainly Wikipedia and Higher Transcendental Functions II (page 306 onward).
The integral I have to calculate is, for a 4th-degree polynomial $P$ :
$$I_1 = \int^1_0 \frac{dx}{\sqrt{P(x)}}$$
which in the book is called $I_1$. I have read up on the Legendre transformation, which allows one via rational transformations to "write" said $P$ as $(1 - \xi^2)(1-k^2\xi^2) = \eta^2$, whatever that means. My understanding of it is that this $k^2$ parameter is somehow linked to the starting polynomial $P$, so that aforementionned integral can be computed as the complete elliptic integral of the first kind :
$$K(k) = \int^1_0 \frac{d\xi}{\sqrt{(1 - \xi^2)(1-k^2\xi^2)}}$$
I was able to successfully compute $k(P)$, but before I go any further I want to be sure that I am understanding this correctly. Do I really have $I_1 = K(k(P))$, just like that ? I know that I can easily compute $K(k(P))$ using the arithmetic-geometric mean, so that's not a problem.