The discussion revolves around the mathematical problem of expressing \( y \) as a function of \( x \) given the equation \( \partial x/\partial y = A * [B/y + C/(y-D)]^{1/2} \). The focus is on the challenges associated with solving this equation, particularly in the context of elliptic integrals.
Participants express a general agreement on the complexity of the problem and the potential connection to elliptic integrals, but there is no consensus on the methods or solutions available for expressing \( y \) as a function of \( x \).
The discussion highlights the unresolved nature of the problem, with participants acknowledging the historical challenges without providing definitive solutions or methods. There are also references to broader mathematical difficulties that remain unsolved.
DeaconJohn said:Replacing the partial of x with respect to y with dx/dy, and moving the dy from the denominator of the l.h.s. to the numerator of the r.h.s., and then taking the indefinite integral of both sides, does not that express x in terms of an (incomplete) elliptic integral of y?
DeaconJohn said:Well, if it is an elliptic integral like I suspect, then that is a classic and difficult problem. Many very good mathematicians were unable to make any progress on it for many years. However, there is now a known method of solving it. That is, if I remember correctly. In any case, it's the kind of stuff Ramanujan was good at. I post a reference next time I run across one.
Nobody said mathematics was easy. Many elementary things are beyond our current grasp. Mathematicians don't even understand which integers in cyclotomic fields are units. It's because there are a lot of hard problems.