(from Bill Ackerman) ----- Date: Sat, 08 Aug 2015 00:29:18 -0400 Subject: Re: [math-fun] Here I go again. From: Bill Ackerman <wbackerman@gmail.com> On 07-Aug-15 22:07, Bill Gosper wrote: [Hide Quoted Text] Both http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html (18) and https://en.wikipedia.org/wiki/Elliptic_integral#Differential_equation give K'[k] == E[k]/k/(1 - k^2) - K[k]/k . Both sources use the k (modulus) notation vs Mathematica's m (parameter) notation. But I can't find *any* interpretation that makes this work. I see no similar claim in DLMF nor Borwein*2. ?? --rwg A long long time ago, when I was interested in writing super-precision math libraries, I had to slog through all the funny notations, including Abramowitz & Stegun and Whittaker & Watson. In order to maintain what was left of my sanity while trying to write routines for E and K, I wrote up the block of comments below. It might possibly be of some help. Or maybe not. But please don't ask me to explain this stuff in further detail, it's long forgotten. /* NOMENCLATURE This is in terms of the "complementary parameter", commonly denoted "m1". This is just 1-m, where m is the parameter. (Using the complement puts the bad behavior at zero rather than one.) Now the parameter is the square of the modulus, and the modulus is the sine of the modular angle, so the parameter is the square of the sine of the modular angle. The complementary parameter is the square of the cosine of the modular angle. It is also the square of the sine of the complementary modular angle, where the complementary modular angle is (pi/2 - modular angle). The normal "complete elliptic integral of the second kind" (which is what this is) is generally denoted "E", and the complementary one (which is what we are actually computing) is denoted "E'" (where the prime is not to be confused with the derivative.) When written in terms of the parameter, normal functional notation is used, as in "E(m)". The complementary function is E'(m), or E(1-m), or E(m1). There is other notation, involving vertical bars and backslashes (I'm not making this up! It's in all the classical textbooks!) for denoting the functions of the modulus and modular angle, respectively. There is also notation involving the use of brackets instead of parentheses to denote composition with the inverse of K, the elliptic integral of the first kind. All in all, the classical notation is incredibly befuddled. */ /* To summarize: parameter = modulus^2 complementary parameter = (complementary modulus)^2 complementary parameter = 1 - parameter complementary modulus = sqrt(1 - modulus^2) elpkcm(x) = elpkc(x^2) elpecm(x) = elpec(x^2) */ // Get pi itself. Argument is ignored. // This is the Brent-Salamin algorithm. const arith arith::getpi(const arith &) { ...... (That last thing is for you, Gene.) <eom>