Elliptic function - different definitions

[csopi]

Elliptic function -- different definitions

Hi,

I have recently discovered, that the definition of the complete elliptic integral of the first kind in Wolfram Mathematica (EllipticK[m]) is different from the usual (K(m)), given in Abramowitz-Stegun.

Their domains are not the same. In Abramowitz-Stegun, K is defined for |m| < 1, however in Mathematica, the domain is -Infinity < m<1.

My question is, the following. How EllipticK[m] is related to K, when m < -1? Is it related at all?

 

[Ben Niehoff]

Did you look here?

http://mathworld.wolfram.com/CompleteEllipticIntegraloftheFirstKind.html

It looks like Mathematica's version has two branch cuts on the real axis, one for m < -1, one for m > 1. So the functions ought to be the same up to a choice of which branch...

 

[csopi]

Thanks for the fast reply.

Yes, I have checked it, although I did not really understand it. For m > 1, EllipticK is complex, while for m < -1 it is real, and I cannot see, how they are related.

To be specific, my problem is that I am writing a numerical code in GSL, that uses elliptic functions. The formulas to be calculated numerically were derivedwith the help of Mathematica, so they contain EllipticK. The problem is, that (following Abramowitz) the GSL version of K(m) is defined only for |m| < 1, and when I'm passing a large negative argument to it (e.g. -4, that is a valid value for EllipticK), it dies with an out of domain error. So I need to express EllipticK[m] in terms of K.