[OctDev] ellipke with negative arguments

[Jaakko R <jaa...@we...>]

Hi,

Many years ago, after calculating the magnetic field produced by a 
circular loop, I was wondering why Matlab doesn't support complete 
elliptic integrals with negative arguments.

I would suggest to improve octave-forge's version.  Could someone step 
in and drop a patch for a vectorized version, or should I give it a try?

It should be something like this for a scalar version 
if m<0,
   [k, e] = ellipke(-m./(1-m))
   k = 1/sqrt(1-m)*k;
   e = sqrt(1-m)*e;
end

The following equations can be seen to be correct by the substitution 
theta=pi/2-theta'.  Comments?

Cheerio,
-Jaakko


Complete elliptic integral of first kind
\begin{equation}
   \label{eq:ellkint}
   K(m) = \int_0^{\pi/2} [1-m\sin^2(\theta)]^{-1/2}\,d\theta.
\end{equation}

\begin{equation}
   \label{eq:ellk}
   K(-m) = \frac{1}{\sqrt{1+m}}\,K\left(\frac{m}{1+m}\right)
\end{equation}

Complete elliptic integral of second kind
\begin{equation}
   \label{eq:elleint}
   E(m) = \int_0^{\pi/2} [1-m\sin^2(\theta)]^{1/2}\,d\theta.
\end{equation}

\begin{equation}
   \label{eq:elle}
   E(-m) = \sqrt{1+m}\,E\left(\frac{m}{1+m}\right)
\end{equation}