NavList:

John,
In the paper and subsequent ones I adopted the notation of reference (1) (Nevannlinna & Paatero) as they are the original source of the derivation you refer to. Throughout I use z-bar to represent the complex conjugate. z* in the derivation is used to denote the point diametrically opposite to z (see the diagram in the paper). Admittedly the choice of this notation is a bit unfortunate and can be source of confusion. It might have been better to use z-prime or z-hat.

If I understand the question in your last paragraph correctly, you are asking why the point diametrically opposite z isn't just 1/z. The reason is that if z = r exp(+i phi) then the point diametrically opposite satisfies z* (or z-pole in your notation) = (1/r) exp(+i (phi + pi)) = - (1/r) exp(+i phi) = 1/z-bar. As an example to show why 1/z doesn't work, consider a position on the prime meridian, phi = 0. On the complex plane this corresponds to a point, z, lying on the positive real axis. But 1/z also lies on the positive real axis whereas the point diametrically opposite (i.e. z* or z-pole) clearly should lie on the negative real axis.

I hope this helps,

Regards,
Robin

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