ACOS logarithmic form

[Claudio L.]

(04-29-2016 06:06 AM)Ángel Martin Wrote:  That's not logical; once a branch of the logarithm is used it should apply to all your functions and provide the same criteria across. The Ln is the root cause of every multi-value here, including the square root which is nothing more that another logarithm if you use the expression SQRT(z) = exp [ ln(z) / 2]. The ACOS function is more of the same, in this instance with the rule applied twice since the ln appears twice in its expression - or even three times if you'd use Z^2 = exp [ 2 ln(z) ] ...

The branch selection is therefore critical. I remember in complex analysis classes we sometimes needed to change the branch to avoid function singularities during integration, as in the integration path crossing the cut where the function wasn't analytical. It's been a while since that so I may also remember wrong though I suspect the basic concepts are still clear in my memory.

You are right, once you select the right convention for which branch to take on sqrt() and ln(), acos() should be automatic... or that's what I thought, that's the whole point of this thread.
The formula is supposed to take the branch cut of the sqrt(), then shift it, then ln() remaps it and adds its own branch cuts.
The whole point is that, like you, I expected this to be taken care of automatically, but that's not the case with the formula from Wikipedia or the 15C manual. It does work great when you use i*sqrt(1-Z^2), rather than sqrt(z^2-1).
It seems the 15C and the 41 (thank to all who provided the results) agree with the branch cuts of i*sqrt(1-Z^2), so why is the formula in the manual showing sqrt(Z^2-1)? same question to the Wikipedia folks and a couple of other websites I found.

There's very good agreement in the results between all calculators and major CAS systems (thanks to the Maple check in this thread), it's the docs that cause the discrepancy.