ACOS logarithmic form

[Ángel Martin]

(04-28-2016 08:29 PM)Claudio L. Wrote:  

(04-28-2016 06:42 PM)Ángel Martin Wrote:  It's commonly accepted to define as "principal" branch the solution with argument between -pi and pi. This is covered with good explanations in the 15C Special Functions Manual, I'm sure somebody here will be able to point at its URL...

I know it might seem trivial, but once you get into the subject in depth it's a mess.
Each transcendental function has its own weird branch cuts, not so trivial as you can see here.
EDIT: What you mentioned works for ln() and sqrt(), acos() is a different animal.

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.