ACOS logarithmic form

[Ángel Martin]

(04-29-2016 09:15 PM)ljubo Wrote:  If I'm interpreting correctly the ACOS illustration from HP 15C Advanced Functions Handbook, page 61, then the positive real axis for x>1 (blue line) is mapped to the positive imaginary axis - means documentation and implementation are consistent. Note that the principal value is denoted with all cap letters (on top of right-hand illustration), where lowercase letters are denoting the multi-value inverse, see explanation on the page 59.

Interesting, I never looked at those diagrams as showing mappings between domain and results regions but as defining the ranges of applicability, i.e. showing where will the results be; yet what you're saying appears to be accurate.

What's intriguing is the sentence in pg. 62: "The principal branches in the last four graphs above are obtained from the equations shown, but don't necessarily use the principal branches of ln(z) and sqr(z)"

So this leads us to believe that in some cases they had another criteria for choosing the principal branches, like keeping symmetries or other properties to make them more analogous to the behavior in the real domain.

Oh and I remember better now: for contour integrals involving multi-valued (or other) functions the method wasn't to change the branch but to make the integration path "elude" the singularities - thus those funny "C" shapes and circles around the poles.


Perhaps a mathematician in the audience could stop our poking the beast and provide a more rigorous clarification?