There are many calculations that use only the differential or algebraic
properties of
log, sqrt, and etc. Often in mid-stream, such calculations choose a
particular branch
and integrate along one or both sides of the branch cut.
The Digital Library of Mathematical Functions (DLMF) recognizes the
utility of defining
both a principal value and a general value for log-like functions. And
sometimes (maybe
always) the DLMF gives different functions different names (see
http://dlmf.nist.gov/4.2).
Maxima has both a log and a plog function. I suppose Maxima's plog
function is the
principal value logarithm and log is a general value logarithm. For
numerical evaluation,
Maxima is backwards:
(%i38) log(1.2 + %i);
(%o38) 0.6947382761967*%i+0.44599901965256
(%i39) plog(1.2 + %i);
(%o39) plog(%i+1.2)
Although I recognize the utility have having both a principal value and a
general value
square root function, if we have just one, I would prefer that it be the
principal value
square root (and similarly for the log-like and exp-like functions). I'm
biased, but I
think the to_poly_solver does OK making the assumption that sqrt means the
principal
square root.
For odd roots, there is the complication of the real-branch rule.
Currently, the application
of the real branch rule is controlled by the option variables m1pranch and
domain, I think.
--Barton