The Common Lisp standard is
acosh (x) = 2 log( sqrt((x+1) /2 ) + sqrt((x-1) / 2))
(See
http://www.franz.com/support/documentation/6.2/ansicl/dictentr/sinhcosh.htm
.) But in Maxima
(%i14) acosh(x), logarc;
(%o14) log(sqrt(x^2-1)+x)
I often get these things wrong, but I believe:
(1) The function x -> log(sqrt(x^2 -1) + x) is discontinuous along the
imaginary axis,
(2) Except for the origin, the function x -> = 2 log( sqrt((x+1) /2 )
+ sqrt((x-1) / 2)) is
continuous on the imaginary axis.
(3) Both functions are antiderivaties of x -> 1/sqrt(x^2 -1), but
x -> = 2 log( sqrt((x+1) /2 ) + sqrt((x-1) / 2)) is valid on
a larger set.
(4) Abramowitz and Stegun Equation 4.6.21 clearly makes the restriction x
> 1 for
acosh(z) = log(z + sqrt(z^2 - 1)), z = x + i y. See
http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&Page=87&Submit=Go
(5) Since we last talked about such things, GCL has fixed some bugs in
its acosh function; see
(there may be other important messages, but I can't find them right now)
http://www.math.utexas.edu/pipermail/maxima/2002/002955.htmlhttp://www.math.utexas.edu/pipermail/maxima/2002/002949.html
Maybe 5.9.3 would be a good time to:
(a) make the Maxima acosh function the same as the CL acosh function.
(b) move numerical evaluation for exponetial-like functions to CL. Then
Maxima would
not have to use rectform to evaluate things like sin(1.0 + 5.0 * %i).
(c) check for other branch cut inconsistencies.
Barton