#2312 real and imaginary parts acosh

[Ted Woollett]

despite acosh(x) for real x having both real and
imaginary parts,

(%i1) realpart(acosh(x));

(%o1) acosh(x)

(%i2) imagpart(acosh(x));

(%o2) 0

[Ted Woollett]

system info:
Maxima version: 5.25.1
Maxima build date: 10:2 9/6/2011
Host type: i686-pc-mingw32
Lisp implementation type: GNU Common Lisp (GCL)
Lisp implementation version: GCL 2.6.8

[Kris Katterjohn]

I'm getting the following results now (SBCL 1.2.4 + git HEAD). It looks like these are correct, just glancing at some plots.

Anyone want to confirm? I'm marking this ticket as pending for now.

(%i1) realpart(acosh(x));
(%o1) log((cos(atan2(0,x-1)/2)*sin(atan2(0,x+1)/2)
           +sin(atan2(0,x-1)/2)*cos(atan2(0,x+1)/2))
           ^2
           *abs(x-1)*abs(x+1)
           +((-sin(atan2(0,x-1)/2)*sin(atan2(0,x+1)/2)*sqrt(abs(x-1))
                                  *sqrt(abs(x+1)))
            +cos(atan2(0,x-1)/2)*cos(atan2(0,x+1)/2)*sqrt(abs(x-1))
                                *sqrt(abs(x+1))+x)
            ^2)
 /2
(%i2) imagpart(acosh(x));
(%o2) atan2((cos(atan2(0,x-1)/2)*sin(atan2(0,x+1)/2)
             +sin(atan2(0,x-1)/2)*cos(atan2(0,x+1)/2))
             *sqrt(abs(x-1))*sqrt(abs(x+1)),
            (cos(atan2(0,x-1)/2)*cos(atan2(0,x+1)/2)
             -sin(atan2(0,x-1)/2)*sin(atan2(0,x+1)/2))
             *sqrt(abs(x-1))*sqrt(abs(x+1))
             +x)

[Raymond Toy]

I didn't check your final expressions, but since logarc(acosh(x)) says log(stuff), couldn't we say the realpart is log(abs(stuff)) and imagpart is atan2(0,stuff)? Hmm, maybee the imagpart needs more work since stuff might be a complex number.