Wrong formulae for complex elementary functions

["Richard B. Kreckel" <kreckel-HEL5OUoDxoc@public.gmane.org>]

Hi!

The man pages for cacos, cacosf, cacosl, catan, catanf, catanl, cacosh, 
cacoshf, cacoshl, catanh, catanhf, and catanhl contain wrong maths.

cacos, cacosf, cacosl:
   The formula given in the man page
     cacos(z) = -i clog(z + csqrt(z * z - 1))
   gives wrong results in second and fourth quadrant of complex plain.
   The formula
     cacos(z) = -i clog(z + I*csqrt(1 - z * z))
   gives correct results.

catan, catanf, catanl:
   The formula given in the man page
     catan(z) = 1 / 2i clog((1 + iz) / (1 - iz))
   gives wrong results on the negative imaginary axis beginning at -I
   (along one of the two branch cuts). Besides, the formula is written
   in an ambiguous way.
   The formula
     catan(z) = (clog(1 + iz) - clog(1 - iz)) / 2i
   gives correct results.

cacosh, cacoshf, cacoshl:
   The formula given in the man page
     cacosh(z) = (0.5) * clog((1 + z) / (1 - z))
   gives wrong results everywhere in the complex plain. (The formula
   seems to be copied from the one for catanh, where it is sometimes
   correct.)
   The formula
     cacosh(z) = 2 * clog(csqrt((z + 1)/2) + csqrt((z - 1)/2))
   gives correct results.

catanh, catanhf, catanhl:
   The formula given in the man page
     catanh(z) = 0.5 * clog((1 + z) / (1 - z))
   gives wrong results on the positive real axis beginning at 1 (along
   one of the two branch cuts).
   The formula
     catanh(z) = 0.5 * (clog(1 + z) - clog(1 - z))
   gives correct results.

I've also checked casin, casinf, casinl, casinh, casinhf, and casinhl 
and the formulae given there
     casin(z) = -i clog(iz + csqrt(1 - z * z))
     casinh(z) = clog(z + csqrt(z * z + 1))
are actually correct.

I suspect that some of these errors are due to somebody trying to 
"simplify" these formulae. Since this can ruin the behavior, I recommend 
to add a comment to the upstream sources that warns against attempt of 
simplification.

For the sake of reference, I am looking at manpages-dev 3.25-1 on 
Debian/squeeze.

Bye!
   -richy.
-- 
Richard B. Kreckel
<http://www.ginac.de/~kreckel/>
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