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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