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cacos.3, cacosh.3, catan.3, catanh.3: Fix formula describing function 

The man pages for cacos(), cacosh(), catan(), catanh()
contain incorrect formulae describing the functions.

As reported by Richard B. Kreckel:

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.

Reported-by: Richard B. Kreckel <kreckel@ginac.de>
Reviewed-by: Andries Brouwer <Andries.Brouwer@cwi.nl>
Signed-off-by: Michael Kerrisk <mtk.manpages@gmail.com>
This commit is contained in:
Michael Kerrisk 2011-09-15 12:35:56 +02:00
parent 3d5be2aa85
commit 4c3fa19d23
4 changed files with 8 additions and 8 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

4
man3/cacos.3
View File

@ -1,7 +1,7 @@
.\" Copyright 2002 Walter Harms (walter.harms@informatik.uni-oldenburg.de)
.\" Distributed under GPL
.\"
.TH CACOS 3 2008-08-11 "" "Linux Programmer's Manual"
.TH CACOS 3 2011-09-15 "" "Linux Programmer's Manual"
.SH NAME
cacos, cacosf, cacosl \- complex arc cosine
.SH SYNOPSIS
@ -27,7 +27,7 @@ is chosen in the interval [0,pi].
One has:
.nf
cacos(z) = \-i clog(z + csqrt(z * z \- 1))
cacos(z) = \-i * clog(z + i * csqrt(1 \- z * z))
.fi
.SH VERSIONS
These functions first appeared in glibc in version 2.1.

4
man3/cacosh.3
View File

@ -1,7 +1,7 @@
.\" Copyright 2002 Walter Harms(walter.harms@informatik.uni-oldenburg.de)
.\" Distributed under GPL
.\"
.TH CACOSH 3 2008-08-11 "" "Linux Programmer's Manual"
.TH CACOSH 3 2011-09-15 "" "Linux Programmer's Manual"
.SH NAME
cacosh, cacoshf, cacoshl \- complex arc hyperbolic cosine
.SH SYNOPSIS
@ -30,7 +30,7 @@ is chosen nonnegative.
One has:
.nf
cacosh(z) = (0.5) * clog((1 + z) / (1 \- z))
cacosh(z) = 2 * clog(csqrt((z + 1) / 2) + csqrt((z \- 1) / 2))
.fi
.SH VERSIONS
These functions first appeared in glibc in version 2.1.

4
man3/catan.3
View File

@ -1,7 +1,7 @@
.\" Copyright 2002 Walter Harms (walter.harms@informatik.uni-oldenburg.de)
.\" Distributed under GPL
.\"
.TH CATAN 3 2008-08-11 "" "Linux Programmer's Manual"
.TH CATAN 3 2011-09-15 "" "Linux Programmer's Manual"
.SH NAME
catan, catanf, catanl \- complex arc tangents
.SH SYNOPSIS
@ -25,7 +25,7 @@ The real part of y is chosen in the interval [\-pi/2,pi/2].
One has:
.nf
catan(z) = 1 / 2i clog((1 + iz) / (1 \- iz))
catan(z) = (clog(1 + i * z) \- clog(1 \- i * z)) / (2 * i)
.fi
.SH VERSIONS
These functions first appeared in glibc in version 2.1.

4
man3/catanh.3
View File

@ -1,7 +1,7 @@
.\" Copyright 2002 Walter Harms (walter.harms@informatik.uni-oldenburg.de)
.\" Distributed under GPL
.\"
.TH CATANH 3 2008-08-11 "" "Linux Programmer's Manual"
.TH CATANH 3 2011-09-15 "" "Linux Programmer's Manual"
.SH NAME
catanh, catanhf, catanhl \- complex arc tangents hyperbolic
.SH SYNOPSIS
@ -27,7 +27,7 @@ is chosen in the interval [\-pi/2,pi/2].
One has:
.nf
catanh(z) = 0.5 * clog((1 + z) / (1 \- z))
catanh(z) = 0.5 * (clog(1 + z) \- clog(1 \- z))
.fi
.SH VERSIONS
These functions first appeared in glibc in version 2.1.