From: Jarkko Hietaniemi Date: Fri, 5 Sep 1997 00:00:00 +0000 (+0000) Subject: 5.004_02: Complex/Trig: update X-Git-Url: http://git.shadowcat.co.uk/gitweb/gitweb.cgi?a=commitdiff_plain;h=8c03c583a5f470c68c67a27898643a5dafca2d66;p=p5sagit%2Fp5-mst-13.2.git 5.004_02: Complex/Trig: update The following patches do not fix actual grave errors but they do: - make the code more robust (more discontinuities catched) (e.g. atan(-i), atanh(-1)) - make the results agree on signs and/or conjugate forms with the results MATLAB gives: the results were already correct thanks to the periodicity of trig funcs but now they are also consistent. (e.g. acos(x) did have an unnecessary discontinuity at x = 0) - for some pure real arguments short-circuit the calculation to avoid rounding errors (which make epsilons appear where clear zeros should reign) Tested on NetBSD 1.2G i686, Linux 2.0.25 i686, Digital UNIX 4.0 EV56. p5p-msgid: 199708081842.VAA31214@alpha.hut.fi --- diff --git a/lib/Math/Complex.pm b/lib/Math/Complex.pm index 7a4617c..33c6023 100644 --- a/lib/Math/Complex.pm +++ b/lib/Math/Complex.pm @@ -511,6 +511,27 @@ sub exp { } # +# _logofzero +# +# Die on division by zero. +# +sub _logofzero { + my $mess = "$_[0]: Logarithm of zero.\n"; + + if (defined $_[1]) { + $mess .= "(Because in the definition of $_[0], the argument "; + $mess .= "$_[1] " unless ($_[1] eq '0'); + $mess .= "is 0)\n"; + } + + my @up = caller(1); + + $mess .= "Died at $up[1] line $up[2].\n"; + + die $mess; +} + +# # (log) # # Compute log(z). @@ -659,7 +680,19 @@ sub cotan { Math::Complex::cot(@_) } sub acos { my ($z) = @_; $z = cplx($z, 0) unless ref $z; - return ~i * log($z + (Re($z) * Im($z) > 0 ? 1 : -1) * sqrt($z*$z - 1)); + my ($re, $im) = @{$z->cartesian}; + return atan2(sqrt(1 - $re * $re), $re) + if ($im == 0 and abs($re) <= 1.0); + my $acos = ~i * log($z + sqrt($z*$z - 1)); + if ($im == 0 || + (abs($re) < 1 && abs($im) < 1) || + (abs($re) > 1 && abs($im) > 1 + && !($re > 1 && $im > 1) + && !($re < -1 && $im < -1))) { + # this rule really, REALLY, must be simpler + return -$acos; + } + return $acos; } # @@ -670,6 +703,9 @@ sub acos { sub asin { my ($z) = @_; $z = cplx($z, 0) unless ref $z; + my ($re, $im) = @{$z->cartesian}; + return atan2($re, sqrt(1 - $re * $re)) + if ($im == 0 and abs($re) <= 1.0); return ~i * log(i * $z + sqrt(1 - $z*$z)); } @@ -681,7 +717,8 @@ sub asin { sub atan { my ($z) = @_; $z = cplx($z, 0) unless ref $z; - _divbyzero "atan($z)", "i - $z" if ($z == i); + _divbyzero "atan(i)" if ( $z == i); + _divbyzero "atan(-i)" if (-$z == i); return i/2*log((i + $z) / (i - $z)); } @@ -693,18 +730,35 @@ sub atan { sub asec { my ($z) = @_; _divbyzero "asec($z)", $z if ($z == 0); - return acos(1 / $z); + $z = cplx($z, 0) unless ref $z; + my ($re, $im) = @{$z->cartesian}; + if ($im == 0 && abs($re) >= 1.0) { + my $ire = 1 / $re; + return atan2(sqrt(1 - $ire * $ire), $ire); + } + my $asec = acos(1 / $z); + return ~$asec if $re < 0 && $re > -1 && $im == 0; + return -$asec if $im && !($re > 0 && $im > 0) && !($re < 0 && $im < 0); + return $asec; } # # acsc # -# Computes the arc cosecant sec(z) = asin(1 / z). +# Computes the arc cosecant acsc(z) = asin(1 / z). # sub acsc { my ($z) = @_; _divbyzero "acsc($z)", $z if ($z == 0); - return asin(1 / $z); + $z = cplx($z, 0) unless ref $z; + my ($re, $im) = @{$z->cartesian}; + if ($im == 0 && abs($re) >= 1.0) { + my $ire = 1 / $re; + return atan2($ire, sqrt(1 - $ire * $ire)); + } + my $acsc = asin(1 / $z); + return ~$acsc if $re < 0 && $re > -1 && $im == 0; + return $acsc; } # @@ -717,13 +771,15 @@ sub acosec { Math::Complex::acsc(@_) } # # acot # -# Computes the arc cotangent acot(z) = -i/2 log((i+z) / (z-i)) +# Computes the arc cotangent acot(z) = atan(1 / z) # sub acot { my ($z) = @_; + _divbyzero "acot($z)" if ($z == 0); $z = cplx($z, 0) unless ref $z; - _divbyzero "acot($z)", "$z - i" if ($z == i); - return i/-2 * log((i + $z) / ($z - i)); + _divbyzero "acot(i)", if ( $z == i); + _divbyzero "acot(-i)" if (-$z == i); + return atan(1 / $z); } # @@ -838,11 +894,14 @@ sub cotanh { Math::Complex::coth(@_) } # # acosh # -# Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)). +# Computes the arc hyperbolic cosine acosh(z) = log(z +- sqrt(z*z-1)). # sub acosh { my ($z) = @_; $z = cplx($z, 0) unless ref $z; + my ($re, $im) = @{$z->cartesian}; + return log($re + sqrt(cplx($re*$re - 1, 0))) + if ($im == 0 && $re < 0); return log($z + sqrt($z*$z - 1)); } @@ -864,10 +923,14 @@ sub asinh { # sub atanh { my ($z) = @_; - _divbyzero 'atanh(1)', "1 - $z" if ($z == 1); + _divbyzero 'atanh(1)', "1 - $z" if ($z == 1); + _logofzero 'atanh(-1)' if ($z == -1); $z = cplx($z, 0) unless ref $z; - my $cz = (1 + $z) / (1 - $z); - return log($cz) / 2; + my ($re, $im) = @{$z->cartesian}; + if ($im == 0 && $re > 1) { + return cplx(atanh(1 / $re), pi/2); + } + return log((1 + $z) / (1 - $z)) / 2; } # @@ -878,6 +941,12 @@ sub atanh { sub asech { my ($z) = @_; _divbyzero 'asech(0)', $z if ($z == 0); + $z = cplx($z, 0) unless ref $z; + my ($re, $im) = @{$z->cartesian}; + if ($im == 0 && $re < 0) { + my $ire = 1 / $re; + return log($ire + sqrt(cplx($ire*$ire - 1, 0))); + } return acosh(1 / $z); } @@ -906,10 +975,14 @@ sub acosech { Math::Complex::acsch(@_) } # sub acoth { my ($z) = @_; - _divbyzero 'acoth(1)', "$z - 1" if ($z == 1); + _divbyzero 'acoth(1)', "$z - 1" if ($z == 1); + _logofzero 'acoth(-1)' if ($z == -1); $z = cplx($z, 0) unless ref $z; - my $cz = (1 + $z) / ($z - 1); - return log($cz) / 2; + my ($re, $im) = @{$z->cartesian}; + if ($im == 0 and abs($re) < 1) { + return cplx(acoth(1/$re) , pi/2); + } + return log((1 + $z) / ($z - 1)) / 2; } # @@ -1295,7 +1368,7 @@ numbers: acsc(z) = asin(1 / z) asec(z) = acos(1 / z) - acot(z) = -i/2 * log((i+z) / (z-i)) + acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i)) sinh(z) = 1/2 (exp(z) - exp(-z)) cosh(z) = 1/2 (exp(z) + exp(-z)) @@ -1437,18 +1510,26 @@ The division (/) and the following functions acoth cannot be computed for all arguments because that would mean dividing -by zero. These situations cause fatal runtime errors looking like this +by zero or taking logarithm of zero. These situations cause fatal +runtime errors looking like this cot(0): Division by zero. (Because in the definition of cot(0), the divisor sin(0) is 0) Died at ... -For the C, C, C, C, C, C, C, -C, the argument cannot be C<0> (zero). For the C, -C, the argument cannot be C<1> (one). For the C, C, -the argument cannot be C (the imaginary unit). For the C, -C, C, C, the argument cannot be I, where -I is any integer. +or + + atanh(-1): Logarithm of zero. + Died at... + +For the C, C, C, C, C, C, C, +C, C, the argument cannot be C<0> (zero). For the +C, C, the argument cannot be C<1> (one). For the +C, C, the argument cannot be C<-1> (minus one). For the +C, C, the argument cannot be C (the imaginary unit). +For the C, C, the argument cannot be C<-i> (the negative +imaginary unit). For the C, C, C, C, the +argument cannot be I, where I is any integer. =head1 BUGS diff --git a/lib/Math/Trig.pm b/lib/Math/Trig.pm index c9c045d..a1cbb07 100644 --- a/lib/Math/Trig.pm +++ b/lib/Math/Trig.pm @@ -150,17 +150,24 @@ The following functions acoth cannot be computed for all arguments because that would mean dividing -by zero. These situations cause fatal runtime errors looking like this +by zero or taking logarithm of zero. These situations cause fatal +runtime errors looking like this cot(0): Division by zero. (Because in the definition of cot(0), the divisor sin(0) is 0) Died at ... -For the C, C, C, C, C, C, C, -C, the argument cannot be C<0> (zero). For the C, -C, the argument cannot be C<1> (one). For the C, C, -C, C, the argument cannot be I, where I is -any integer. +or + + atanh(-1): Logarithm of zero. + Died at... + +For the C, C, C, C, C, C, C, +C, C, the argument cannot be C<0> (zero). For the +C, C, the argument cannot be C<1> (one). For the +C, C, the argument cannot be C<-1> (minus one). For the +C, C, C, C, the argument cannot be I, where I is any integer. =head2 SIMPLE (REAL) ARGUMENTS, COMPLEX RESULTS diff --git a/t/lib/complex.t b/t/lib/complex.t index 80a5625..c05f40f 100755 --- a/t/lib/complex.t +++ b/t/lib/complex.t @@ -62,6 +62,21 @@ sub test_dbz { } } +# test the logofzeros + +sub test_loz { + for my $op (@_) { + $test++; + +# push(@script, qq(print "# '$op'\n";)); + push(@script, qq(eval '$op';)); + push(@script, qq(print 'not ' unless (\$@ =~ /Logarithm of zero/);)); + push(@script, qq(print "ok $test\n";)); + } +} + +my $minusi = cplx(0, -1); + test_dbz( 'i/0', # 'tan(pi/2)', # may succeed thanks to floating point inaccuracies @@ -69,9 +84,11 @@ test_dbz( 'csc(0)', 'cot(0)', 'atan(i)', + 'atan($minusi)', 'asec(0)', 'acsc(0)', 'acot(i)', + 'acot($minusi)', # 'tanh(pi/2)', # may succeed thanks to floating point inaccuracies # 'sech(pi/2)', # may succeed thanks to floating point inaccuracies 'csch(0)', @@ -79,7 +96,12 @@ test_dbz( 'atanh(1)', 'asech(0)', 'acsch(0)', - 'acoth(1)' + 'acoth(1)', + ); + +test_loz( + 'atanh(-1)', + 'acoth(-1)', ); # test the 0**0 @@ -342,7 +364,7 @@ __END__ |'z - ~z':'2*i*Im(z)' |'z * ~z':'abs(z) * abs(z)' -{ (2,3); [3,2]; (-3,2); (0,2); 3; 1.2; (-3, 0); (-2, -1); [2,1] } +{ (0.5, 0); (-0.5, 0); (2,3); [3,2]; (-3,2); (0,2); 3; 1.2; (-3, 0); (-2, -1); [2,1] } |'(root(z, 4))[1] ** 4':'z' |'(root(z, 5))[3] ** 5':'z' @@ -350,8 +372,8 @@ __END__ |'abs(z)':'r' |'acot(z)':'acotan(z)' |'acsc(z)':'acosec(z)' -|'acsc(z)':'asin(1 / z)' -|'asec(z)':'acos(1 / z)' +|'abs(acsc(z))':'abs(asin(1 / z))' +|'abs(asec(z))':'abs(acos(1 / z))' |'cbrt(z)':'cbrt(r) * exp(i * t/3)' |'cos(acos(z))':'z' |'cos(z) ** 2 + sin(z) ** 2':1 @@ -409,144 +431,346 @@ __END__ |'atanh(tanh(z))':'z' &sin +(-2.0,0):( -0.90929742682568, 0 ) +(-1.0,0):( -0.84147098480790, 0 ) +(-0.5,0):( -0.47942553860420, 0 ) +( 0.0,0):( 0 , 0 ) +( 0.5,0):( 0.47942553860420, 0 ) +( 1.0,0):( 0.84147098480790, 0 ) +( 2.0,0):( 0.90929742682568, 0 ) + +&sin ( 2, 3):( 9.15449914691143, -4.16890695996656) (-2, 3):( -9.15449914691143, -4.16890695996656) (-2,-3):( -9.15449914691143, 4.16890695996656) ( 2,-3):( 9.15449914691143, 4.16890695996656) &cos +(-2.0,0):( -0.41614683654714, 0 ) +(-1.0,0):( 0.54030230586814, 0 ) +(-0.5,0):( 0.87758256189037, 0 ) +( 0.0,0):( 1 , 0 ) +( 0.5,0):( 0.87758256189037, 0 ) +( 1.0,0):( 0.54030230586814, 0 ) +( 2.0,0):( -0.41614683654714, 0 ) + +&cos ( 2, 3):( -4.18962569096881, -9.10922789375534) (-2, 3):( -4.18962569096881, 9.10922789375534) (-2,-3):( -4.18962569096881, -9.10922789375534) ( 2,-3):( -4.18962569096881, 9.10922789375534) &tan +(-2.0,0):( 2.18503986326152, 0 ) +(-1.0,0):( -1.55740772465490, 0 ) +(-0.5,0):( -0.54630248984379, 0 ) +( 0.0,0):( 0 , 0 ) +( 0.5,0):( 0.54630248984379, 0 ) +( 1.0,0):( 1.55740772465490, 0 ) +( 2.0,0):( -2.18503986326152, 0 ) + +&tan ( 2, 3):( -0.00376402564150, 1.00323862735361) (-2, 3):( 0.00376402564150, 1.00323862735361) (-2,-3):( 0.00376402564150, -1.00323862735361) ( 2,-3):( -0.00376402564150, -1.00323862735361) &sec +(-2.0,0):( -2.40299796172238, 0 ) +(-1.0,0):( 1.85081571768093, 0 ) +(-0.5,0):( 1.13949392732455, 0 ) +( 0.0,0):( 1 , 0 ) +( 0.5,0):( 1.13949392732455, 0 ) +( 1.0,0):( 1.85081571768093, 0 ) +( 2.0,0):( -2.40299796172238, 0 ) + +&sec ( 2, 3):( -0.04167496441114, 0.09061113719624) (-2, 3):( -0.04167496441114, -0.09061113719624) (-2,-3):( -0.04167496441114, 0.09061113719624) ( 2,-3):( -0.04167496441114, -0.09061113719624) &csc +(-2.0,0):( -1.09975017029462, 0 ) +(-1.0,0):( -1.18839510577812, 0 ) +(-0.5,0):( -2.08582964293349, 0 ) +( 0.5,0):( 2.08582964293349, 0 ) +( 1.0,0):( 1.18839510577812, 0 ) +( 2.0,0):( 1.09975017029462, 0 ) + +&csc ( 2, 3):( 0.09047320975321, 0.04120098628857) (-2, 3):( -0.09047320975321, 0.04120098628857) (-2,-3):( -0.09047320975321, -0.04120098628857) ( 2,-3):( 0.09047320975321, -0.04120098628857) &cot +(-2.0,0):( 0.45765755436029, 0 ) +(-1.0,0):( -0.64209261593433, 0 ) +(-0.5,0):( -1.83048772171245, 0 ) +( 0.5,0):( 1.83048772171245, 0 ) +( 1.0,0):( 0.64209261593433, 0 ) +( 2.0,0):( -0.45765755436029, 0 ) + +&cot ( 2, 3):( -0.00373971037634, -0.99675779656936) (-2, 3):( 0.00373971037634, -0.99675779656936) (-2,-3):( 0.00373971037634, 0.99675779656936) ( 2,-3):( -0.00373971037634, 0.99675779656936) &asin +(-2.0,0):( -1.57079632679490, 1.31695789692482) +(-1.0,0):( -1.57079632679490, 0 ) +(-0.5,0):( -0.52359877559830, 0 ) +( 0.0,0):( 0 , 0 ) +( 0.5,0):( 0.52359877559830, 0 ) +( 1.0,0):( 1.57079632679490, 0 ) +( 2.0,0):( 1.57079632679490, -1.31695789692482) + +&asin ( 2, 3):( 0.57065278432110, 1.98338702991654) (-2, 3):( -0.57065278432110, 1.98338702991654) (-2,-3):( -0.57065278432110, -1.98338702991654) ( 2,-3):( 0.57065278432110, -1.98338702991654) &acos +(-2.0,0):( 3.14159265358979, -1.31695789692482) +(-1.0,0):( 3.14159265358979, 0 ) +(-0.5,0):( 2.09439510239320, 0 ) +( 0.0,0):( 1.57079632679490, 0 ) +( 0.5,0):( 1.04719755119660, 0 ) +( 1.0,0):( 0 , 0 ) +( 2.0,0):( 0 , 1.31695789692482) + +&acos ( 2, 3):( 1.00014354247380, -1.98338702991654) (-2, 3):( 2.14144911111600, -1.98338702991654) (-2,-3):( 2.14144911111600, 1.98338702991654) ( 2,-3):( 1.00014354247380, 1.98338702991654) &atan +(-2.0,0):( -1.10714871779409, 0 ) +(-1.0,0):( -0.78539816339745, 0 ) +(-0.5,0):( -0.46364760900081, 0 ) +( 0.0,0):( 0 , 0 ) +( 0.5,0):( 0.46364760900081, 0 ) +( 1.0,0):( 0.78539816339745, 0 ) +( 2.0,0):( 1.10714871779409, 0 ) + +&atan ( 2, 3):( 1.40992104959658, 0.22907268296854) (-2, 3):( -1.40992104959658, 0.22907268296854) (-2,-3):( -1.40992104959658, -0.22907268296854) ( 2,-3):( 1.40992104959658, -0.22907268296854) &asec +(-2.0,0):( 2.09439510239320, 0 ) +(-1.0,0):( 3.14159265358979, 0 ) +(-0.5,0):( 3.14159265358979, -1.31695789692482) +( 0.5,0):( 0 , 1.31695789692482) +( 1.0,0):( 0 , 0 ) +( 2.0,0):( 1.04719755119660, 0 ) + +&asec ( 2, 3):( 1.42041072246703, 0.23133469857397) (-2, 3):( 1.72118193112276, 0.23133469857397) (-2,-3):( 1.72118193112276, -0.23133469857397) ( 2,-3):( 1.42041072246703, -0.23133469857397) &acsc +(-2.0,0):( -0.52359877559830, 0 ) +(-1.0,0):( -1.57079632679490, 0 ) +(-0.5,0):( -1.57079632679490, 1.31695789692482) +( 0.5,0):( 1.57079632679490, -1.31695789692482) +( 1.0,0):( 1.57079632679490, 0 ) +( 2.0,0):( 0.52359877559830, 0 ) + +&acsc ( 2, 3):( 0.15038560432786, -0.23133469857397) (-2, 3):( -0.15038560432786, -0.23133469857397) (-2,-3):( -0.15038560432786, 0.23133469857397) ( 2,-3):( 0.15038560432786, 0.23133469857397) &acot +(-2.0,0):( -0.46364760900081, 0 ) +(-1.0,0):( -0.78539816339745, 0 ) +(-0.5,0):( -1.10714871779409, 0 ) +( 0.5,0):( 1.10714871779409, 0 ) +( 1.0,0):( 0.78539816339745, 0 ) +( 2.0,0):( 0.46364760900081, 0 ) + +&acot ( 2, 3):( 0.16087527719832, -0.22907268296854) (-2, 3):( -0.16087527719832, -0.22907268296854) (-2,-3):( -0.16087527719832, 0.22907268296854) ( 2,-3):( 0.16087527719832, 0.22907268296854) &sinh +(-2.0,0):( -3.62686040784702, 0 ) +(-1.0,0):( -1.17520119364380, 0 ) +(-0.5,0):( -0.52109530549375, 0 ) +( 0.0,0):( 0 , 0 ) +( 0.5,0):( 0.52109530549375, 0 ) +( 1.0,0):( 1.17520119364380, 0 ) +( 2.0,0):( 3.62686040784702, 0 ) + +&sinh ( 2, 3):( -3.59056458998578, 0.53092108624852) (-2, 3):( 3.59056458998578, 0.53092108624852) (-2,-3):( 3.59056458998578, -0.53092108624852) ( 2,-3):( -3.59056458998578, -0.53092108624852) &cosh +(-2.0,0):( 3.76219569108363, 0 ) +(-1.0,0):( 1.54308063481524, 0 ) +(-0.5,0):( 1.12762596520638, 0 ) +( 0.0,0):( 1 , 0 ) +( 0.5,0):( 1.12762596520638, 0 ) +( 1.0,0):( 1.54308063481524, 0 ) +( 2.0,0):( 3.76219569108363, 0 ) + +&cosh ( 2, 3):( -3.72454550491532, 0.51182256998738) (-2, 3):( -3.72454550491532, -0.51182256998738) (-2,-3):( -3.72454550491532, 0.51182256998738) ( 2,-3):( -3.72454550491532, -0.51182256998738) &tanh +(-2.0,0):( -0.96402758007582, 0 ) +(-1.0,0):( -0.76159415595576, 0 ) +(-0.5,0):( -0.46211715726001, 0 ) +( 0.0,0):( 0 , 0 ) +( 0.5,0):( 0.46211715726001, 0 ) +( 1.0,0):( 0.76159415595576, 0 ) +( 2.0,0):( 0.96402758007582, 0 ) + +&tanh ( 2, 3):( 0.96538587902213, -0.00988437503832) (-2, 3):( -0.96538587902213, -0.00988437503832) (-2,-3):( -0.96538587902213, 0.00988437503832) ( 2,-3):( 0.96538587902213, 0.00988437503832) &sech +(-2.0,0):( 0.26580222883408, 0 ) +(-1.0,0):( 0.64805427366389, 0 ) +(-0.5,0):( 0.88681888397007, 0 ) +( 0.0,0):( 1 , 0 ) +( 0.5,0):( 0.88681888397007, 0 ) +( 1.0,0):( 0.64805427366389, 0 ) +( 2.0,0):( 0.26580222883408, 0 ) + +&sech ( 2, 3):( -0.26351297515839, -0.03621163655877) (-2, 3):( -0.26351297515839, 0.03621163655877) (-2,-3):( -0.26351297515839, -0.03621163655877) ( 2,-3):( -0.26351297515839, 0.03621163655877) &csch +(-2.0,0):( -0.27572056477178, 0 ) +(-1.0,0):( -0.85091812823932, 0 ) +(-0.5,0):( -1.91903475133494, 0 ) +( 0.5,0):( 1.91903475133494, 0 ) +( 1.0,0):( 0.85091812823932, 0 ) +( 2.0,0):( 0.27572056477178, 0 ) + +&csch ( 2, 3):( -0.27254866146294, -0.04030057885689) (-2, 3):( 0.27254866146294, -0.04030057885689) (-2,-3):( 0.27254866146294, 0.04030057885689) ( 2,-3):( -0.27254866146294, 0.04030057885689) &coth +(-2.0,0):( -1.03731472072755, 0 ) +(-1.0,0):( -1.31303528549933, 0 ) +(-0.5,0):( -2.16395341373865, 0 ) +( 0.5,0):( 2.16395341373865, 0 ) +( 1.0,0):( 1.31303528549933, 0 ) +( 2.0,0):( 1.03731472072755, 0 ) + +&coth ( 2, 3):( 1.03574663776500, 0.01060478347034) (-2, 3):( -1.03574663776500, 0.01060478347034) (-2,-3):( -1.03574663776500, -0.01060478347034) ( 2,-3):( 1.03574663776500, -0.01060478347034) &asinh +(-2.0,0):( -1.44363547517881, 0 ) +(-1.0,0):( -0.88137358701954, 0 ) +(-0.5,0):( -0.48121182505960, 0 ) +( 0.0,0):( 0 , 0 ) +( 0.5,0):( 0.48121182505960, 0 ) +( 1.0,0):( 0.88137358701954, 0 ) +( 2.0,0):( 1.44363547517881, 0 ) + +&asinh ( 2, 3):( 1.96863792579310, 0.96465850440760) (-2, 3):( -1.96863792579310, 0.96465850440761) (-2,-3):( -1.96863792579310, -0.96465850440761) ( 2,-3):( 1.96863792579310, -0.96465850440760) &acosh +(-2.0,0):( -1.31695789692482, 3.14159265358979) +(-1.0,0):( 0, 3.14159265358979) +(-0.5,0):( 0, 2.09439510239320) +( 0.0,0):( 0, 1.57079632679490) +( 0.5,0):( 0, 1.04719755119660) +( 1.0,0):( 0 , 0 ) +( 2.0,0):( 1.31695789692482, 0 ) + +&acosh ( 2, 3):( 1.98338702991654, 1.00014354247380) (-2, 3):( -1.98338702991653, -2.14144911111600) (-2,-3):( -1.98338702991653, 2.14144911111600) ( 2,-3):( 1.98338702991654, -1.00014354247380) &atanh +(-2.0,0):( -0.54930614433405, 1.57079632679490) +(-0.5,0):( -0.54930614433405, 0 ) +( 0.0,0):( 0 , 0 ) +( 0.5,0):( 0.54930614433405, 0 ) +( 2.0,0):( 0.54930614433405, 1.57079632679490) + +&atanh ( 2, 3):( 0.14694666622553, 1.33897252229449) (-2, 3):( -0.14694666622553, 1.33897252229449) (-2,-3):( -0.14694666622553, -1.33897252229449) ( 2,-3):( 0.14694666622553, -1.33897252229449) &asech +(-2.0,0):( 0 , 2.09439510239320) +(-1.0,0):( 0 , 3.14159265358979) +(-0.5,0):( -1.31695789692482, 3.14159265358979) +( 0.5,0):( 1.31695789692482, 0 ) +( 1.0,0):( 0 , 0 ) +( 2.0,0):( 0 , 1.04719755119660) + +&asech ( 2, 3):( 0.23133469857397, -1.42041072246703) (-2, 3):( -0.23133469857397, 1.72118193112276) (-2,-3):( -0.23133469857397, -1.72118193112276) ( 2,-3):( 0.23133469857397, 1.42041072246703) &acsch +(-2.0,0):( -0.48121182505960, 0 ) +(-1.0,0):( -0.88137358701954, 0 ) +(-0.5,0):( -1.44363547517881, 0 ) +( 0.5,0):( 1.44363547517881, 0 ) +( 1.0,0):( 0.88137358701954, 0 ) +( 2.0,0):( 0.48121182505960, 0 ) + +&acsch ( 2, 3):( 0.15735549884499, -0.22996290237721) (-2, 3):( -0.15735549884499, -0.22996290237721) (-2,-3):( -0.15735549884499, 0.22996290237721) ( 2,-3):( 0.15735549884499, 0.22996290237721) &acoth +(-2.0,0):( -0.54930614433405, 0 ) +(-0.5,0):( -0.54930614433405, 1.57079632679490) +( 0.5,0):( 0.54930614433405, 1.57079632679490) +( 2.0,0):( 0.54930614433405, 0 ) + +&acoth ( 2, 3):( 0.14694666622553, -0.23182380450040) (-2, 3):( -0.14694666622553, -0.23182380450040) (-2,-3):( -0.14694666622553, 0.23182380450040)