Initial import of Perl5. The king is dead; long live the king!

1 files changed, 1775 insertions, 0 deletions

diff --git a/contrib/perl5/lib/Math/Complex.pm b/contrib/perl5/lib/Math/Complex.pm
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+#
+# Complex numbers and associated mathematical functions
+# -- Raphael Manfredi	Since Sep 1996
+# -- Jarkko Hietaniemi	Since Mar 1997
+# -- Daniel S. Lewart	Since Sep 1997
+#
+
+require Exporter;
+package Math::Complex;
+
+use strict;
+
+use vars qw($VERSION @ISA @EXPORT %EXPORT_TAGS);
+
+my ( $i, $ip2, %logn );
+
+$VERSION = sprintf("%s", q$Id: Complex.pm,v 1.25 1998/02/05 16:07:37 jhi Exp $ =~ /(\d+\.\d+)/);
+
+@ISA = qw(Exporter);
+
+my @trig = qw(
+	      pi
+	      tan
+	      csc cosec sec cot cotan
+	      asin acos atan
+	      acsc acosec asec acot acotan
+	      sinh cosh tanh
+	      csch cosech sech coth cotanh
+	      asinh acosh atanh
+	      acsch acosech asech acoth acotanh
+	     );
+
+@EXPORT = (qw(
+	     i Re Im rho theta arg
+	     sqrt log ln
+	     log10 logn cbrt root
+	     cplx cplxe
+	     ),
+	   @trig);
+
+%EXPORT_TAGS = (
+    'trig' => [@trig],
+);
+
+use overload
+	'+'	=> \&plus,
+	'-'	=> \&minus,
+	'*'	=> \&multiply,
+	'/'	=> \&divide,
+	'**'	=> \&power,
+	'<=>'	=> \&spaceship,
+	'neg'	=> \&negate,
+	'~'	=> \&conjugate,
+	'abs'	=> \&abs,
+	'sqrt'	=> \&sqrt,
+	'exp'	=> \&exp,
+	'log'	=> \&log,
+	'sin'	=> \&sin,
+	'cos'	=> \&cos,
+	'tan'	=> \&tan,
+	'atan2'	=> \&atan2,
+	qw("" stringify);
+
+#
+# Package "privates"
+#
+
+my $package = 'Math::Complex';		# Package name
+my $display = 'cartesian';		# Default display format
+my $eps     = 1e-14;			# Epsilon
+
+#
+# Object attributes (internal):
+#	cartesian	[real, imaginary] -- cartesian form
+#	polar		[rho, theta] -- polar form
+#	c_dirty		cartesian form not up-to-date
+#	p_dirty		polar form not up-to-date
+#	display		display format (package's global when not set)
+#
+
+# Die on bad *make() arguments.
+
+sub _cannot_make {
+    die "@{[(caller(1))[3]]}: Cannot take $_[0] of $_[1].\n";
+}
+
+#
+# ->make
+#
+# Create a new complex number (cartesian form)
+#
+sub make {
+	my $self = bless {}, shift;
+	my ($re, $im) = @_;
+	my $rre = ref $re;
+	if ( $rre ) {
+	    if ( $rre eq ref $self ) {
+		$re = Re($re);
+	    } else {
+		_cannot_make("real part", $rre);
+	    }
+	}
+	my $rim = ref $im;
+	if ( $rim ) {
+	    if ( $rim eq ref $self ) {
+		$im = Im($im);
+	    } else {
+		_cannot_make("imaginary part", $rim);
+	    }
+	}
+	$self->{'cartesian'} = [ $re, $im ];
+	$self->{c_dirty} = 0;
+	$self->{p_dirty} = 1;
+	$self->display_format('cartesian');
+	return $self;
+}
+
+#
+# ->emake
+#
+# Create a new complex number (exponential form)
+#
+sub emake {
+	my $self = bless {}, shift;
+	my ($rho, $theta) = @_;
+	my $rrh = ref $rho;
+	if ( $rrh ) {
+	    if ( $rrh eq ref $self ) {
+		$rho = rho($rho);
+	    } else {
+		_cannot_make("rho", $rrh);
+	    }
+	}
+	my $rth = ref $theta;
+	if ( $rth ) {
+	    if ( $rth eq ref $self ) {
+		$theta = theta($theta);
+	    } else {
+		_cannot_make("theta", $rth);
+	    }
+	}
+	if ($rho < 0) {
+	    $rho   = -$rho;
+	    $theta = ($theta <= 0) ? $theta + pi() : $theta - pi();
+	}
+	$self->{'polar'} = [$rho, $theta];
+	$self->{p_dirty} = 0;
+	$self->{c_dirty} = 1;
+	$self->display_format('polar');
+	return $self;
+}
+
+sub new { &make }		# For backward compatibility only.
+
+#
+# cplx
+#
+# Creates a complex number from a (re, im) tuple.
+# This avoids the burden of writing Math::Complex->make(re, im).
+#
+sub cplx {
+	my ($re, $im) = @_;
+	return $package->make($re, defined $im ? $im : 0);
+}
+
+#
+# cplxe
+#
+# Creates a complex number from a (rho, theta) tuple.
+# This avoids the burden of writing Math::Complex->emake(rho, theta).
+#
+sub cplxe {
+	my ($rho, $theta) = @_;
+	return $package->emake($rho, defined $theta ? $theta : 0);
+}
+
+#
+# pi
+#
+# The number defined as pi = 180 degrees
+#
+use constant pi => 4 * CORE::atan2(1, 1);
+
+#
+# pit2
+#
+# The full circle
+#
+use constant pit2 => 2 * pi;
+
+#
+# pip2
+#
+# The quarter circle
+#
+use constant pip2 => pi / 2;
+
+#
+# deg1
+#
+# One degree in radians, used in stringify_polar.
+#
+
+use constant deg1 => pi / 180;
+
+#
+# uplog10
+#
+# Used in log10().
+#
+use constant uplog10 => 1 / CORE::log(10);
+
+#
+# i
+#
+# The number defined as i*i = -1;
+#
+sub i () {
+        return $i if ($i);
+	$i = bless {};
+	$i->{'cartesian'} = [0, 1];
+	$i->{'polar'}     = [1, pip2];
+	$i->{c_dirty} = 0;
+	$i->{p_dirty} = 0;
+	return $i;
+}
+
+#
+# Attribute access/set routines
+#
+
+sub cartesian {$_[0]->{c_dirty} ?
+		   $_[0]->update_cartesian : $_[0]->{'cartesian'}}
+sub polar     {$_[0]->{p_dirty} ?
+		   $_[0]->update_polar : $_[0]->{'polar'}}
+
+sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] }
+sub set_polar     { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] }
+
+#
+# ->update_cartesian
+#
+# Recompute and return the cartesian form, given accurate polar form.
+#
+sub update_cartesian {
+	my $self = shift;
+	my ($r, $t) = @{$self->{'polar'}};
+	$self->{c_dirty} = 0;
+	return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)];
+}
+
+#
+#
+# ->update_polar
+#
+# Recompute and return the polar form, given accurate cartesian form.
+#
+sub update_polar {
+	my $self = shift;
+	my ($x, $y) = @{$self->{'cartesian'}};
+	$self->{p_dirty} = 0;
+	return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0;
+	return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y), CORE::atan2($y, $x)];
+}
+
+#
+# (plus)
+#
+# Computes z1+z2.
+#
+sub plus {
+	my ($z1, $z2, $regular) = @_;
+	my ($re1, $im1) = @{$z1->cartesian};
+	$z2 = cplx($z2) unless ref $z2;
+	my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+	unless (defined $regular) {
+		$z1->set_cartesian([$re1 + $re2, $im1 + $im2]);
+		return $z1;
+	}
+	return (ref $z1)->make($re1 + $re2, $im1 + $im2);
+}
+
+#
+# (minus)
+#
+# Computes z1-z2.
+#
+sub minus {
+	my ($z1, $z2, $inverted) = @_;
+	my ($re1, $im1) = @{$z1->cartesian};
+	$z2 = cplx($z2) unless ref $z2;
+	my ($re2, $im2) = @{$z2->cartesian};
+	unless (defined $inverted) {
+		$z1->set_cartesian([$re1 - $re2, $im1 - $im2]);
+		return $z1;
+	}
+	return $inverted ?
+		(ref $z1)->make($re2 - $re1, $im2 - $im1) :
+		(ref $z1)->make($re1 - $re2, $im1 - $im2);
+
+}
+
+#
+# (multiply)
+#
+# Computes z1*z2.
+#
+sub multiply {
+        my ($z1, $z2, $regular) = @_;
+	if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
+	    # if both polar better use polar to avoid rounding errors
+	    my ($r1, $t1) = @{$z1->polar};
+	    my ($r2, $t2) = @{$z2->polar};
+	    my $t = $t1 + $t2;
+	    if    ($t >   pi()) { $t -= pit2 }
+	    elsif ($t <= -pi()) { $t += pit2 }
+	    unless (defined $regular) {
+		$z1->set_polar([$r1 * $r2, $t]);
+		return $z1;
+	    }
+	    return (ref $z1)->emake($r1 * $r2, $t);
+	} else {
+	    my ($x1, $y1) = @{$z1->cartesian};
+	    if (ref $z2) {
+		my ($x2, $y2) = @{$z2->cartesian};
+		return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2);
+	    } else {
+		return (ref $z1)->make($x1*$z2, $y1*$z2);
+	    }
+	}
+}
+
+#
+# _divbyzero
+#
+# Die on division by zero.
+#
+sub _divbyzero {
+    my $mess = "$_[0]: Division by zero.\n";
+
+    if (defined $_[1]) {
+	$mess .= "(Because in the definition of $_[0], the divisor ";
+	$mess .= "$_[1] " unless ($_[1] eq '0');
+	$mess .= "is 0)\n";
+    }
+
+    my @up = caller(1);
+
+    $mess .= "Died at $up[1] line $up[2].\n";
+
+    die $mess;
+}
+
+#
+# (divide)
+#
+# Computes z1/z2.
+#
+sub divide {
+	my ($z1, $z2, $inverted) = @_;
+	if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
+	    # if both polar better use polar to avoid rounding errors
+	    my ($r1, $t1) = @{$z1->polar};
+	    my ($r2, $t2) = @{$z2->polar};
+	    my $t;
+	    if ($inverted) {
+		_divbyzero "$z2/0" if ($r1 == 0);
+		$t = $t2 - $t1;
+		if    ($t >   pi()) { $t -= pit2 }
+		elsif ($t <= -pi()) { $t += pit2 }
+		return (ref $z1)->emake($r2 / $r1, $t);
+	    } else {
+		_divbyzero "$z1/0" if ($r2 == 0);
+		$t = $t1 - $t2;
+		if    ($t >   pi()) { $t -= pit2 }
+		elsif ($t <= -pi()) { $t += pit2 }
+		return (ref $z1)->emake($r1 / $r2, $t);
+	    }
+	} else {
+	    my ($d, $x2, $y2);
+	    if ($inverted) {
+		($x2, $y2) = @{$z1->cartesian};
+		$d = $x2*$x2 + $y2*$y2;
+		_divbyzero "$z2/0" if $d == 0;
+		return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d);
+	    } else {
+		my ($x1, $y1) = @{$z1->cartesian};
+		if (ref $z2) {
+		    ($x2, $y2) = @{$z2->cartesian};
+		    $d = $x2*$x2 + $y2*$y2;
+		    _divbyzero "$z1/0" if $d == 0;
+		    my $u = ($x1*$x2 + $y1*$y2)/$d;
+		    my $v = ($y1*$x2 - $x1*$y2)/$d;
+		    return (ref $z1)->make($u, $v);
+		} else {
+		    _divbyzero "$z1/0" if $z2 == 0;
+		    return (ref $z1)->make($x1/$z2, $y1/$z2);
+		}
+	    }
+	}
+}
+
+#
+# _zerotozero
+#
+# Die on zero raised to the zeroth.
+#
+sub _zerotozero {
+    my $mess = "The zero raised to the zeroth power is not defined.\n";
+
+    my @up = caller(1);
+
+    $mess .= "Died at $up[1] line $up[2].\n";
+
+    die $mess;
+}
+
+#
+# (power)
+#
+# Computes z1**z2 = exp(z2 * log z1)).
+#
+sub power {
+	my ($z1, $z2, $inverted) = @_;
+	my $z1z = $z1 == 0;
+	my $z2z = $z2 == 0;
+	_zerotozero if ($z1z and $z2z);
+	if ($inverted) {
+	    return 0 if ($z2z);
+	    return 1 if ($z1z or $z2 == 1);
+	} else {
+	    return 0 if ($z1z);
+	    return 1 if ($z2z or $z1 == 1);
+	}
+	my $w = $inverted ? CORE::exp($z1 * CORE::log($z2)) : CORE::exp($z2 * CORE::log($z1));
+	# If both arguments cartesian, return cartesian, else polar.
+	return $z1->{c_dirty} == 0 &&
+	       (not ref $z2 or $z2->{c_dirty} == 0) ?
+	       cplx(@{$w->cartesian}) : $w;
+}
+
+#
+# (spaceship)
+#
+# Computes z1 <=> z2.
+# Sorts on the real part first, then on the imaginary part. Thus 2-4i > 3+8i.
+#
+sub spaceship {
+	my ($z1, $z2, $inverted) = @_;
+	my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
+	my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+	my $sgn = $inverted ? -1 : 1;
+	return $sgn * ($re1 <=> $re2) if $re1 != $re2;
+	return $sgn * ($im1 <=> $im2);
+}
+
+#
+# (negate)
+#
+# Computes -z.
+#
+sub negate {
+	my ($z) = @_;
+	if ($z->{c_dirty}) {
+		my ($r, $t) = @{$z->polar};
+		$t = ($t <= 0) ? $t + pi : $t - pi;
+		return (ref $z)->emake($r, $t);
+	}
+	my ($re, $im) = @{$z->cartesian};
+	return (ref $z)->make(-$re, -$im);
+}
+
+#
+# (conjugate)
+#
+# Compute complex's conjugate.
+#
+sub conjugate {
+	my ($z) = @_;
+	if ($z->{c_dirty}) {
+		my ($r, $t) = @{$z->polar};
+		return (ref $z)->emake($r, -$t);
+	}
+	my ($re, $im) = @{$z->cartesian};
+	return (ref $z)->make($re, -$im);
+}
+
+#
+# (abs)
+#
+# Compute or set complex's norm (rho).
+#
+sub abs {
+	my ($z, $rho) = @_;
+	return $z unless ref $z;
+	if (defined $rho) {
+	    $z->{'polar'} = [ $rho, ${$z->polar}[1] ];
+	    $z->{p_dirty} = 0;
+	    $z->{c_dirty} = 1;
+	    return $rho;
+	} else {
+	    return ${$z->polar}[0];
+	}
+}
+
+sub _theta {
+    my $theta = $_[0];
+
+    if    ($$theta >   pi()) { $$theta -= pit2 }
+    elsif ($$theta <= -pi()) { $$theta += pit2 }
+}
+
+#
+# arg
+#
+# Compute or set complex's argument (theta).
+#
+sub arg {
+	my ($z, $theta) = @_;
+	return $z unless ref $z;
+	if (defined $theta) {
+	    _theta(\$theta);
+	    $z->{'polar'} = [ ${$z->polar}[0], $theta ];
+	    $z->{p_dirty} = 0;
+	    $z->{c_dirty} = 1;
+	} else {
+	    $theta = ${$z->polar}[1];
+	    _theta(\$theta);
+	}
+	return $theta;
+}
+
+#
+# (sqrt)
+#
+# Compute sqrt(z).
+#
+# It is quite tempting to use wantarray here so that in list context
+# sqrt() would return the two solutions.  This, however, would
+# break things like
+#
+#	print "sqrt(z) = ", sqrt($z), "\n";
+#
+# The two values would be printed side by side without no intervening
+# whitespace, quite confusing.
+# Therefore if you want the two solutions use the root().
+#
+sub sqrt {
+	my ($z) = @_;
+	my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0);
+	return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re) if $im == 0;
+	my ($r, $t) = @{$z->polar};
+	return (ref $z)->emake(CORE::sqrt($r), $t/2);
+}
+
+#
+# cbrt
+#
+# Compute cbrt(z) (cubic root).
+#
+# Why are we not returning three values?  The same answer as for sqrt().
+#
+sub cbrt {
+	my ($z) = @_;
+	return $z < 0 ? -CORE::exp(CORE::log(-$z)/3) : ($z > 0 ? CORE::exp(CORE::log($z)/3): 0)
+	    unless ref $z;
+	my ($r, $t) = @{$z->polar};
+	return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3);
+}
+
+#
+# _rootbad
+#
+# Die on bad root.
+#
+sub _rootbad {
+    my $mess = "Root $_[0] not defined, root must be positive integer.\n";
+
+    my @up = caller(1);
+
+    $mess .= "Died at $up[1] line $up[2].\n";
+
+    die $mess;
+}
+
+#
+# root
+#
+# Computes all nth root for z, returning an array whose size is n.
+# `n' must be a positive integer.
+#
+# The roots are given by (for k = 0..n-1):
+#
+# z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n))
+#
+sub root {
+	my ($z, $n) = @_;
+	_rootbad($n) if ($n < 1 or int($n) != $n);
+	my ($r, $t) = ref $z ? @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
+	my @root;
+	my $k;
+	my $theta_inc = pit2 / $n;
+	my $rho = $r ** (1/$n);
+	my $theta;
+	my $cartesian = ref $z && $z->{c_dirty} == 0;
+	for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) {
+	    my $w = cplxe($rho, $theta);
+	    # Yes, $cartesian is loop invariant.
+	    push @root, $cartesian ? cplx(@{$w->cartesian}) : $w;
+	}
+	return @root;
+}
+
+#
+# Re
+#
+# Return or set Re(z).
+#
+sub Re {
+	my ($z, $Re) = @_;
+	return $z unless ref $z;
+	if (defined $Re) {
+	    $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ];
+	    $z->{c_dirty} = 0;
+	    $z->{p_dirty} = 1;
+	} else {
+	    return ${$z->cartesian}[0];
+	}
+}
+
+#
+# Im
+#
+# Return or set Im(z).
+#
+sub Im {
+	my ($z, $Im) = @_;
+	return $z unless ref $z;
+	if (defined $Im) {
+	    $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ];
+	    $z->{c_dirty} = 0;
+	    $z->{p_dirty} = 1;
+	} else {
+	    return ${$z->cartesian}[1];
+	}
+}
+
+#
+# rho
+#
+# Return or set rho(w).
+#
+sub rho {
+    Math::Complex::abs(@_);
+}
+
+#
+# theta
+#
+# Return or set theta(w).
+#
+sub theta {
+    Math::Complex::arg(@_);
+}
+
+#
+# (exp)
+#
+# Computes exp(z).
+#
+sub exp {
+	my ($z) = @_;
+	my ($x, $y) = @{$z->cartesian};
+	return (ref $z)->emake(CORE::exp($x), $y);
+}
+
+#
+# _logofzero
+#
+# Die on logarithm of 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 $mess;
+}
+
+#
+# (log)
+#
+# Compute log(z).
+#
+sub log {
+	my ($z) = @_;
+	unless (ref $z) {
+	    _logofzero("log") if $z == 0;
+	    return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi);
+	}
+	my ($r, $t) = @{$z->polar};
+	_logofzero("log") if $r == 0;
+	if    ($t >   pi()) { $t -= pit2 }
+	elsif ($t <= -pi()) { $t += pit2 }
+	return (ref $z)->make(CORE::log($r), $t);
+}
+
+#
+# ln
+#
+# Alias for log().
+#
+sub ln { Math::Complex::log(@_) }
+
+#
+# log10
+#
+# Compute log10(z).
+#
+
+sub log10 {
+	return Math::Complex::log($_[0]) * uplog10;
+}
+
+#
+# logn
+#
+# Compute logn(z,n) = log(z) / log(n)
+#
+sub logn {
+	my ($z, $n) = @_;
+	$z = cplx($z, 0) unless ref $z;
+	my $logn = $logn{$n};
+	$logn = $logn{$n} = CORE::log($n) unless defined $logn;	# Cache log(n)
+	return CORE::log($z) / $logn;
+}
+
+#
+# (cos)
+#
+# Compute cos(z) = (exp(iz) + exp(-iz))/2.
+#
+sub cos {
+	my ($z) = @_;
+	my ($x, $y) = @{$z->cartesian};
+	my $ey = CORE::exp($y);
+	my $ey_1 = 1 / $ey;
+	return (ref $z)->make(CORE::cos($x) * ($ey + $ey_1)/2,
+			      CORE::sin($x) * ($ey_1 - $ey)/2);
+}
+
+#
+# (sin)
+#
+# Compute sin(z) = (exp(iz) - exp(-iz))/2.
+#
+sub sin {
+	my ($z) = @_;
+	my ($x, $y) = @{$z->cartesian};
+	my $ey = CORE::exp($y);
+	my $ey_1 = 1 / $ey;
+	return (ref $z)->make(CORE::sin($x) * ($ey + $ey_1)/2,
+			      CORE::cos($x) * ($ey - $ey_1)/2);
+}
+
+#
+# tan
+#
+# Compute tan(z) = sin(z) / cos(z).
+#
+sub tan {
+	my ($z) = @_;
+	my $cz = CORE::cos($z);
+	_divbyzero "tan($z)", "cos($z)" if (CORE::abs($cz) < $eps);
+	return CORE::sin($z) / $cz;
+}
+
+#
+# sec
+#
+# Computes the secant sec(z) = 1 / cos(z).
+#
+sub sec {
+	my ($z) = @_;
+	my $cz = CORE::cos($z);
+	_divbyzero "sec($z)", "cos($z)" if ($cz == 0);
+	return 1 / $cz;
+}
+
+#
+# csc
+#
+# Computes the cosecant csc(z) = 1 / sin(z).
+#
+sub csc {
+	my ($z) = @_;
+	my $sz = CORE::sin($z);
+	_divbyzero "csc($z)", "sin($z)" if ($sz == 0);
+	return 1 / $sz;
+}
+
+#
+# cosec
+#
+# Alias for csc().
+#
+sub cosec { Math::Complex::csc(@_) }
+
+#
+# cot
+#
+# Computes cot(z) = cos(z) / sin(z).
+#
+sub cot {
+	my ($z) = @_;
+	my $sz = CORE::sin($z);
+	_divbyzero "cot($z)", "sin($z)" if ($sz == 0);
+	return CORE::cos($z) / $sz;
+}
+
+#
+# cotan
+#
+# Alias for cot().
+#
+sub cotan { Math::Complex::cot(@_) }
+
+#
+# acos
+#
+# Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
+#
+sub acos {
+	my $z = $_[0];
+	return CORE::atan2(CORE::sqrt(1-$z*$z), $z) if (! ref $z) && CORE::abs($z) <= 1;
+	my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
+	my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
+	my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
+	my $alpha = ($t1 + $t2)/2;
+	my $beta  = ($t1 - $t2)/2;
+	$alpha = 1 if $alpha < 1;
+	if    ($beta >  1) { $beta =  1 }
+	elsif ($beta < -1) { $beta = -1 }
+	my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta);
+	my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
+	$v = -$v if $y > 0 || ($y == 0 && $x < -1);
+	return $package->make($u, $v);
+}
+
+#
+# asin
+#
+# Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)).
+#
+sub asin {
+	my $z = $_[0];
+	return CORE::atan2($z, CORE::sqrt(1-$z*$z)) if (! ref $z) && CORE::abs($z) <= 1;
+	my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
+	my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
+	my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
+	my $alpha = ($t1 + $t2)/2;
+	my $beta  = ($t1 - $t2)/2;
+	$alpha = 1 if $alpha < 1;
+	if    ($beta >  1) { $beta =  1 }
+	elsif ($beta < -1) { $beta = -1 }
+	my $u =  CORE::atan2($beta, CORE::sqrt(1-$beta*$beta));
+	my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
+	$v = -$v if $y > 0 || ($y == 0 && $x < -1);
+	return $package->make($u, $v);
+}
+
+#
+# atan
+#
+# Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)).
+#
+sub atan {
+	my ($z) = @_;
+	return CORE::atan2($z, 1) unless ref $z;
+	_divbyzero "atan(i)"  if ( $z == i);
+	_divbyzero "atan(-i)" if (-$z == i);
+	my $log = CORE::log((i + $z) / (i - $z));
+	$ip2 = 0.5 * i unless defined $ip2;
+	return $ip2 * $log;
+}
+
+#
+# asec
+#
+# Computes the arc secant asec(z) = acos(1 / z).
+#
+sub asec {
+	my ($z) = @_;
+	_divbyzero "asec($z)", $z if ($z == 0);
+	return acos(1 / $z);
+}
+
+#
+# acsc
+#
+# Computes the arc cosecant acsc(z) = asin(1 / z).
+#
+sub acsc {
+	my ($z) = @_;
+	_divbyzero "acsc($z)", $z if ($z == 0);
+	return asin(1 / $z);
+}
+
+#
+# acosec
+#
+# Alias for acsc().
+#
+sub acosec { Math::Complex::acsc(@_) }
+
+#
+# acot
+#
+# Computes the arc cotangent acot(z) = atan(1 / z)
+#
+sub acot {
+	my ($z) = @_;
+	_divbyzero "acot(0)"  if (CORE::abs($z)     < $eps);
+	return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z) unless ref $z;
+	_divbyzero "acot(i)"  if (CORE::abs($z - i) < $eps);
+	_logofzero "acot(-i)" if (CORE::abs($z + i) < $eps);
+	return atan(1 / $z);
+}
+
+#
+# acotan
+#
+# Alias for acot().
+#
+sub acotan { Math::Complex::acot(@_) }
+
+#
+# cosh
+#
+# Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2.
+#
+sub cosh {
+	my ($z) = @_;
+	my $ex;
+	unless (ref $z) {
+	    $ex = CORE::exp($z);
+	    return ($ex + 1/$ex)/2;
+	}
+	my ($x, $y) = @{$z->cartesian};
+	$ex = CORE::exp($x);
+	my $ex_1 = 1 / $ex;
+	return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2,
+			      CORE::sin($y) * ($ex - $ex_1)/2);
+}
+
+#
+# sinh
+#
+# Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2.
+#
+sub sinh {
+	my ($z) = @_;
+	my $ex;
+	unless (ref $z) {
+	    $ex = CORE::exp($z);
+	    return ($ex - 1/$ex)/2;
+	}
+	my ($x, $y) = @{$z->cartesian};
+	$ex = CORE::exp($x);
+	my $ex_1 = 1 / $ex;
+	return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2,
+			      CORE::sin($y) * ($ex + $ex_1)/2);
+}
+
+#
+# tanh
+#
+# Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z).
+#
+sub tanh {
+	my ($z) = @_;
+	my $cz = cosh($z);
+	_divbyzero "tanh($z)", "cosh($z)" if ($cz == 0);
+	return sinh($z) / $cz;
+}
+
+#
+# sech
+#
+# Computes the hyperbolic secant sech(z) = 1 / cosh(z).
+#
+sub sech {
+	my ($z) = @_;
+	my $cz = cosh($z);
+	_divbyzero "sech($z)", "cosh($z)" if ($cz == 0);
+	return 1 / $cz;
+}
+
+#
+# csch
+#
+# Computes the hyperbolic cosecant csch(z) = 1 / sinh(z).
+#
+sub csch {
+	my ($z) = @_;
+	my $sz = sinh($z);
+	_divbyzero "csch($z)", "sinh($z)" if ($sz == 0);
+	return 1 / $sz;
+}
+
+#
+# cosech
+#
+# Alias for csch().
+#
+sub cosech { Math::Complex::csch(@_) }
+
+#
+# coth
+#
+# Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z).
+#
+sub coth {
+	my ($z) = @_;
+	my $sz = sinh($z);
+	_divbyzero "coth($z)", "sinh($z)" if ($sz == 0);
+	return cosh($z) / $sz;
+}
+
+#
+# cotanh
+#
+# Alias for coth().
+#
+sub cotanh { Math::Complex::coth(@_) }
+
+#
+# acosh
+#
+# Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).
+#
+sub acosh {
+	my ($z) = @_;
+	unless (ref $z) {
+	    return CORE::log($z + CORE::sqrt($z*$z-1)) if $z >= 1;
+	    $z = cplx($z, 0);
+	}
+	my ($re, $im) = @{$z->cartesian};
+	if ($im == 0) {
+	    return cplx(CORE::log($re + CORE::sqrt($re*$re - 1)), 0) if $re >= 1;
+	    return cplx(0, CORE::atan2(CORE::sqrt(1-$re*$re), $re)) if CORE::abs($re) <= 1;
+	}
+	return CORE::log($z + CORE::sqrt($z*$z - 1));
+}
+
+#
+# asinh
+#
+# Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z-1))
+#
+sub asinh {
+	my ($z) = @_;
+	return CORE::log($z + CORE::sqrt($z*$z + 1));
+}
+
+#
+# atanh
+#
+# Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)).
+#
+sub atanh {
+	my ($z) = @_;
+	unless (ref $z) {
+	    return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1;
+	    $z = cplx($z, 0);
+	}
+	_divbyzero 'atanh(1)',  "1 - $z" if ($z ==  1);
+	_logofzero 'atanh(-1)'           if ($z == -1);
+	return 0.5 * CORE::log((1 + $z) / (1 - $z));
+}
+
+#
+# asech
+#
+# Computes the hyperbolic arc secant asech(z) = acosh(1 / z).
+#
+sub asech {
+	my ($z) = @_;
+	_divbyzero 'asech(0)', $z if ($z == 0);
+	return acosh(1 / $z);
+}
+
+#
+# acsch
+#
+# Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z).
+#
+sub acsch {
+	my ($z) = @_;
+	_divbyzero 'acsch(0)', $z if ($z == 0);
+	return asinh(1 / $z);
+}
+
+#
+# acosech
+#
+# Alias for acosh().
+#
+sub acosech { Math::Complex::acsch(@_) }
+
+#
+# acoth
+#
+# Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)).
+#
+sub acoth {
+	my ($z) = @_;
+	_divbyzero 'acoth(0)'            if (CORE::abs($z)     < $eps);
+	unless (ref $z) {
+	    return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1;
+	    $z = cplx($z, 0);
+	}
+	_divbyzero 'acoth(1)',  "$z - 1" if (CORE::abs($z - 1) < $eps);
+	_logofzero 'acoth(-1)', "1 / $z" if (CORE::abs($z + 1) < $eps);
+	return CORE::log((1 + $z) / ($z - 1)) / 2;
+}
+
+#
+# acotanh
+#
+# Alias for acot().
+#
+sub acotanh { Math::Complex::acoth(@_) }
+
+#
+# (atan2)
+#
+# Compute atan(z1/z2).
+#
+sub atan2 {
+	my ($z1, $z2, $inverted) = @_;
+	my ($re1, $im1, $re2, $im2);
+	if ($inverted) {
+	    ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+	    ($re2, $im2) = @{$z1->cartesian};
+	} else {
+	    ($re1, $im1) = @{$z1->cartesian};
+	    ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+	}
+	if ($im2 == 0) {
+	    return cplx(CORE::atan2($re1, $re2), 0) if $im1 == 0;
+	    return cplx(($im1<=>0) * pip2, 0) if $re2 == 0;
+	}
+	my $w = atan($z1/$z2);
+	my ($u, $v) = ref $w ? @{$w->cartesian} : ($w, 0);
+	$u += pi   if $re2 < 0;
+	$u -= pit2 if $u > pi;
+	return cplx($u, $v);
+}
+
+#
+# display_format
+# ->display_format
+#
+# Set (fetch if no argument) display format for all complex numbers that
+# don't happen to have overridden it via ->display_format
+#
+# When called as a method, this actually sets the display format for
+# the current object.
+#
+# Valid object formats are 'c' and 'p' for cartesian and polar. The first
+# letter is used actually, so the type can be fully spelled out for clarity.
+#
+sub display_format {
+	my $self = shift;
+	my $format = undef;
+
+	if (ref $self) {			# Called as a method
+		$format = shift;
+	} else {				# Regular procedure call
+		$format = $self;
+		undef $self;
+	}
+
+	if (defined $self) {
+		return defined $self->{display} ? $self->{display} : $display
+			unless defined $format;
+		return $self->{display} = $format;
+	}
+
+	return $display unless defined $format;
+	return $display = $format;
+}
+
+#
+# (stringify)
+#
+# Show nicely formatted complex number under its cartesian or polar form,
+# depending on the current display format:
+#
+# . If a specific display format has been recorded for this object, use it.
+# . Otherwise, use the generic current default for all complex numbers,
+#   which is a package global variable.
+#
+sub stringify {
+	my ($z) = shift;
+	my $format;
+
+	$format = $display;
+	$format = $z->{display} if defined $z->{display};
+
+	return $z->stringify_polar if $format =~ /^p/i;
+	return $z->stringify_cartesian;
+}
+
+#
+# ->stringify_cartesian
+#
+# Stringify as a cartesian representation 'a+bi'.
+#
+sub stringify_cartesian {
+	my $z  = shift;
+	my ($x, $y) = @{$z->cartesian};
+	my ($re, $im);
+
+	$x = int($x + ($x < 0 ? -1 : 1) * $eps)
+		if int(CORE::abs($x)) != int(CORE::abs($x) + $eps);
+	$y = int($y + ($y < 0 ? -1 : 1) * $eps)
+		if int(CORE::abs($y)) != int(CORE::abs($y) + $eps);
+
+	$re = "$x" if CORE::abs($x) >= $eps;
+        if ($y == 1)                           { $im = 'i' }
+        elsif ($y == -1)                       { $im = '-i' }
+        elsif (CORE::abs($y) >= $eps)                { $im = $y . "i" }
+
+	my $str = '';
+	$str = $re if defined $re;
+	$str .= "+$im" if defined $im;
+	$str =~ s/\+-/-/;
+	$str =~ s/^\+//;
+	$str =~ s/([-+])1i/$1i/; # Not redundant with the above 1/-1 tests.
+	$str = '0' unless $str;
+
+	return $str;
+}
+
+
+# Helper for stringify_polar, a Greatest Common Divisor with a memory.
+
+sub _gcd {
+    my ($a, $b) = @_;
+
+    use integer;
+
+    # Loops forever if given negative inputs.
+
+    if    ($b and $a > $b) { return gcd($a % $b, $b) }
+    elsif ($a and $b > $a) { return gcd($b % $a, $a) }
+    else                   { return $a ? $a : $b     }
+}
+
+my %gcd;
+
+sub gcd {
+    my ($a, $b) = @_;
+
+    my $id = "$a $b";
+    
+    unless (exists $gcd{$id}) {
+	$gcd{$id} = _gcd($a, $b);
+	$gcd{"$b $a"} = $gcd{$id};
+    }
+
+    return $gcd{$id};
+}
+
+#
+# ->stringify_polar
+#
+# Stringify as a polar representation '[r,t]'.
+#
+sub stringify_polar {
+	my $z  = shift;
+	my ($r, $t) = @{$z->polar};
+	my $theta;
+
+	return '[0,0]' if $r <= $eps;
+
+	my $nt = $t / pit2;
+	$nt = ($nt - int($nt)) * pit2;
+	$nt += pit2 if $nt < 0;			# Range [0, 2pi]
+
+	if (CORE::abs($nt) <= $eps)		{ $theta = 0 }
+	elsif (CORE::abs(pi-$nt) <= $eps)	{ $theta = 'pi' }
+
+	if (defined $theta) {
+		$r = int($r + ($r < 0 ? -1 : 1) * $eps)
+			if int(CORE::abs($r)) != int(CORE::abs($r) + $eps);
+		$theta = int($theta + ($theta < 0 ? -1 : 1) * $eps)
+			if ($theta ne 'pi' and
+			    int(CORE::abs($theta)) != int(CORE::abs($theta) + $eps));
+		return "\[$r,$theta\]";
+	}
+
+	#
+	# Okay, number is not a real. Try to identify pi/n and friends...
+	#
+
+	$nt -= pit2 if $nt > pi;
+
+	if (CORE::abs($nt) >= deg1) {
+	    my ($n, $k, $kpi);
+
+	    for ($k = 1, $kpi = pi; $k < 10; $k++, $kpi += pi) {
+		$n = int($kpi / $nt + ($nt > 0 ? 1 : -1) * 0.5);
+		if (CORE::abs($kpi/$n - $nt) <= $eps) {
+		    $n = CORE::abs($n);
+		    my $gcd = gcd($k, $n);
+		    if ($gcd > 1) {
+			$k /= $gcd;
+			$n /= $gcd;
+		    }
+		    next if $n > 360;
+		    $theta = ($nt < 0 ? '-':'').
+			     ($k == 1 ? 'pi':"${k}pi");
+		    $theta .= '/'.$n if $n > 1;
+		    last;
+		}
+	    }
+	}
+
+	$theta = $nt unless defined $theta;
+
+	$r = int($r + ($r < 0 ? -1 : 1) * $eps)
+		if int(CORE::abs($r)) != int(CORE::abs($r) + $eps);
+	$theta = int($theta + ($theta < 0 ? -1 : 1) * $eps)
+		if ($theta !~ m(^-?\d*pi/\d+$) and
+		    int(CORE::abs($theta)) != int(CORE::abs($theta) + $eps));
+
+	return "\[$r,$theta\]";
+}
+
+1;
+__END__
+
+=head1 NAME
+
+Math::Complex - complex numbers and associated mathematical functions
+
+=head1 SYNOPSIS
+
+	use Math::Complex;
+
+	$z = Math::Complex->make(5, 6);
+	$t = 4 - 3*i + $z;
+	$j = cplxe(1, 2*pi/3);
+
+=head1 DESCRIPTION
+
+This package lets you create and manipulate complex numbers. By default,
+I<Perl> limits itself to real numbers, but an extra C<use> statement brings
+full complex support, along with a full set of mathematical functions
+typically associated with and/or extended to complex numbers.
+
+If you wonder what complex numbers are, they were invented to be able to solve
+the following equation:
+
+	x*x = -1
+
+and by definition, the solution is noted I<i> (engineers use I<j> instead since
+I<i> usually denotes an intensity, but the name does not matter). The number
+I<i> is a pure I<imaginary> number.
+
+The arithmetics with pure imaginary numbers works just like you would expect
+it with real numbers... you just have to remember that
+
+	i*i = -1
+
+so you have:
+
+	5i + 7i = i * (5 + 7) = 12i
+	4i - 3i = i * (4 - 3) = i
+	4i * 2i = -8
+	6i / 2i = 3
+	1 / i = -i
+
+Complex numbers are numbers that have both a real part and an imaginary
+part, and are usually noted:
+
+	a + bi
+
+where C<a> is the I<real> part and C<b> is the I<imaginary> part. The
+arithmetic with complex numbers is straightforward. You have to
+keep track of the real and the imaginary parts, but otherwise the
+rules used for real numbers just apply:
+
+	(4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
+	(2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i
+
+A graphical representation of complex numbers is possible in a plane
+(also called the I<complex plane>, but it's really a 2D plane).
+The number
+
+	z = a + bi
+
+is the point whose coordinates are (a, b). Actually, it would
+be the vector originating from (0, 0) to (a, b). It follows that the addition
+of two complex numbers is a vectorial addition.
+
+Since there is a bijection between a point in the 2D plane and a complex
+number (i.e. the mapping is unique and reciprocal), a complex number
+can also be uniquely identified with polar coordinates:
+
+	[rho, theta]
+
+where C<rho> is the distance to the origin, and C<theta> the angle between
+the vector and the I<x> axis. There is a notation for this using the
+exponential form, which is:
+
+	rho * exp(i * theta)
+
+where I<i> is the famous imaginary number introduced above. Conversion
+between this form and the cartesian form C<a + bi> is immediate:
+
+	a = rho * cos(theta)
+	b = rho * sin(theta)
+
+which is also expressed by this formula:
+
+	z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
+
+In other words, it's the projection of the vector onto the I<x> and I<y>
+axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta>
+the I<argument> of the complex number. The I<norm> of C<z> will be
+noted C<abs(z)>.
+
+The polar notation (also known as the trigonometric
+representation) is much more handy for performing multiplications and
+divisions of complex numbers, whilst the cartesian notation is better
+suited for additions and subtractions. Real numbers are on the I<x>
+axis, and therefore I<theta> is zero or I<pi>.
+
+All the common operations that can be performed on a real number have
+been defined to work on complex numbers as well, and are merely
+I<extensions> of the operations defined on real numbers. This means
+they keep their natural meaning when there is no imaginary part, provided
+the number is within their definition set.
+
+For instance, the C<sqrt> routine which computes the square root of
+its argument is only defined for non-negative real numbers and yields a
+non-negative real number (it is an application from B<R+> to B<R+>).
+If we allow it to return a complex number, then it can be extended to
+negative real numbers to become an application from B<R> to B<C> (the
+set of complex numbers):
+
+	sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
+
+It can also be extended to be an application from B<C> to B<C>,
+whilst its restriction to B<R> behaves as defined above by using
+the following definition:
+
+	sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)
+
+Indeed, a negative real number can be noted C<[x,pi]> (the modulus
+I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative
+number) and the above definition states that
+
+	sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i
+
+which is exactly what we had defined for negative real numbers above.
+The C<sqrt> returns only one of the solutions: if you want the both,
+use the C<root> function.
+
+All the common mathematical functions defined on real numbers that
+are extended to complex numbers share that same property of working
+I<as usual> when the imaginary part is zero (otherwise, it would not
+be called an extension, would it?).
+
+A I<new> operation possible on a complex number that is
+the identity for real numbers is called the I<conjugate>, and is noted
+with an horizontal bar above the number, or C<~z> here.
+
+	 z = a + bi
+	~z = a - bi
+
+Simple... Now look:
+
+	z * ~z = (a + bi) * (a - bi) = a*a + b*b
+
+We saw that the norm of C<z> was noted C<abs(z)> and was defined as the
+distance to the origin, also known as:
+
+	rho = abs(z) = sqrt(a*a + b*b)
+
+so
+
+	z * ~z = abs(z) ** 2
+
+If z is a pure real number (i.e. C<b == 0>), then the above yields:
+
+	a * a = abs(a) ** 2
+
+which is true (C<abs> has the regular meaning for real number, i.e. stands
+for the absolute value). This example explains why the norm of C<z> is
+noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet
+is the regular C<abs> we know when the complex number actually has no
+imaginary part... This justifies I<a posteriori> our use of the C<abs>
+notation for the norm.
+
+=head1 OPERATIONS
+
+Given the following notations:
+
+	z1 = a + bi = r1 * exp(i * t1)
+	z2 = c + di = r2 * exp(i * t2)
+	z = <any complex or real number>
+
+the following (overloaded) operations are supported on complex numbers:
+
+	z1 + z2 = (a + c) + i(b + d)
+	z1 - z2 = (a - c) + i(b - d)
+	z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
+	z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
+	z1 ** z2 = exp(z2 * log z1)
+	~z = a - bi
+	abs(z) = r1 = sqrt(a*a + b*b)
+	sqrt(z) = sqrt(r1) * exp(i * t/2)
+	exp(z) = exp(a) * exp(i * b)
+	log(z) = log(r1) + i*t
+	sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
+	cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
+	atan2(z1, z2) = atan(z1/z2)
+
+The following extra operations are supported on both real and complex
+numbers:
+
+	Re(z) = a
+	Im(z) = b
+	arg(z) = t
+	abs(z) = r
+
+	cbrt(z) = z ** (1/3)
+	log10(z) = log(z) / log(10)
+	logn(z, n) = log(z) / log(n)
+
+	tan(z) = sin(z) / cos(z)
+
+	csc(z) = 1 / sin(z)
+	sec(z) = 1 / cos(z)
+	cot(z) = 1 / tan(z)
+
+	asin(z) = -i * log(i*z + sqrt(1-z*z))
+	acos(z) = -i * log(z + i*sqrt(1-z*z))
+	atan(z) = i/2 * log((i+z) / (i-z))
+
+	acsc(z) = asin(1 / z)
+	asec(z) = acos(1 / z)
+	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))
+	tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z))
+
+	csch(z) = 1 / sinh(z)
+	sech(z) = 1 / cosh(z)
+	coth(z) = 1 / tanh(z)
+
+	asinh(z) = log(z + sqrt(z*z+1))
+	acosh(z) = log(z + sqrt(z*z-1))
+	atanh(z) = 1/2 * log((1+z) / (1-z))
+
+	acsch(z) = asinh(1 / z)
+	asech(z) = acosh(1 / z)
+	acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))
+
+I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>,
+I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>,
+I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>,
+I<acosech>, I<acotanh>, respectively.  C<Re>, C<Im>, C<arg>, C<abs>,
+C<rho>, and C<theta> can be used also also mutators.  The C<cbrt>
+returns only one of the solutions: if you want all three, use the
+C<root> function.
+
+The I<root> function is available to compute all the I<n>
+roots of some complex, where I<n> is a strictly positive integer.
+There are exactly I<n> such roots, returned as a list. Getting the
+number mathematicians call C<j> such that:
+
+	1 + j + j*j = 0;
+
+is a simple matter of writing:
+
+	$j = ((root(1, 3))[1];
+
+The I<k>th root for C<z = [r,t]> is given by:
+
+	(root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
+
+The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In
+order to ensure its restriction to real numbers is conform to what you
+would expect, the comparison is run on the real part of the complex
+number first, and imaginary parts are compared only when the real
+parts match.
+
+=head1 CREATION
+
+To create a complex number, use either:
+
+	$z = Math::Complex->make(3, 4);
+	$z = cplx(3, 4);
+
+if you know the cartesian form of the number, or
+
+	$z = 3 + 4*i;
+
+if you like. To create a number using the polar form, use either:
+
+	$z = Math::Complex->emake(5, pi/3);
+	$x = cplxe(5, pi/3);
+
+instead. The first argument is the modulus, the second is the angle
+(in radians, the full circle is 2*pi).  (Mnemonic: C<e> is used as a
+notation for complex numbers in the polar form).
+
+It is possible to write:
+
+	$x = cplxe(-3, pi/4);
+
+but that will be silently converted into C<[3,-3pi/4]>, since the modulus
+must be non-negative (it represents the distance to the origin in the complex
+plane).
+
+It is also possible to have a complex number as either argument of
+either the C<make> or C<emake>: the appropriate component of
+the argument will be used.
+
+	$z1 = cplx(-2,  1);
+	$z2 = cplx($z1, 4);
+
+=head1 STRINGIFICATION
+
+When printed, a complex number is usually shown under its cartesian
+form I<a+bi>, but there are legitimate cases where the polar format
+I<[r,t]> is more appropriate.
+
+By calling the routine C<Math::Complex::display_format> and supplying either
+C<"polar"> or C<"cartesian">, you override the default display format,
+which is C<"cartesian">. Not supplying any argument returns the current
+setting.
+
+This default can be overridden on a per-number basis by calling the
+C<display_format> method instead. As before, not supplying any argument
+returns the current display format for this number. Otherwise whatever you
+specify will be the new display format for I<this> particular number.
+
+For instance:
+
+	use Math::Complex;
+
+	Math::Complex::display_format('polar');
+	$j = ((root(1, 3))[1];
+	print "j = $j\n";		# Prints "j = [1,2pi/3]
+	$j->display_format('cartesian');
+	print "j = $j\n";		# Prints "j = -0.5+0.866025403784439i"
+
+The polar format attempts to emphasize arguments like I<k*pi/n>
+(where I<n> is a positive integer and I<k> an integer within [-9,+9]).
+
+=head1 USAGE
+
+Thanks to overloading, the handling of arithmetics with complex numbers
+is simple and almost transparent.
+
+Here are some examples:
+
+	use Math::Complex;
+
+	$j = cplxe(1, 2*pi/3);	# $j ** 3 == 1
+	print "j = $j, j**3 = ", $j ** 3, "\n";
+	print "1 + j + j**2 = ", 1 + $j + $j**2, "\n";
+
+	$z = -16 + 0*i;			# Force it to be a complex
+	print "sqrt($z) = ", sqrt($z), "\n";
+
+	$k = exp(i * 2*pi/3);
+	print "$j - $k = ", $j - $k, "\n";
+
+	$z->Re(3);			# Re, Im, arg, abs,
+	$j->arg(2);			# (the last two aka rho, theta)
+					# can be used also as mutators.
+
+=head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
+
+The division (/) and the following functions
+
+	log	ln	log10	logn
+	tan	sec 	csc	cot
+	atan	asec	acsc	acot
+	tanh	sech	csch	coth
+	atanh	asech	acsch	acoth
+
+cannot be computed for all arguments because that would mean dividing
+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 ...
+
+or
+
+	atanh(-1): Logarithm of zero.
+	Died at...
+
+For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>,
+C<asech>, C<acsch>, the argument cannot be C<0> (zero).  For the the
+logarithmic functions and the C<atanh>, C<acoth>, the argument cannot
+be C<1> (one).  For the C<atanh>, C<acoth>, the argument cannot be
+C<-1> (minus one).  For the C<atan>, C<acot>, the argument cannot be
+C<i> (the imaginary unit).  For the C<atan>, C<acoth>, the argument
+cannot be C<-i> (the negative imaginary unit).  For the C<tan>,
+C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k>
+is any integer.
+
+Note that because we are operating on approximations of real numbers,
+these errors can happen when merely `too close' to the singularities
+listed above.  For example C<tan(2*atan2(1,1)+1e-15)> will die of
+division by zero.
+
+=head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS
+
+The C<make> and C<emake> accept both real and complex arguments.
+When they cannot recognize the arguments they will die with error
+messages like the following
+
+    Math::Complex::make: Cannot take real part of ...
+    Math::Complex::make: Cannot take real part of ...
+    Math::Complex::emake: Cannot take rho of ...
+    Math::Complex::emake: Cannot take theta of ...
+
+=head1 BUGS
+
+Saying C<use Math::Complex;> exports many mathematical routines in the
+caller environment and even overrides some (C<sqrt>, C<log>).
+This is construed as a feature by the Authors, actually... ;-)
+
+All routines expect to be given real or complex numbers. Don't attempt to
+use BigFloat, since Perl has currently no rule to disambiguate a '+'
+operation (for instance) between two overloaded entities.
+
+In Cray UNICOS there is some strange numerical instability that results
+in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast.  Beware.
+The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex.
+Whatever it is, it does not manifest itself anywhere else where Perl runs.
+
+=head1 AUTHORS
+
+Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>> and
+Jarkko Hietaniemi <F<jhi@iki.fi>>.
+
+Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>.
+
+=cut
+
+1;
+
+# eof