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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
+#
+
+package Math::Complex;
+
+use strict;
+
+use vars qw($VERSION @ISA @EXPORT @EXPORT_OK %EXPORT_TAGS $Inf $ExpInf);
+
+$VERSION = 1.56;
+
+use Config;
+
+BEGIN {
+ my %DBL_MAX =
+ (
+ 4 => '1.70141183460469229e+38',
+ 8 => '1.7976931348623157e+308',
+ # AFAICT the 10, 12, and 16-byte long doubles
+ # all have the same maximum.
+ 10 => '1.1897314953572317650857593266280070162E+4932',
+ 12 => '1.1897314953572317650857593266280070162E+4932',
+ 16 => '1.1897314953572317650857593266280070162E+4932',
+ );
+ my $nvsize = $Config{nvsize} ||
+ ($Config{uselongdouble} && $Config{longdblsize}) ||
+ $Config{doublesize};
+ die "Math::Complex: Could not figure out nvsize\n"
+ unless defined $nvsize;
+ die "Math::Complex: Cannot not figure out max nv (nvsize = $nvsize)\n"
+ unless defined $DBL_MAX{$nvsize};
+ my $DBL_MAX = eval $DBL_MAX{$nvsize};
+ die "Math::Complex: Could not figure out max nv (nvsize = $nvsize)\n"
+ unless defined $DBL_MAX;
+ my $BIGGER_THAN_THIS = 1e30; # Must find something bigger than this.
+ if ($^O eq 'unicosmk') {
+ $Inf = $DBL_MAX;
+ } else {
+ local $SIG{FPE} = { };
+ local $!;
+ # We do want an arithmetic overflow, Inf INF inf Infinity.
+ for my $t (
+ 'exp(99999)', # Enough even with 128-bit long doubles.
+ 'inf',
+ 'Inf',
+ 'INF',
+ 'infinity',
+ 'Infinity',
+ 'INFINITY',
+ '1e99999',
+ ) {
+ local $^W = 0;
+ my $i = eval "$t+1.0";
+ if (defined $i && $i > $BIGGER_THAN_THIS) {
+ $Inf = $i;
+ last;
+ }
+ }
+ $Inf = $DBL_MAX unless defined $Inf; # Oh well, close enough.
+ die "Math::Complex: Could not get Infinity"
+ unless $Inf > $BIGGER_THAN_THIS;
+ $ExpInf = exp(99999);
+ }
+ # print "# On this machine, Inf = '$Inf'\n";
+}
+
+use Scalar::Util qw(set_prototype);
+
+use warnings;
+no warnings 'syntax'; # To avoid the (_) warnings.
+
+BEGIN {
+ # For certain functions that we override, in 5.10 or better
+ # we can set a smarter prototype that will handle the lexical $_
+ # (also a 5.10+ feature).
+ if ($] >= 5.010000) {
+ set_prototype \&abs, '_';
+ set_prototype \&cos, '_';
+ set_prototype \&exp, '_';
+ set_prototype \&log, '_';
+ set_prototype \&sin, '_';
+ set_prototype \&sqrt, '_';
+ }
+}
+
+my $i;
+my %LOGN;
+
+# Regular expression for floating point numbers.
+# These days we could use Scalar::Util::lln(), I guess.
+my $gre = qr'\s*([\+\-]?(?:(?:(?:\d+(?:_\d+)*(?:\.\d*(?:_\d+)*)?|\.\d+(?:_\d+)*)(?:[eE][\+\-]?\d+(?:_\d+)*)?))|inf)'i;
+
+require Exporter;
+
+@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
+ atan2
+ ),
+ @trig);
+
+my @pi = qw(pi pi2 pi4 pip2 pip4 Inf);
+
+@EXPORT_OK = @pi;
+
+%EXPORT_TAGS = (
+ 'trig' => [@trig],
+ 'pi' => [@pi],
+);
+
+use overload
+ '+' => \&_plus,
+ '-' => \&_minus,
+ '*' => \&_multiply,
+ '/' => \&_divide,
+ '**' => \&_power,
+ '==' => \&_numeq,
+ '<=>' => \&_spaceship,
+ 'neg' => \&_negate,
+ '~' => \&_conjugate,
+ 'abs' => \&abs,
+ 'sqrt' => \&sqrt,
+ 'exp' => \&exp,
+ 'log' => \&log,
+ 'sin' => \&sin,
+ 'cos' => \&cos,
+ 'tan' => \&tan,
+ 'atan2' => \&atan2,
+ '""' => \&_stringify;
+
+#
+# Package "privates"
+#
+
+my %DISPLAY_FORMAT = ('style' => 'cartesian',
+ 'polar_pretty_print' => 1);
+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";
+}
+
+sub _make {
+ my $arg = shift;
+ my ($p, $q);
+
+ if ($arg =~ /^$gre$/) {
+ ($p, $q) = ($1, 0);
+ } elsif ($arg =~ /^(?:$gre)?$gre\s*i\s*$/) {
+ ($p, $q) = ($1 || 0, $2);
+ } elsif ($arg =~ /^\s*\(\s*$gre\s*(?:,\s*$gre\s*)?\)\s*$/) {
+ ($p, $q) = ($1, $2 || 0);
+ }
+
+ if (defined $p) {
+ $p =~ s/^\+//;
+ $p =~ s/^(-?)inf$/"${1}9**9**9"/e;
+ $q =~ s/^\+//;
+ $q =~ s/^(-?)inf$/"${1}9**9**9"/e;
+ }
+
+ return ($p, $q);
+}
+
+sub _emake {
+ my $arg = shift;
+ my ($p, $q);
+
+ if ($arg =~ /^\s*\[\s*$gre\s*(?:,\s*$gre\s*)?\]\s*$/) {
+ ($p, $q) = ($1, $2 || 0);
+ } elsif ($arg =~ m!^\s*\[\s*$gre\s*(?:,\s*([-+]?\d*\s*)?pi(?:/\s*(\d+))?\s*)?\]\s*$!) {
+ ($p, $q) = ($1, ($2 eq '-' ? -1 : ($2 || 1)) * pi() / ($3 || 1));
+ } elsif ($arg =~ /^\s*\[\s*$gre\s*\]\s*$/) {
+ ($p, $q) = ($1, 0);
+ } elsif ($arg =~ /^\s*$gre\s*$/) {
+ ($p, $q) = ($1, 0);
+ }
+
+ if (defined $p) {
+ $p =~ s/^\+//;
+ $q =~ s/^\+//;
+ $p =~ s/^(-?)inf$/"${1}9**9**9"/e;
+ $q =~ s/^(-?)inf$/"${1}9**9**9"/e;
+ }
+
+ return ($p, $q);
+}
+
+#
+# ->make
+#
+# Create a new complex number (cartesian form)
+#
+sub make {
+ my $self = bless {}, shift;
+ my ($re, $im);
+ if (@_ == 0) {
+ ($re, $im) = (0, 0);
+ } elsif (@_ == 1) {
+ return (ref $self)->emake($_[0])
+ if ($_[0] =~ /^\s*\[/);
+ ($re, $im) = _make($_[0]);
+ } elsif (@_ == 2) {
+ ($re, $im) = @_;
+ }
+ if (defined $re) {
+ _cannot_make("real part", $re) unless $re =~ /^$gre$/;
+ }
+ $im ||= 0;
+ _cannot_make("imaginary part", $im) unless $im =~ /^$gre$/;
+ $self->_set_cartesian([$re, $im ]);
+ $self->display_format('cartesian');
+
+ return $self;
+}
+
+#
+# ->emake
+#
+# Create a new complex number (exponential form)
+#
+sub emake {
+ my $self = bless {}, shift;
+ my ($rho, $theta);
+ if (@_ == 0) {
+ ($rho, $theta) = (0, 0);
+ } elsif (@_ == 1) {
+ return (ref $self)->make($_[0])
+ if ($_[0] =~ /^\s*\(/ || $_[0] =~ /i\s*$/);
+ ($rho, $theta) = _emake($_[0]);
+ } elsif (@_ == 2) {
+ ($rho, $theta) = @_;
+ }
+ if (defined $rho && defined $theta) {
+ if ($rho < 0) {
+ $rho = -$rho;
+ $theta = ($theta <= 0) ? $theta + pi() : $theta - pi();
+ }
+ }
+ if (defined $rho) {
+ _cannot_make("rho", $rho) unless $rho =~ /^$gre$/;
+ }
+ $theta ||= 0;
+ _cannot_make("theta", $theta) unless $theta =~ /^$gre$/;
+ $self->_set_polar([$rho, $theta]);
+ $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 {
+ return __PACKAGE__->make(@_);
+}
+
+#
+# cplxe
+#
+# Creates a complex number from a (rho, theta) tuple.
+# This avoids the burden of writing Math::Complex->emake(rho, theta).
+#
+sub cplxe {
+ return __PACKAGE__->emake(@_);
+}
+
+#
+# pi
+#
+# The number defined as pi = 180 degrees
+#
+sub pi () { 4 * CORE::atan2(1, 1) }
+
+#
+# pi2
+#
+# The full circle
+#
+sub pi2 () { 2 * pi }
+
+#
+# pi4
+#
+# The full circle twice.
+#
+sub pi4 () { 4 * pi }
+
+#
+# pip2
+#
+# The quarter circle
+#
+sub pip2 () { pi / 2 }
+
+#
+# pip4
+#
+# The eighth circle.
+#
+sub pip4 () { pi / 4 }
+
+#
+# _uplog10
+#
+# Used in log10().
+#
+sub _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 $i;
+}
+
+#
+# _ip2
+#
+# Half of i.
+#
+sub _ip2 () { i / 2 }
+
+#
+# Attribute access/set routines
+#
+
+sub _cartesian {$_[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]->{c_dirty} = 0;
+ $_[0]->{'cartesian'} = $_[1] }
+sub _set_polar { $_[0]->{c_dirty}++; $_[0]->{p_dirty} = 0;
+ $_[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()) { $t -= pi2 }
+ elsif ($t <= -pi()) { $t += pi2 }
+ 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()) { $t -= pi2 }
+ elsif ($t <= -pi()) { $t += pi2 }
+ return (ref $z1)->emake($r2 / $r1, $t);
+ } else {
+ _divbyzero "$z1/0" if ($r2 == 0);
+ $t = $t1 - $t2;
+ if ($t > pi()) { $t -= pi2 }
+ elsif ($t <= -pi()) { $t += pi2 }
+ 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);
+ }
+ }
+ }
+}
+
+#
+# (_power)
+#
+# Computes z1**z2 = exp(z2 * log z1)).
+#
+sub _power {
+ my ($z1, $z2, $inverted) = @_;
+ if ($inverted) {
+ return 1 if $z1 == 0 || $z2 == 1;
+ return 0 if $z2 == 0 && Re($z1) > 0;
+ } else {
+ return 1 if $z2 == 0 || $z1 == 1;
+ return 0 if $z1 == 0 && Re($z2) > 0;
+ }
+ my $w = $inverted ? &exp($z1 * &log($z2))
+ : &exp($z2 * &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 <=> $im2);
+}
+
+#
+# (_numeq)
+#
+# Computes z1 == z2.
+#
+# (Required in addition to _spaceship() because of NaNs.)
+sub _numeq {
+ my ($z1, $z2, $inverted) = @_;
+ my ($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
+ my ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
+ return $re1 == $re2 && $im1 == $im2 ? 1 : 0;
+}
+
+#
+# (_negate)
+#
+# Computes -z.
+#
+sub _negate {
+ my ($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) = @_ ? @_ : $_;
+ unless (ref $z) {
+ if (@_ == 2) {
+ $_[0] = $_[1];
+ } else {
+ return CORE::abs($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 -= pi2 }
+ elsif ($$theta <= -pi()) { $$theta += pi2 }
+}
+
+#
+# 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) = @_ ? $_[0] : $_;
+ my ($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 0 if $r == 0;
+ return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3);
+}
+
+#
+# _rootbad
+#
+# Die on bad root.
+#
+sub _rootbad {
+ my $mess = "Root '$_[0]' illegal, root rank 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, $k) = @_;
+ _rootbad($n) if ($n < 1 or int($n) != $n);
+ my ($r, $t) = ref $z ?
+ @{$z->_polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
+ my $theta_inc = pi2 / $n;
+ my $rho = $r ** (1/$n);
+ my $cartesian = ref $z && $z->{c_dirty} == 0;
+ if (@_ == 2) {
+ my @root;
+ for (my $i = 0, my $theta = $t / $n;
+ $i < $n;
+ $i++, $theta += $theta_inc) {
+ my $w = cplxe($rho, $theta);
+ # Yes, $cartesian is loop invariant.
+ push @root, $cartesian ? cplx(@{$w->_cartesian}) : $w;
+ }
+ return @root;
+ } elsif (@_ == 3) {
+ my $w = cplxe($rho, $t / $n + $k * $theta_inc);
+ return $cartesian ? cplx(@{$w->_cartesian}) : $w;
+ }
+}
+
+#
+# Re
+#
+# Return or set Re(z).
+#
+sub Re {
+ my ($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 0 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) = @_ ? @_ : $_;
+ return CORE::exp($z) unless ref $z;
+ my ($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()) { $t -= pi2 }
+ elsif ($t <= -pi()) { $t += pi2 }
+ 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 $logn; # Cache log(n)
+ return &log($z) / $logn;
+}
+
+#
+# (cos)
+#
+# Compute cos(z) = (exp(iz) + exp(-iz))/2.
+#
+sub cos {
+ my ($z) = @_ ? @_ : $_;
+ return CORE::cos($z) unless ref $z;
+ my ($x, $y) = @{$z->_cartesian};
+ my $ey = CORE::exp($y);
+ my $sx = CORE::sin($x);
+ my $cx = CORE::cos($x);
+ my $ey_1 = $ey ? 1 / $ey : Inf();
+ return (ref $z)->make($cx * ($ey + $ey_1)/2,
+ $sx * ($ey_1 - $ey)/2);
+}
+
+#
+# (sin)
+#
+# Compute sin(z) = (exp(iz) - exp(-iz))/2.
+#
+sub sin {
+ my ($z) = @_ ? @_ : $_;
+ return CORE::sin($z) unless ref $z;
+ my ($x, $y) = @{$z->_cartesian};
+ my $ey = CORE::exp($y);
+ my $sx = CORE::sin($x);
+ my $cx = CORE::cos($x);
+ my $ey_1 = $ey ? 1 / $ey : Inf();
+ return (ref $z)->make($sx * ($ey + $ey_1)/2,
+ $cx * ($ey - $ey_1)/2);
+}
+
+#
+# tan
+#
+# Compute tan(z) = sin(z) / cos(z).
+#
+sub tan {
+ my ($z) = @_;
+ my $cz = &cos($z);
+ _divbyzero "tan($z)", "cos($z)" if $cz == 0;
+ return &sin($z) / $cz;
+}
+
+#
+# sec
+#
+# Computes the secant sec(z) = 1 / cos(z).
+#
+sub sec {
+ my ($z) = @_;
+ my $cz = &cos($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 $sz = &sin($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 $sz = &sin($z);
+ _divbyzero "cot($z)", "sin($z)" if ($sz == 0);
+ return &cos($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;
+ $z = cplx($z, 0) unless ref $z;
+ my ($x, $y) = @{$z->_cartesian};
+ return 0 if $x == 1 && $y == 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 (ref $z)->make($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;
+ $z = cplx($z, 0) unless ref $z;
+ my ($x, $y) = @{$z->_cartesian};
+ return 0 if $x == 0 && $y == 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 (ref $z)->make($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 $z;
+ my ($x, $y) = ref $z ? @{$z->_cartesian} : ($z, 0);
+ return 0 if $x == 0 && $y == 0;
+ _divbyzero "atan(i)" if ( $z == i);
+ _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test...
+ my $log = &log((i + $z) / (i - $z));
+ 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 $z == 0;
+ return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z)
+ unless ref $z;
+ _divbyzero "acot(i)" if ($z - i == 0);
+ _logofzero "acot(-i)" if ($z + i == 0);
+ 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 $ex ? ($ex == $ExpInf ? Inf() : ($ex + 1/$ex)/2) : Inf();
+ }
+ my ($x, $y) = @{$z->_cartesian};
+ $ex = CORE::exp($x);
+ my $ex_1 = $ex ? 1 / $ex : Inf();
+ 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) {
+ return 0 if $z == 0;
+ $ex = CORE::exp($z);
+ return $ex ? ($ex == $ExpInf ? Inf() : ($ex - 1/$ex)/2) : -Inf();
+ }
+ my ($x, $y) = @{$z->_cartesian};
+ my $cy = CORE::cos($y);
+ my $sy = CORE::sin($y);
+ $ex = CORE::exp($x);
+ my $ex_1 = $ex ? 1 / $ex : Inf();
+ 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);
+ my $sz = sinh($z);
+ return 1 if $cz == $sz;
+ return -1 if $cz == -$sz;
+ return $sz / $cz;
+}
+
+#
+# sech
+#
+# Computes the hyperbolic secant sech(z) = 1 / cosh(z).
+#
+sub sech {
+ my ($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;
+ my $cz = cosh($z);
+ return 1 if $cz == $sz;
+ return -1 if $cz == -$sz;
+ return $cz / $sz;
+}
+
+#
+# cotanh
+#
+# Alias for coth().
+#
+sub cotanh { Math::Complex::coth(@_) }
+
+#
+# acosh
+#
+# Computes the area/inverse hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).
+#
+sub acosh {
+ my ($z) = @_;
+ unless (ref $z) {
+ $z = cplx($z, 0);
+ }
+ my ($re, $im) = @{$z->_cartesian};
+ if ($im == 0) {
+ return CORE::log($re + CORE::sqrt($re*$re - 1))
+ if $re >= 1;
+ return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re))
+ if CORE::abs($re) < 1;
+ }
+ my $t = &sqrt($z * $z - 1) + $z;
+ # Try Taylor if looking bad (this usually means that
+ # $z was large negative, therefore the sqrt is really
+ # close to abs(z), summing that with z...)
+ $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
+ if $t == 0;
+ my $u = &log($t);
+ $u->Im(-$u->Im) if $re < 0 && $im == 0;
+ return $re < 0 ? -$u : $u;
+}
+
+#
+# asinh
+#
+# Computes the area/inverse hyperbolic sine asinh(z) = log(z + sqrt(z*z+1))
+#
+sub asinh {
+ my ($z) = @_;
+ unless (ref $z) {
+ my $t = $z + CORE::sqrt($z*$z + 1);
+ return CORE::log($t) if $t;
+ }
+ my $t = &sqrt($z * $z + 1) + $z;
+ # Try Taylor if looking bad (this usually means that
+ # $z was large negative, therefore the sqrt is really
+ # close to abs(z), summing that with z...)
+ $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
+ if $t == 0;
+ return &log($t);
+}
+
+#
+# atanh
+#
+# Computes the area/inverse 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 (1 - $z == 0);
+ _logofzero 'atanh(-1)' if (1 + $z == 0);
+ return 0.5 * &log((1 + $z) / (1 - $z));
+}
+
+#
+# asech
+#
+# Computes the area/inverse hyperbolic secant asech(z) = acosh(1 / z).
+#
+sub asech {
+ my ($z) = @_;
+ _divbyzero 'asech(0)', "$z" if ($z == 0);
+ return acosh(1 / $z);
+}
+
+#
+# acsch
+#
+# Computes the area/inverse hyperbolic cosecant acsch(z) = asinh(1 / z).
+#
+sub acsch {
+ my ($z) = @_;
+ _divbyzero 'acsch(0)', $z if ($z == 0);
+ return asinh(1 / $z);
+}
+
+#
+# acosech
+#
+# Alias for acosh().
+#
+sub acosech { Math::Complex::acsch(@_) }
+
+#
+# acoth
+#
+# Computes the area/inverse hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)).
+#
+sub acoth {
+ my ($z) = @_;
+ _divbyzero 'acoth(0)' if ($z == 0);
+ unless (ref $z) {
+ return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1;
+ $z = cplx($z, 0);
+ }
+ _divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0);
+ _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0);
+ return &log((1 + $z) / ($z - 1)) / 2;
+}
+
+#
+# acotanh
+#
+# Alias for acot().
+#
+sub acotanh { Math::Complex::acoth(@_) }
+
+#
+# (atan2)
+#
+# Compute atan(z1/z2), minding the right quadrant.
+#
+sub atan2 {
+ my ($z1, $z2, $inverted) = @_;
+ my ($re1, $im1, $re2, $im2);
+ if ($inverted) {
+ ($re1, $im1) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
+ ($re2, $im2) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
+ } else {
+ ($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
+ ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
+ }
+ if ($im1 || $im2) {
+ # In MATLAB the imaginary parts are ignored.
+ # warn "atan2: Imaginary parts ignored";
+ # http://documents.wolfram.com/mathematica/functions/ArcTan
+ # NOTE: Mathematica ArcTan[x,y] while atan2(y,x)
+ my $s = $z1 * $z1 + $z2 * $z2;
+ _divbyzero("atan2") if $s == 0;
+ my $i = &i;
+ my $r = $z2 + $z1 * $i;
+ return -$i * &log($r / &sqrt( $s ));
+ }
+ return CORE::atan2($re1, $re2);
+}
+
+#
+# display_format
+# ->display_format
+#
+# Set (get if no argument) the display format for all complex numbers that
+# don't happen to have overridden it via ->display_format
+#
+# When called as an object 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 %display_format = %DISPLAY_FORMAT;
+
+ if (ref $self) { # Called as an object method
+ if (exists $self->{display_format}) {
+ my %obj = %{$self->{display_format}};
+ @display_format{keys %obj} = values %obj;
+ }
+ }
+ if (@_ == 1) {
+ $display_format{style} = shift;
+ } else {
+ my %new = @_;
+ @display_format{keys %new} = values %new;
+ }
+
+ if (ref $self) { # Called as an object method
+ $self->{display_format} = { %display_format };
+ return
+ wantarray ?
+ %{$self->{display_format}} :
+ $self->{display_format}->{style};
+ }
+
+ # Called as a class method
+ %DISPLAY_FORMAT = %display_format;
+ return
+ wantarray ?
+ %DISPLAY_FORMAT :
+ $DISPLAY_FORMAT{style};
+}
+
+#
+# (_stringify)
+#
+# Show nicely formatted complex number under its cartesian or polar form,
+# depending on the current display format:
+#
+# . If a specific display format has been recorded for this object, use it.
+# . Otherwise, use the generic current default for all complex numbers,
+# which is a package global variable.
+#
+sub _stringify {
+ my ($z) = shift;
+
+ my $style = $z->display_format;
+
+ $style = $DISPLAY_FORMAT{style} unless defined $style;
+
+ return $z->_stringify_polar if $style =~ /^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);
+
+ my %format = $z->display_format;
+ my $format = $format{format};
+
+ if ($x) {
+ if ($x =~ /^NaN[QS]?$/i) {
+ $re = $x;
+ } else {
+ if ($x =~ /^-?\Q$Inf\E$/oi) {
+ $re = $x;
+ } else {
+ $re = defined $format ? sprintf($format, $x) : $x;
+ }
+ }
+ } else {
+ undef $re;
+ }
+
+ if ($y) {
+ if ($y =~ /^(NaN[QS]?)$/i) {
+ $im = $y;
+ } else {
+ if ($y =~ /^-?\Q$Inf\E$/oi) {
+ $im = $y;
+ } else {
+ $im =
+ defined $format ?
+ sprintf($format, $y) :
+ ($y == 1 ? "" : ($y == -1 ? "-" : $y));
+ }
+ }
+ $im .= "i";
+ } else {
+ undef $im;
+ }
+
+ my $str = $re;
+
+ if (defined $im) {
+ if ($y < 0) {
+ $str .= $im;
+ } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) {
+ $str .= "+" if defined $re;
+ $str .= $im;
+ }
+ } elsif (!defined $re) {
+ $str = "0";
+ }
+
+ return $str;
+}
+
+
+#
+# ->_stringify_polar
+#
+# Stringify as a polar representation '[r,t]'.
+#
+sub _stringify_polar {
+ my $z = shift;
+ my ($r, $t) = @{$z->_polar};
+ my $theta;
+
+ my %format = $z->display_format;
+ my $format = $format{format};
+
+ if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?\Q$Inf\E$/oi) {
+ $theta = $t;
+ } elsif ($t == pi) {
+ $theta = "pi";
+ } elsif ($r == 0 || $t == 0) {
+ $theta = defined $format ? sprintf($format, $t) : $t;
+ }
+
+ return "[$r,$theta]" if defined $theta;
+
+ #
+ # Try to identify pi/n and friends.
+ #
+
+ $t -= int(CORE::abs($t) / pi2) * pi2;
+
+ if ($format{polar_pretty_print} && $t) {
+ my ($a, $b);
+ for $a (2..9) {
+ $b = $t * $a / pi;
+ if ($b =~ /^-?\d+$/) {
+ $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1;
+ $theta = "${b}pi/$a";
+ last;
+ }
+ }
+ }
+
+ if (defined $format) {
+ $r = sprintf($format, $r);
+ $theta = sprintf($format, $theta) unless defined $theta;
+ } else {
+ $theta = $t unless defined $theta;
+ }
+
+ return "[$r,$theta]";
+}
+
+sub Inf {
+ return $Inf;
+}
+
+1;
+__END__
+
+=pod
+
+=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> is
+marked here as 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<y> or 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 a 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(y, x) = atan(y / x) # Minding the right quadrant, note the order.
+
+The definition used for complex arguments of atan2() is
+
+ -i log((x + iy)/sqrt(x*x+y*y))
+
+Note that atan2(0, 0) is not well-defined.
+
+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 as 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)
+
+You can return the I<k>th root directly by C<root(z, n, k)>,
+indexing starting from I<zero> and ending at I<n - 1>.
+
+The I<spaceship> numeric 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 the
+C<make>, C<emake>, C<cplx>, and C<cplxe>: the appropriate component of
+the argument will be used.
+
+ $z1 = cplx(-2, 1);
+ $z2 = cplx($z1, 4);
+
+The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also
+understand a single (string) argument of the forms
+
+ 2-3i
+ -3i
+ [2,3]
+ [2,-3pi/4]
+ [2]
+
+in which case the appropriate cartesian and exponential components
+will be parsed from the string and used to create new complex numbers.
+The imaginary component and the theta, respectively, will default to zero.
+
+The C<new>, C<make>, C<emake>, C<cplx>, and C<cplxe> will also
+understand the case of no arguments: this means plain zero or (0, 0).
+
+=head1 DISPLAYING
+
+When printed, a complex number is usually shown under its cartesian
+style I<a+bi>, but there are legitimate cases where the polar style
+I<[r,t]> is more appropriate. The process of converting the complex
+number into a string that can be displayed is known as I<stringification>.
+
+By calling the class method C<Math::Complex::display_format> and
+supplying either C<"polar"> or C<"cartesian"> as an argument, you
+override the default display style, which is C<"cartesian">. Not
+supplying any argument returns the current settings.
+
+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 style for this number. Otherwise whatever you
+specify will be the new display style 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 style attempts to emphasize arguments like I<k*pi/n>
+(where I<n> is a positive integer and I<k> an integer within [-9, +9]),
+this is called I<polar pretty-printing>.
+
+For the reverse of stringifying, see the C<make> and C<emake>.
+
+=head2 CHANGED IN PERL 5.6
+
+The C<display_format> class method and the corresponding
+C<display_format> object method can now be called using
+a parameter hash instead of just a one parameter.
+
+The old display format style, which can have values C<"cartesian"> or
+C<"polar">, can be changed using the C<"style"> parameter.
+
+ $j->display_format(style => "polar");
+
+The one parameter calling convention also still works.
+
+ $j->display_format("polar");
+
+There are two new display parameters.
+
+The first one is C<"format">, which is a sprintf()-style format string
+to be used for both numeric parts of the complex number(s). The is
+somewhat system-dependent but most often it corresponds to C<"%.15g">.
+You can revert to the default by setting the C<format> to C<undef>.
+
+ # the $j from the above example
+
+ $j->display_format('format' => '%.5f');
+ print "j = $j\n"; # Prints "j = -0.50000+0.86603i"
+ $j->display_format('format' => undef);
+ print "j = $j\n"; # Prints "j = -0.5+0.86603i"
+
+Notice that this affects also the return values of the
+C<display_format> methods: in list context the whole parameter hash
+will be returned, as opposed to only the style parameter value.
+This is a potential incompatibility with earlier versions if you
+have been calling the C<display_format> method in list context.
+
+The second new display parameter is C<"polar_pretty_print">, which can
+be set to true or false, the default being true. See the previous
+section for what this means.
+
+=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 CONSTANTS
+
+=head2 PI
+
+The constant C<pi> and some handy multiples of it (pi2, pi4,
+and pip2 (pi/2) and pip4 (pi/4)) are also available if separately
+exported:
+
+ use Math::Complex ':pi';
+ $third_of_circle = pi2 / 3;
+
+=head2 Inf
+
+The floating point infinity can be exported as a subroutine Inf():
+
+ use Math::Complex qw(Inf sinh);
+ my $AlsoInf = Inf() + 42;
+ my $AnotherInf = sinh(1e42);
+ print "$AlsoInf is $AnotherInf\n" if $AlsoInf == $AnotherInf;
+
+Note that the stringified form of infinity varies between platforms:
+it can be for example any of
+
+ inf
+ infinity
+ INF
+ 1.#INF
+
+or it can be something else.
+
+Also note that in some platforms trying to use the infinity in
+arithmetic operations may result in Perl crashing because using
+an infinity causes SIGFPE or its moral equivalent to be sent.
+The way to ignore this is
+
+ local $SIG{FPE} = sub { };
+
+=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
+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. atan2(0, 0) is undefined, and if the complex arguments
+are used for atan2(), a division by zero will happen if z1**2+z2**2 == 0.
+
+Note that because we are operating on approximations of real numbers,
+these errors can happen when merely `too close' to the singularities
+listed above.
+
+=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>, C<atan2>).
+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 SEE ALSO
+
+L<Math::Trig>
+
+=head1 AUTHORS
+
+Daniel S. Lewart <F<lewart!at!uiuc.edu>>
+Jarkko Hietaniemi <F<jhi!at!iki.fi>>
+Raphael Manfredi <F<Raphael_Manfredi!at!pobox.com>>
+
+=head1 LICENSE
+
+This library is free software; you can redistribute it and/or modify
+it under the same terms as Perl itself.
+
+=cut
+
+1;
+
+# eof
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