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-rw-r--r--contrib/perl5/lib/Math/Complex.pm1889
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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
-#
-
-package Math::Complex;
-
-our($VERSION, @ISA, @EXPORT, %EXPORT_TAGS, $Inf);
-
-$VERSION = 1.31;
-
-BEGIN {
- unless ($^O eq 'unicosmk') {
- my $e = $!;
- # We do want an arithmetic overflow, Inf INF inf Infinity:.
- undef $Inf unless eval <<'EOE' and $Inf =~ /^inf(?:inity)?$/i;
- local $SIG{FPE} = sub {die};
- my $t = CORE::exp 30;
- $Inf = CORE::exp $t;
-EOE
- if (!defined $Inf) { # Try a different method
- undef $Inf unless eval <<'EOE' and $Inf =~ /^inf(?:inity)?$/i;
- local $SIG{FPE} = sub {die};
- my $t = 1;
- $Inf = $t + "1e99999999999999999999999999999999";
-EOE
- }
- $! = $e; # Clear ERANGE.
- }
- $Inf = "Inf" if !defined $Inf || !($Inf > 0); # Desperation.
-}
-
-use strict;
-
-my $i;
-my %LOGN;
-
-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
- ),
- @trig);
-
-%EXPORT_TAGS = (
- 'trig' => [@trig],
-);
-
-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,
- qw("" 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";
-}
-
-#
-# ->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
-#
-sub pi () { 4 * CORE::atan2(1, 1) }
-
-#
-# pit2
-#
-# The full circle
-#
-sub pit2 () { 2 * pi }
-
-#
-# pip2
-#
-# The quarter circle
-#
-sub pip2 () { pi / 2 }
-
-#
-# deg1
-#
-# One degree in radians, used in stringify_polar.
-#
-
-sub deg1 () { pi / 180 }
-
-#
-# 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]->{'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);
- }
- }
- }
-}
-
-#
-# (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 -= 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 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, $n) = @_;
- _rootbad($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 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 ($z) = @_;
- my ($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 $logn; # Cache log(n)
- return &log($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 + 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 - 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 ($cz == 0);
- return sinh($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) {
- $z = cplx($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 : $u;
-}
-
-#
-# asinh
-#
-# Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1))
-#
-sub asinh {
- my ($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 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 (1 - $z == 0);
- _logofzero 'atanh(-1)' if (1 + $z == 0);
- return 0.5 * &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 ($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).
-#
-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 CORE::atan2($re1, $re2) if $im1 == 0;
- return ($im1<=>0) * pip2 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 (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 =~ /^-?$Inf$/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 =~ /^-?$Inf$/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 || $t =~ /^-?$Inf$/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) / pit2) * pit2;
-
- 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]";
-}
-
-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> 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
-style I<a+bi>, but there are legitimate cases where the polar style
-I<[r,t]> is more appropriate.
-
-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>.
-
-=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 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.
-
-=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@pobox.com>> and
-Jarkko Hietaniemi <F<jhi@iki.fi>>.
-
-Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>.
-
-=cut
-
-1;
-
-# eof
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