-# $RCSFile$
#
# Complex numbers and associated mathematical functions
-# -- Raphael Manfredi, September 1996
-# -- Jarkko Hietaniemi, March-April 1997
+# -- Raphael Manfredi September 1996
+# -- Jarkko Hietaniemi March-October 1997
+# -- Daniel S. Lewart September-October 1997
+#
require Exporter;
package Math::Complex;
+$VERSION = 1.05;
+
+# $Id: Complex.pm,v 1.2 1997/10/15 10:08:39 jhi Exp $
+
use strict;
use vars qw($VERSION @ISA
@EXPORT %EXPORT_TAGS
$package $display
- $i $logn %logn);
+ $i $ip2 $logn %logn);
@ISA = qw(Exporter);
-$VERSION = 1.01;
-
my @trig = qw(
pi
- sin cos tan
+ tan
csc cosec sec cot cotan
asin acos atan
acsc acosec asec acot acotan
@EXPORT = (qw(
i Re Im arg
- sqrt exp log ln
+ sqrt log ln
log10 logn cbrt root
cplx cplxe
),
sub emake {
my $self = bless {}, shift;
my ($rho, $theta) = @_;
- $theta += pi() if $rho < 0;
- $self->{'polar'} = [abs($rho), $theta];
+ if ($rho < 0) {
+ $rho = -$rho;
+ $theta = ($theta <= 0) ? $theta + pi() : $theta - pi();
+ }
+ $self->{'polar'} = [$rho, $theta];
$self->{p_dirty} = 0;
$self->{c_dirty} = 1;
return $self;
#
# pi
#
-# The number defined as 2 * pi = 360 degrees
+# The number defined as pi = 180 degrees
#
-
use constant pi => 4 * atan2(1, 1);
#
-# log2inv
+# pit2
#
-# Used in log10().
+# The full circle
+#
+use constant pit2 => 2 * pi;
+
#
+# pip2
+#
+# The quarter circle
+#
+use constant pip2 => pi / 2;
-use constant log10inv => 1 / log(10);
+#
+# uplog10
+#
+# Used in log10().
+#
+use constant uplog10 => 1 / log(10);
#
# i
return $i if ($i);
$i = bless {};
$i->{'cartesian'} = [0, 1];
- $i->{'polar'} = [1, pi/2];
+ $i->{'polar'} = [1, pip2];
$i->{c_dirty} = 0;
$i->{p_dirty} = 0;
return $i;
# Computes z1*z2.
#
sub multiply {
- my ($z1, $z2, $regular) = @_;
- my ($r1, $t1) = @{$z1->polar};
- $z2 = cplxe(abs($z2), $z2 >= 0 ? 0 : pi) unless ref $z2;
- my ($r2, $t2) = @{$z2->polar};
- unless (defined $regular) {
- $z1->set_polar([$r1 * $r2, $t1 + $t2]);
+ my ($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);
+ }
}
- return (ref $z1)->emake($r1 * $r2, $t1 + $t2);
}
#
}
my @up = caller(1);
-
+
$mess .= "Died at $up[1] line $up[2].\n";
die $mess;
#
sub divide {
my ($z1, $z2, $inverted) = @_;
- my ($r1, $t1) = @{$z1->polar};
- $z2 = cplxe(abs($z2), $z2 >= 0 ? 0 : pi) unless ref $z2;
- my ($r2, $t2) = @{$z2->polar};
- unless (defined $inverted) {
- _divbyzero "$z1/0" if ($r2 == 0);
- $z1->set_polar([$r1 / $r2, $t1 - $t2]);
- return $z1;
- }
- if ($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);
- return (ref $z1)->emake($r2 / $r1, $t2 - $t1);
- } else {
+ $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);
- return (ref $z1)->emake($r1 / $r2, $t1 - $t2);
+ $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);
+ }
+ }
}
}
my $mess = "The zero raised to the zeroth power is not defined.\n";
my @up = caller(1);
-
+
$mess .= "Died at $up[1] line $up[2].\n";
die $mess;
return 0 if ($z1z);
return 1 if ($z2z or $z1 == 1);
}
- $z2 = cplx($z2) unless ref $z2;
- unless (defined $inverted) {
- my $z3 = exp($z2 * log $z1);
- $z1->set_cartesian([@{$z3->cartesian}]);
- return $z1;
- }
- return exp($z2 * log $z1) unless $inverted;
- return exp($z1 * log $z2);
+ return $inverted ? exp($z1 * log $z2) : exp($z2 * log $z1);
}
#
my ($z) = @_;
if ($z->{c_dirty}) {
my ($r, $t) = @{$z->polar};
- return (ref $z)->emake($r, pi + $t);
+ $t = ($t <= 0) ? $t + pi : $t - pi;
+ return (ref $z)->emake($r, $t);
}
my ($re, $im) = @{$z->cartesian};
return (ref $z)->make(-$re, -$im);
#
sub abs {
my ($z) = @_;
- return abs($z) unless ref $z;
my ($r, $t) = @{$z->polar};
- return abs($r);
+ return $r;
}
#
my ($z) = @_;
return ($z < 0 ? pi : 0) unless ref $z;
my ($r, $t) = @{$z->polar};
+ if ($t > pi()) { $t -= pit2 }
+ elsif ($t <= -pi()) { $t += pit2 }
return $t;
}
#
sub sqrt {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
+ return $z >= 0 ? sqrt($z) : cplx(0, sqrt(-$z)) unless ref $z;
+ my ($re, $im) = @{$z->cartesian};
+ return cplx($re < 0 ? (0, sqrt(-$re)) : (sqrt($re), 0)) if $im == 0;
my ($r, $t) = @{$z->polar};
return (ref $z)->emake(sqrt($r), $t/2);
}
#
sub cbrt {
my ($z) = @_;
- return cplx($z, 0) ** (1/3) unless ref $z;
+ return $z < 0 ? -exp(log(-$z)/3) : ($z > 0 ? exp(log($z)/3): 0)
+ unless ref $z;
my ($r, $t) = @{$z->polar};
- return (ref $z)->emake($r**(1/3), $t/3);
+ return (ref $z)->emake(exp(log($r)/3), $t/3);
}
#
my $mess = "Root $_[0] not defined, root must be positive integer.\n";
my @up = caller(1);
-
+
$mess .= "Died at $up[1] line $up[2].\n";
die $mess;
my ($r, $t) = ref $z ? @{$z->polar} : (abs($z), $z >= 0 ? 0 : pi);
my @root;
my $k;
- my $theta_inc = 2 * pi / $n;
+ my $theta_inc = pit2 / $n;
my $rho = $r ** (1/$n);
my $theta;
my $complex = ref($z) || $package;
#
sub exp {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
my ($x, $y) = @{$z->cartesian};
return (ref $z)->emake(exp($x), $y);
}
#
# _logofzero
#
-# Die on division by zero.
+# Die on logarithm of zero.
#
sub _logofzero {
my $mess = "$_[0]: Logarithm of zero.\n";
}
my @up = caller(1);
-
+
$mess .= "Died at $up[1] line $up[2].\n";
die $mess;
#
sub log {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
- my ($x, $y) = @{$z->cartesian};
+ unless (ref $z) {
+ _logofzero("log") if $z == 0;
+ return $z > 0 ? log($z) : cplx(log(-$z), pi);
+ }
my ($r, $t) = @{$z->polar};
- $t -= 2 * pi if ($t > pi() and $x < 0);
- $t += 2 * pi if ($t < -pi() and $x < 0);
+ _logofzero("log") if $r == 0;
+ if ($t > pi()) { $t -= pit2 }
+ elsif ($t <= -pi()) { $t += pit2 }
return (ref $z)->make(log($r), $t);
}
#
sub log10 {
- my ($z) = @_;
-
- return log(cplx($z, 0)) * log10inv unless ref $z;
- my ($r, $t) = @{$z->polar};
- return (ref $z)->make(log($r) * log10inv, $t * log10inv);
+ return Math::Complex::log($_[0]) * uplog10;
}
#
#
sub cos {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
my ($x, $y) = @{$z->cartesian};
my $ey = exp($y);
my $ey_1 = 1 / $ey;
#
sub sin {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
my ($x, $y) = @{$z->cartesian};
my $ey = exp($y);
my $ey_1 = 1 / $ey;
#
# cot
#
-# Computes cot(z) = 1 / tan(z).
+# Computes cot(z) = cos(z) / sin(z).
#
sub cot {
my ($z) = @_;
# Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
#
sub acos {
- my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- return atan2(sqrt(1 - $re * $re), $re)
- if ($im == 0 and abs($re) <= 1.0);
- my $acos = ~i * log($z + sqrt($z*$z - 1));
- if ($im == 0 ||
- (abs($re) < 1 && abs($im) < 1) ||
- (abs($re) > 1 && abs($im) > 1
- && !($re > 1 && $im > 1)
- && !($re < -1 && $im < -1))) {
- # this rule really, REALLY, must be simpler
- return -$acos;
- }
- return $acos;
+ my $z = $_[0];
+ return atan2(sqrt(1-$z*$z), $z) if (! ref $z) && abs($z) <= 1;
+ my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
+ my $t1 = sqrt(($x+1)*($x+1) + $y*$y);
+ my $t2 = sqrt(($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 $u = atan2(sqrt(1-$beta*$beta), $beta);
+ my $v = log($alpha + sqrt($alpha*$alpha-1));
+ $v = -$v if $y > 0 || ($y == 0 && $x < -1);
+ return $package->make($u, $v);
}
#
# Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)).
#
sub asin {
- my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- return atan2($re, sqrt(1 - $re * $re))
- if ($im == 0 and abs($re) <= 1.0);
- return ~i * log(i * $z + sqrt(1 - $z*$z));
+ my $z = $_[0];
+ return atan2($z, sqrt(1-$z*$z)) if (! ref $z) && abs($z) <= 1;
+ my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
+ my $t1 = sqrt(($x+1)*($x+1) + $y*$y);
+ my $t2 = sqrt(($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 $u = atan2($beta, sqrt(1-$beta*$beta));
+ my $v = -log($alpha + sqrt($alpha*$alpha-1));
+ $v = -$v if $y > 0 || ($y == 0 && $x < -1);
+ return $package->make($u, $v);
}
#
#
sub atan {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
+ return atan2($z, 1) unless ref $z;
_divbyzero "atan(i)" if ( $z == i);
_divbyzero "atan(-i)" if (-$z == i);
- return i/2*log((i + $z) / (i - $z));
+ my $log = log((i + $z) / (i - $z));
+ $ip2 = 0.5 * i unless defined $ip2;
+ return $ip2 * $log;
}
#
sub asec {
my ($z) = @_;
_divbyzero "asec($z)", $z if ($z == 0);
- $z = cplx($z, 0) unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- if ($im == 0 && abs($re) >= 1.0) {
- my $ire = 1 / $re;
- return atan2(sqrt(1 - $ire * $ire), $ire);
- }
- my $asec = acos(1 / $z);
- return ~$asec if $re < 0 && $re > -1 && $im == 0;
- return -$asec if $im && !($re > 0 && $im > 0) && !($re < 0 && $im < 0);
- return $asec;
+ return acos(1 / $z);
}
#
sub acsc {
my ($z) = @_;
_divbyzero "acsc($z)", $z if ($z == 0);
- $z = cplx($z, 0) unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- if ($im == 0 && abs($re) >= 1.0) {
- my $ire = 1 / $re;
- return atan2($ire, sqrt(1 - $ire * $ire));
- }
- my $acsc = asin(1 / $z);
- return ~$acsc if $re < 0 && $re > -1 && $im == 0;
- return $acsc;
+ return asin(1 / $z);
}
#
#
sub acot {
my ($z) = @_;
- _divbyzero "acot($z)" if ($z == 0);
- $z = cplx($z, 0) unless ref $z;
+ return ($z >= 0) ? atan2(1, $z) : atan2(-1, -$z) unless ref $z;
_divbyzero "acot(i)", if ( $z == i);
_divbyzero "acot(-i)" if (-$z == i);
return atan(1 / $z);
#
sub cosh {
my ($z) = @_;
- my $real;
+ my $ex;
unless (ref $z) {
- $z = cplx($z, 0);
- $real = 1;
+ $ex = exp($z);
+ return ($ex + 1/$ex)/2;
}
my ($x, $y) = @{$z->cartesian};
- my $ex = exp($x);
+ $ex = exp($x);
my $ex_1 = 1 / $ex;
- return cplx(0.5 * ($ex + $ex_1), 0) if $real;
return (ref $z)->make(cos($y) * ($ex + $ex_1)/2,
sin($y) * ($ex - $ex_1)/2);
}
#
sub sinh {
my ($z) = @_;
- my $real;
+ my $ex;
unless (ref $z) {
- $z = cplx($z, 0);
- $real = 1;
+ $ex = exp($z);
+ return ($ex - 1/$ex)/2;
}
my ($x, $y) = @{$z->cartesian};
- my $ex = exp($x);
+ $ex = exp($x);
my $ex_1 = 1 / $ex;
- return cplx(0.5 * ($ex - $ex_1), 0) if $real;
return (ref $z)->make(cos($y) * ($ex - $ex_1)/2,
sin($y) * ($ex + $ex_1)/2);
}
#
# acosh
#
-# Computes the arc hyperbolic cosine acosh(z) = log(z +- sqrt(z*z-1)).
+# Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).
#
sub acosh {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
+ unless (ref $z) {
+ return log($z + sqrt($z*$z-1)) if $z >= 1;
+ $z = cplx($z, 0);
+ }
my ($re, $im) = @{$z->cartesian};
- return log($re + sqrt(cplx($re*$re - 1, 0)))
- if ($im == 0 && $re < 0);
+ if ($im == 0) {
+ return cplx(log($re + sqrt($re*$re - 1)), 0) if $re >= 1;
+ return cplx(0, atan2(sqrt(1-$re*$re), $re)) if abs($re) <= 1;
+ }
return log($z + sqrt($z*$z - 1));
}
#
sub asinh {
my ($z) = @_;
- $z = cplx($z, 0) unless ref $z;
return log($z + sqrt($z*$z + 1));
}
#
sub atanh {
my ($z) = @_;
+ unless (ref $z) {
+ return log((1 + $z)/(1 - $z))/2 if abs($z) < 1;
+ $z = cplx($z, 0);
+ }
_divbyzero 'atanh(1)', "1 - $z" if ($z == 1);
_logofzero 'atanh(-1)' if ($z == -1);
- $z = cplx($z, 0) unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- if ($im == 0 && $re > 1) {
- return cplx(atanh(1 / $re), pi/2);
- }
- return log((1 + $z) / (1 - $z)) / 2;
+ return 0.5 * log((1 + $z) / (1 - $z));
}
#
sub asech {
my ($z) = @_;
_divbyzero 'asech(0)', $z if ($z == 0);
- $z = cplx($z, 0) unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- if ($im == 0 && $re < 0) {
- my $ire = 1 / $re;
- return log($ire + sqrt(cplx($ire*$ire - 1, 0)));
- }
return acosh(1 / $z);
}
#
sub acoth {
my ($z) = @_;
+ unless (ref $z) {
+ return log(($z + 1)/($z - 1))/2 if abs($z) > 1;
+ $z = cplx($z, 0);
+ }
_divbyzero 'acoth(1)', "$z - 1" if ($z == 1);
_logofzero 'acoth(-1)' if ($z == -1);
- $z = cplx($z, 0) unless ref $z;
- my ($re, $im) = @{$z->cartesian};
- if ($im == 0 and abs($re) < 1) {
- return cplx(acoth(1/$re) , pi/2);
- }
return log((1 + $z) / ($z - 1)) / 2;
}
#
sub atan2 {
my ($z1, $z2, $inverted) = @_;
- my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
- my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
- my $tan;
- if (defined $inverted && $inverted) { # atan(z2/z1)
- return pi * ($re2 > 0 ? 1 : -1) if $re1 == 0 && $im1 == 0;
- $tan = $z2 / $z1;
+ my ($re1, $im1, $re2, $im2);
+ if ($inverted) {
+ ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+ ($re2, $im2) = @{$z1->cartesian};
} else {
- return pi * ($re1 > 0 ? 1 : -1) if $re2 == 0 && $im2 == 0;
- $tan = $z1 / $z2;
+ ($re1, $im1) = @{$z1->cartesian};
+ ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+ }
+ if ($im2 == 0) {
+ return cplx(atan2($re1, $re2), 0) if $im1 == 0;
+ return cplx(($im1<=>0) * pip2, 0) if $re2 == 0;
}
- return atan($tan);
+ 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
#
# Set (fetch if no argument) display format for all complex numbers that
-# don't happen to have overrriden it via ->display_format
+# don't happen to have overridden it via ->display_format
#
# When called as a method, this actually sets the display format for
# the current object.
my $z = shift;
my ($x, $y) = @{$z->cartesian};
my ($re, $im);
+ my $eps = 1e-14;
- $x = int($x + ($x < 0 ? -1 : 1) * 1e-14)
- if int(abs($x)) != int(abs($x) + 1e-14);
- $y = int($y + ($y < 0 ? -1 : 1) * 1e-14)
- if int(abs($y)) != int(abs($y) + 1e-14);
+ $x = int($x + ($x < 0 ? -1 : 1) * $eps)
+ if int(abs($x)) != int(abs($x) + $eps);
+ $y = int($y + ($y < 0 ? -1 : 1) * $eps)
+ if int(abs($y)) != int(abs($y) + $eps);
- $re = "$x" if abs($x) >= 1e-14;
- if ($y == 1) { $im = 'i' }
- elsif ($y == -1) { $im = '-i' }
- elsif (abs($y) >= 1e-14) { $im = $y . "i" }
+ $re = "$x" if abs($x) >= $eps;
+ if ($y == 1) { $im = 'i' }
+ elsif ($y == -1) { $im = '-i' }
+ elsif (abs($y) >= $eps) { $im = $y . "i" }
my $str = '';
$str = $re if defined $re;
return '[0,0]' if $r <= $eps;
- my $tpi = 2 * pi;
- my $nt = $t / $tpi;
- $nt = ($nt - int($nt)) * $tpi;
- $nt += $tpi if $nt < 0; # Range [0, 2pi]
+ my $nt = $t / pit2;
+ $nt = ($nt - int($nt)) * pit2;
+ $nt += pit2 if $nt < 0; # Range [0, 2pi]
if (abs($nt) <= $eps) { $theta = 0 }
elsif (abs(pi-$nt) <= $eps) { $theta = 'pi' }
# Okay, number is not a real. Try to identify pi/n and friends...
#
- $nt -= $tpi if $nt > pi;
+ $nt -= pit2 if $nt > pi;
my ($n, $k, $kpi);
-
+
for ($k = 1, $kpi = pi; $k < 10; $k++, $kpi += pi) {
$n = int($kpi / $nt + ($nt > 0 ? 1 : -1) * 0.5);
if (abs($kpi/$n - $nt) <= $eps) {
=head1 SYNOPSIS
use Math::Complex;
-
+
$z = Math::Complex->make(5, 6);
$t = 4 - 3*i + $z;
$j = cplxe(1, 2*pi/3);
which is also expressed by this formula:
- z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
+ 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 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 substractions. Real numbers are on the I<x>
-axis, and therefore I<theta> is zero.
+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
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 positive real numbers and yields a
-positive real number (it is an application from B<R+> to B<R+>).
+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(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 positive, so C<[x,pi]> is really C<-x>, a
-negative number)
-and the above definition states that
+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
log(z1) = log(r1) + i*t1
sin(z1) = 1/2i (exp(i * z1) - exp(-i * z1))
cos(z1) = 1/2 (exp(i * z1) + exp(-i * z1))
- abs(z1) = r1
atan2(z1, z2) = atan(z1/z2)
The following extra operations are supported on both real and complex
cot(z) = 1 / tan(z)
asin(z) = -i * log(i*z + sqrt(1-z*z))
- acos(z) = -i * log(z + sqrt(z*z-1))
+ acos(z) = -i * log(z + i*sqrt(1-z*z))
atan(z) = i/2 * log((i+z) / (i-z))
acsc(z) = asin(1 / 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))
$z = 3 + 4*i;
-if you like. To create a number using the trigonometric form, use either:
+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). (Mnmemonic: C<e> is used as a
-notation for complex numbers in the trigonometric form).
+(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 positive (it represents the distance to the origin in the complex
+must be non-negative (it represents the distance to the origin in the complex
plane).
=head1 STRINGIFICATION
=head1 BUGS
Saying C<use Math::Complex;> exports many mathematical routines in the
-caller environment and even overrides some (C<sin>, C<cos>, C<sqrt>,
-C<log>, C<exp>). This is construed as a feature by the Authors,
-actually... ;-)
-
-The code is not optimized for speed, although we try to use the cartesian
-form for addition-like operators and the trigonometric form for all
-multiplication-like operators.
-
-The arg() routine does not ensure the angle is within the range [-pi,+pi]
-(a side effect caused by multiplication and division using the trigonometric
-representation).
+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 '+'
Raphael Manfredi <F<Raphael_Manfredi@grenoble.hp.com>> and
Jarkko Hietaniemi <F<jhi@iki.fi>>.
+Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>.
+
=cut
# eof
projects / p5sagit/p5-mst-13.2.git / blobdiff