use vars qw($VERSION @ISA @EXPORT @EXPORT_OK %EXPORT_TAGS $Inf);
-$VERSION = 1.35;
+$VERSION = 1.36;
BEGIN {
unless ($^O eq 'unicosmk') {
),
@trig);
-@EXPORT_OK = qw(decplx);
+my @pi = qw(pi pi2 pi4 pip2 pip4);
+
+@EXPORT_OK = @pi;
%EXPORT_TAGS = (
'trig' => [@trig],
+ 'pi' => [@pi],
);
use overload
- '+' => \&plus,
- '-' => \&minus,
- '*' => \&multiply,
- '/' => \÷,
- '**' => \&power,
- '==' => \&numeq,
- '<=>' => \&spaceship,
- 'neg' => \&negate,
- '~' => \&conjugate,
+ '+' => \&_plus,
+ '-' => \&_minus,
+ '*' => \&_multiply,
+ '/' => \&_divide,
+ '**' => \&_power,
+ '==' => \&_numeq,
+ '<=>' => \&_spaceship,
+ 'neg' => \&_negate,
+ '~' => \&_conjugate,
'abs' => \&abs,
'sqrt' => \&sqrt,
'exp' => \&exp,
'cos' => \&cos,
'tan' => \&tan,
'atan2' => \&atan2,
- qw("" stringify);
+ '""' => \&_stringify;
#
# Package "privates"
# 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)
+# bn_cartesian
+# bnc_dirty
#
# Die on bad *make() arguments.
}
$im ||= 0;
_cannot_make("imaginary part", $im) unless $im =~ /^$gre$/;
- $self->set_cartesian([$re, $im ]);
+ $self->_set_cartesian([$re, $im ]);
$self->display_format('cartesian');
return $self;
}
$theta ||= 0;
_cannot_make("theta", $theta) unless $theta =~ /^$gre$/;
- $self->set_polar([$rho, $theta]);
+ $self->_set_polar([$rho, $theta]);
$self->display_format('polar');
return $self;
sub pi () { 4 * CORE::atan2(1, 1) }
#
-# pit2
+# pi2
#
# The full circle
#
-sub pit2 () { 2 * pi }
+sub pi2 () { 2 * pi }
+
+#
+# pi4
+#
+# The full circle twice.
+#
+sub pi4 () { 4 * pi }
#
# pip2
sub pip2 () { pi / 2 }
#
-# deg1
+# pip4
#
-# One degree in radians, used in stringify_polar.
+# The eighth circle.
#
-
-sub deg1 () { pi / 180 }
+sub pip4 () { pi / 4 }
#
-# uplog10
+# _uplog10
#
# Used in log10().
#
-sub uplog10 () { 1 / CORE::log(10) }
+sub _uplog10 () { 1 / CORE::log(10) }
#
# i
}
#
-# ip2
+# _ip2
#
# Half of i.
#
-sub ip2 () { i / 2 }
+sub _ip2 () { i / 2 }
#
# Attribute access/set routines
#
-sub cartesian {$_[0]->{c_dirty} ?
- $_[0]->update_cartesian : $_[0]->{'cartesian'}}
-sub polar {$_[0]->{p_dirty} ?
- $_[0]->update_polar : $_[0]->{'polar'}}
+sub _cartesian {$_[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] }
+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
+# ->_update_cartesian
#
# Recompute and return the cartesian form, given accurate polar form.
#
-sub update_cartesian {
+sub _update_cartesian {
my $self = shift;
my ($r, $t) = @{$self->{'polar'}};
$self->{c_dirty} = 0;
#
#
-# ->update_polar
+# ->_update_polar
#
# Recompute and return the polar form, given accurate cartesian form.
#
-sub update_polar {
+sub _update_polar {
my $self = shift;
my ($x, $y) = @{$self->{'cartesian'}};
$self->{p_dirty} = 0;
}
#
-# (plus)
+# (_plus)
#
# Computes z1+z2.
#
-sub plus {
+sub _plus {
my ($z1, $z2, $regular) = @_;
- my ($re1, $im1) = @{$z1->cartesian};
+ my ($re1, $im1) = @{$z1->_cartesian};
$z2 = cplx($z2) unless ref $z2;
- my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+ my ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
unless (defined $regular) {
- $z1->set_cartesian([$re1 + $re2, $im1 + $im2]);
+ $z1->_set_cartesian([$re1 + $re2, $im1 + $im2]);
return $z1;
}
return (ref $z1)->make($re1 + $re2, $im1 + $im2);
}
#
-# (minus)
+# (_minus)
#
# Computes z1-z2.
#
-sub minus {
+sub _minus {
my ($z1, $z2, $inverted) = @_;
- my ($re1, $im1) = @{$z1->cartesian};
+ my ($re1, $im1) = @{$z1->_cartesian};
$z2 = cplx($z2) unless ref $z2;
- my ($re2, $im2) = @{$z2->cartesian};
+ my ($re2, $im2) = @{$z2->_cartesian};
unless (defined $inverted) {
- $z1->set_cartesian([$re1 - $re2, $im1 - $im2]);
+ $z1->_set_cartesian([$re1 - $re2, $im1 - $im2]);
return $z1;
}
return $inverted ?
}
#
-# (multiply)
+# (_multiply)
#
# Computes z1*z2.
#
-sub multiply {
+sub _multiply {
my ($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 ($r1, $t1) = @{$z1->_polar};
+ my ($r2, $t2) = @{$z2->_polar};
my $t = $t1 + $t2;
- if ($t > pi()) { $t -= pit2 }
- elsif ($t <= -pi()) { $t += pit2 }
+ if ($t > pi()) { $t -= pi2 }
+ elsif ($t <= -pi()) { $t += pi2 }
unless (defined $regular) {
- $z1->set_polar([$r1 * $r2, $t]);
+ $z1->_set_polar([$r1 * $r2, $t]);
return $z1;
}
return (ref $z1)->emake($r1 * $r2, $t);
} else {
- my ($x1, $y1) = @{$z1->cartesian};
+ my ($x1, $y1) = @{$z1->_cartesian};
if (ref $z2) {
- my ($x2, $y2) = @{$z2->cartesian};
+ 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);
}
#
-# (divide)
+# (_divide)
#
# Computes z1/z2.
#
-sub divide {
+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 ($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()) { $t += pit2 }
+ if ($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()) { $t -= pit2 }
- elsif ($t <= -pi()) { $t += pit2 }
+ if ($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};
+ ($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};
+ my ($x1, $y1) = @{$z1->_cartesian};
if (ref $z2) {
- ($x2, $y2) = @{$z2->cartesian};
+ ($x2, $y2) = @{$z2->_cartesian};
$d = $x2*$x2 + $y2*$y2;
_divbyzero "$z1/0" if $d == 0;
my $u = ($x1*$x2 + $y1*$y2)/$d;
}
#
-# (power)
+# (_power)
#
# Computes z1**z2 = exp(z2 * log z1)).
#
-sub power {
+sub _power {
my ($z1, $z2, $inverted) = @_;
if ($inverted) {
return 1 if $z1 == 0 || $z2 == 1;
# If both arguments cartesian, return cartesian, else polar.
return $z1->{c_dirty} == 0 &&
(not ref $z2 or $z2->{c_dirty} == 0) ?
- cplx(@{$w->cartesian}) : $w;
+ cplx(@{$w->_cartesian}) : $w;
}
#
-# (spaceship)
+# (_spaceship)
#
# Computes z1 <=> z2.
# Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i.
#
-sub spaceship {
+sub _spaceship {
my ($z1, $z2, $inverted) = @_;
- my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
- my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+ my ($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 <=> $im2);
}
#
-# (numeq)
+# (_numeq)
#
# Computes z1 == z2.
#
-# (Required in addition to spaceship() because of NaNs.)
-sub numeq {
+# (Required in addition to _spaceship() because of NaNs.)
+sub _numeq {
my ($z1, $z2, $inverted) = @_;
- my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
- my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
+ my ($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
+ my ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
return $re1 == $re2 && $im1 == $im2 ? 1 : 0;
}
#
-# (negate)
+# (_negate)
#
# Computes -z.
#
-sub negate {
+sub _negate {
my ($z) = @_;
if ($z->{c_dirty}) {
- my ($r, $t) = @{$z->polar};
+ my ($r, $t) = @{$z->_polar};
$t = ($t <= 0) ? $t + pi : $t - pi;
return (ref $z)->emake($r, $t);
}
- my ($re, $im) = @{$z->cartesian};
+ my ($re, $im) = @{$z->_cartesian};
return (ref $z)->make(-$re, -$im);
}
#
-# (conjugate)
+# (_conjugate)
#
-# Compute complex's conjugate.
+# Compute complex's _conjugate.
#
-sub conjugate {
+sub _conjugate {
my ($z) = @_;
if ($z->{c_dirty}) {
- my ($r, $t) = @{$z->polar};
+ my ($r, $t) = @{$z->_polar};
return (ref $z)->emake($r, -$t);
}
- my ($re, $im) = @{$z->cartesian};
+ my ($re, $im) = @{$z->_cartesian};
return (ref $z)->make($re, -$im);
}
}
}
if (defined $rho) {
- $z->{'polar'} = [ $rho, ${$z->polar}[1] ];
+ $z->{'polar'} = [ $rho, ${$z->_polar}[1] ];
$z->{p_dirty} = 0;
$z->{c_dirty} = 1;
return $rho;
} else {
- return ${$z->polar}[0];
+ return ${$z->_polar}[0];
}
}
sub _theta {
my $theta = $_[0];
- if ($$theta > pi()) { $$theta -= pit2 }
- elsif ($$theta <= -pi()) { $$theta += pit2 }
+ if ($$theta > pi()) { $$theta -= pi2 }
+ elsif ($$theta <= -pi()) { $$theta += pi2 }
}
#
return $z unless ref $z;
if (defined $theta) {
_theta(\$theta);
- $z->{'polar'} = [ ${$z->polar}[0], $theta ];
+ $z->{'polar'} = [ ${$z->_polar}[0], $theta ];
$z->{p_dirty} = 0;
$z->{c_dirty} = 1;
} else {
- $theta = ${$z->polar}[1];
+ $theta = ${$z->_polar}[1];
_theta(\$theta);
}
return $theta;
#
sub sqrt {
my ($z) = @_;
- my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0);
+ my ($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};
+ my ($r, $t) = @{$z->_polar};
return (ref $z)->emake(CORE::sqrt($r), $t/2);
}
-CORE::exp(CORE::log(-$z)/3) :
($z > 0 ? CORE::exp(CORE::log($z)/3): 0)
unless ref $z;
- my ($r, $t) = @{$z->polar};
+ my ($r, $t) = @{$z->_polar};
return 0 if $r == 0;
return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3);
}
my ($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 = pit2 / $n;
+ @{$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) {
$i++, $theta += $theta_inc) {
my $w = cplxe($rho, $theta);
# Yes, $cartesian is loop invariant.
- push @root, $cartesian ? cplx(@{$w->cartesian}) : $w;
+ 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;
+ return $cartesian ? cplx(@{$w->_cartesian}) : $w;
}
}
my ($z, $Re) = @_;
return $z unless ref $z;
if (defined $Re) {
- $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ];
+ $z->{'cartesian'} = [ $Re, ${$z->_cartesian}[1] ];
$z->{c_dirty} = 0;
$z->{p_dirty} = 1;
} else {
- return ${$z->cartesian}[0];
+ return ${$z->_cartesian}[0];
}
}
my ($z, $Im) = @_;
return 0 unless ref $z;
if (defined $Im) {
- $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ];
+ $z->{'cartesian'} = [ ${$z->_cartesian}[0], $Im ];
$z->{c_dirty} = 0;
$z->{p_dirty} = 1;
} else {
- return ${$z->cartesian}[1];
+ return ${$z->_cartesian}[1];
}
}
#
sub exp {
my ($z) = @_;
- my ($x, $y) = @{$z->cartesian};
+ my ($x, $y) = @{$z->_cartesian};
return (ref $z)->emake(CORE::exp($x), $y);
}
_logofzero("log") if $z == 0;
return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi);
}
- my ($r, $t) = @{$z->polar};
+ my ($r, $t) = @{$z->_polar};
_logofzero("log") if $r == 0;
- if ($t > pi()) { $t -= pit2 }
- elsif ($t <= -pi()) { $t += pit2 }
+ if ($t > pi()) { $t -= pi2 }
+ elsif ($t <= -pi()) { $t += pi2 }
return (ref $z)->make(CORE::log($r), $t);
}
#
sub log10 {
- return Math::Complex::log($_[0]) * uplog10;
+ return Math::Complex::log($_[0]) * _uplog10;
}
#
sub cos {
my ($z) = @_;
return CORE::cos($z) unless ref $z;
- my ($x, $y) = @{$z->cartesian};
+ my ($x, $y) = @{$z->_cartesian};
my $ey = CORE::exp($y);
my $sx = CORE::sin($x);
my $cx = CORE::cos($x);
sub sin {
my ($z) = @_;
return CORE::sin($z) unless ref $z;
- my ($x, $y) = @{$z->cartesian};
+ my ($x, $y) = @{$z->_cartesian};
my $ey = CORE::exp($y);
my $sx = CORE::sin($x);
my $cx = CORE::cos($x);
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};
+ 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);
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};
+ 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);
sub atan {
my ($z) = @_;
return CORE::atan2($z, 1) unless ref $z;
- my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
+ my ($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;
+ return _ip2 * $log;
}
#
$ex = CORE::exp($z);
return $ex ? ($ex + 1/$ex)/2 : $Inf;
}
- my ($x, $y) = @{$z->cartesian};
+ 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,
$ex = CORE::exp($z);
return $ex ? ($ex - 1/$ex)/2 : "-$Inf";
}
- my ($x, $y) = @{$z->cartesian};
+ my ($x, $y) = @{$z->_cartesian};
my $cy = CORE::cos($y);
my $sy = CORE::sin($y);
$ex = CORE::exp($x);
unless (ref $z) {
$z = cplx($z, 0);
}
- my ($re, $im) = @{$z->cartesian};
+ my ($re, $im) = @{$z->_cartesian};
if ($im == 0) {
return CORE::log($re + CORE::sqrt($re*$re - 1))
if $re >= 1;
my ($z1, $z2, $inverted) = @_;
my ($re1, $im1, $re2, $im2);
if ($inverted) {
- ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
- ($re2, $im2) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
+ ($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} : ($z2, 0);
+ ($re1, $im1) = ref $z1 ? @{$z1->_cartesian} : ($z1, 0);
+ ($re2, $im2) = ref $z2 ? @{$z2->_cartesian} : ($z2, 0);
}
if ($im1 || $im2) {
# In MATLAB the imaginary parts are ignored.
}
#
-# (stringify)
+# (_stringify)
#
# Show nicely formatted complex number under its cartesian or polar form,
# depending on the current display format:
# . Otherwise, use the generic current default for all complex numbers,
# which is a package global variable.
#
-sub stringify {
+sub _stringify {
my ($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;
+ return $z->_stringify_polar if $style =~ /^p/i;
+ return $z->_stringify_cartesian;
}
#
-# ->stringify_cartesian
+# ->_stringify_cartesian
#
# Stringify as a cartesian representation 'a+bi'.
#
-sub stringify_cartesian {
+sub _stringify_cartesian {
my $z = shift;
- my ($x, $y) = @{$z->cartesian};
+ my ($x, $y) = @{$z->_cartesian};
my ($re, $im);
my %format = $z->display_format;
#
-# ->stringify_polar
+# ->_stringify_polar
#
# Stringify as a polar representation '[r,t]'.
#
-sub stringify_polar {
+sub _stringify_polar {
my $z = shift;
- my ($r, $t) = @{$z->polar};
+ my ($r, $t) = @{$z->_polar};
my $theta;
my %format = $z->display_format;
# Try to identify pi/n and friends.
#
- $t -= int(CORE::abs($t) / pit2) * pit2;
+ $t -= int(CORE::abs($t) / pi2) * pi2;
if ($format{polar_pretty_print} && $t) {
my ($a, $b);
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 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<theta> is zero or I<pi>.
+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 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:
$j->arg(2); # (the last two aka rho, theta)
# can be used also as mutators.
+=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;
+
=head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
The division (/) and the following functions
=head1 AUTHORS
-Daniel S. Lewart <F<d-lewart@uiuc.edu>>
-
-Original authors Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and
-Jarkko Hietaniemi <F<jhi@iki.fi>>
+Daniel S. Lewart <F<lewart!at!uiuc.edu>>
+Jarkko Hietaniemi <F<jhi!at!iki.fi>>
+Raphael Manfredi <F<Raphael_Manfredi!at!pobox.com>>
=cut
projects / p5sagit/p5-mst-13.2.git / blobdiff