2 # Complex numbers and associated mathematical functions
3 # -- Raphael Manfredi Since Sep 1996
4 # -- Jarkko Hietaniemi Since Mar 1997
5 # -- Daniel S. Lewart Since Sep 1997
12 our($VERSION, @ISA, @EXPORT, %EXPORT_TAGS, $Inf);
15 eval { require POSIX; import POSIX 'HUGE_VAL' };
16 if (exists &HUGE_VAL) {
17 $Inf = sprintf "%g", &HUGE_VAL;
20 $Inf = CORE::exp(CORE::exp(30));
21 $! = $e; # Clear ERANGE.
23 undef $Inf unless $Inf =~ /^inf(?:inity)?$/i; # Inf INF inf Infinity
24 $Inf = "Inf" if !defined $Inf || !($Inf > 0); # Desperation.
39 csc cosec sec cot cotan
41 acsc acosec asec acot acotan
43 csch cosech sech coth cotanh
45 acsch acosech asech acoth acotanh
84 my %DISPLAY_FORMAT = ('style' => 'cartesian',
85 'polar_pretty_print' => 1);
86 my $eps = 1e-14; # Epsilon
89 # Object attributes (internal):
90 # cartesian [real, imaginary] -- cartesian form
91 # polar [rho, theta] -- polar form
92 # c_dirty cartesian form not up-to-date
93 # p_dirty polar form not up-to-date
94 # display display format (package's global when not set)
97 # Die on bad *make() arguments.
100 die "@{[(caller(1))[3]]}: Cannot take $_[0] of $_[1].\n";
106 # Create a new complex number (cartesian form)
109 my $self = bless {}, shift;
113 if ( $rre eq ref $self ) {
116 _cannot_make("real part", $rre);
121 if ( $rim eq ref $self ) {
124 _cannot_make("imaginary part", $rim);
127 $self->{'cartesian'} = [ $re, $im ];
128 $self->{c_dirty} = 0;
129 $self->{p_dirty} = 1;
130 $self->display_format('cartesian');
137 # Create a new complex number (exponential form)
140 my $self = bless {}, shift;
141 my ($rho, $theta) = @_;
144 if ( $rrh eq ref $self ) {
147 _cannot_make("rho", $rrh);
150 my $rth = ref $theta;
152 if ( $rth eq ref $self ) {
153 $theta = theta($theta);
155 _cannot_make("theta", $rth);
160 $theta = ($theta <= 0) ? $theta + pi() : $theta - pi();
162 $self->{'polar'} = [$rho, $theta];
163 $self->{p_dirty} = 0;
164 $self->{c_dirty} = 1;
165 $self->display_format('polar');
169 sub new { &make } # For backward compatibility only.
174 # Creates a complex number from a (re, im) tuple.
175 # This avoids the burden of writing Math::Complex->make(re, im).
179 return __PACKAGE__->make($re, defined $im ? $im : 0);
185 # Creates a complex number from a (rho, theta) tuple.
186 # This avoids the burden of writing Math::Complex->emake(rho, theta).
189 my ($rho, $theta) = @_;
190 return __PACKAGE__->emake($rho, defined $theta ? $theta : 0);
196 # The number defined as pi = 180 degrees
198 sub pi () { 4 * CORE::atan2(1, 1) }
205 sub pit2 () { 2 * pi }
212 sub pip2 () { pi / 2 }
217 # One degree in radians, used in stringify_polar.
220 sub deg1 () { pi / 180 }
227 sub uplog10 () { 1 / CORE::log(10) }
232 # The number defined as i*i = -1;
237 $i->{'cartesian'} = [0, 1];
238 $i->{'polar'} = [1, pip2];
252 # Attribute access/set routines
255 sub cartesian {$_[0]->{c_dirty} ?
256 $_[0]->update_cartesian : $_[0]->{'cartesian'}}
257 sub polar {$_[0]->{p_dirty} ?
258 $_[0]->update_polar : $_[0]->{'polar'}}
260 sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{'cartesian'} = $_[1] }
261 sub set_polar { $_[0]->{c_dirty}++; $_[0]->{'polar'} = $_[1] }
266 # Recompute and return the cartesian form, given accurate polar form.
268 sub update_cartesian {
270 my ($r, $t) = @{$self->{'polar'}};
271 $self->{c_dirty} = 0;
272 return $self->{'cartesian'} = [$r * CORE::cos($t), $r * CORE::sin($t)];
279 # Recompute and return the polar form, given accurate cartesian form.
283 my ($x, $y) = @{$self->{'cartesian'}};
284 $self->{p_dirty} = 0;
285 return $self->{'polar'} = [0, 0] if $x == 0 && $y == 0;
286 return $self->{'polar'} = [CORE::sqrt($x*$x + $y*$y),
287 CORE::atan2($y, $x)];
296 my ($z1, $z2, $regular) = @_;
297 my ($re1, $im1) = @{$z1->cartesian};
298 $z2 = cplx($z2) unless ref $z2;
299 my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
300 unless (defined $regular) {
301 $z1->set_cartesian([$re1 + $re2, $im1 + $im2]);
304 return (ref $z1)->make($re1 + $re2, $im1 + $im2);
313 my ($z1, $z2, $inverted) = @_;
314 my ($re1, $im1) = @{$z1->cartesian};
315 $z2 = cplx($z2) unless ref $z2;
316 my ($re2, $im2) = @{$z2->cartesian};
317 unless (defined $inverted) {
318 $z1->set_cartesian([$re1 - $re2, $im1 - $im2]);
322 (ref $z1)->make($re2 - $re1, $im2 - $im1) :
323 (ref $z1)->make($re1 - $re2, $im1 - $im2);
333 my ($z1, $z2, $regular) = @_;
334 if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
335 # if both polar better use polar to avoid rounding errors
336 my ($r1, $t1) = @{$z1->polar};
337 my ($r2, $t2) = @{$z2->polar};
339 if ($t > pi()) { $t -= pit2 }
340 elsif ($t <= -pi()) { $t += pit2 }
341 unless (defined $regular) {
342 $z1->set_polar([$r1 * $r2, $t]);
345 return (ref $z1)->emake($r1 * $r2, $t);
347 my ($x1, $y1) = @{$z1->cartesian};
349 my ($x2, $y2) = @{$z2->cartesian};
350 return (ref $z1)->make($x1*$x2-$y1*$y2, $x1*$y2+$y1*$x2);
352 return (ref $z1)->make($x1*$z2, $y1*$z2);
360 # Die on division by zero.
363 my $mess = "$_[0]: Division by zero.\n";
366 $mess .= "(Because in the definition of $_[0], the divisor ";
367 $mess .= "$_[1] " unless ("$_[1]" eq '0');
373 $mess .= "Died at $up[1] line $up[2].\n";
384 my ($z1, $z2, $inverted) = @_;
385 if ($z1->{p_dirty} == 0 and ref $z2 and $z2->{p_dirty} == 0) {
386 # if both polar better use polar to avoid rounding errors
387 my ($r1, $t1) = @{$z1->polar};
388 my ($r2, $t2) = @{$z2->polar};
391 _divbyzero "$z2/0" if ($r1 == 0);
393 if ($t > pi()) { $t -= pit2 }
394 elsif ($t <= -pi()) { $t += pit2 }
395 return (ref $z1)->emake($r2 / $r1, $t);
397 _divbyzero "$z1/0" if ($r2 == 0);
399 if ($t > pi()) { $t -= pit2 }
400 elsif ($t <= -pi()) { $t += pit2 }
401 return (ref $z1)->emake($r1 / $r2, $t);
406 ($x2, $y2) = @{$z1->cartesian};
407 $d = $x2*$x2 + $y2*$y2;
408 _divbyzero "$z2/0" if $d == 0;
409 return (ref $z1)->make(($x2*$z2)/$d, -($y2*$z2)/$d);
411 my ($x1, $y1) = @{$z1->cartesian};
413 ($x2, $y2) = @{$z2->cartesian};
414 $d = $x2*$x2 + $y2*$y2;
415 _divbyzero "$z1/0" if $d == 0;
416 my $u = ($x1*$x2 + $y1*$y2)/$d;
417 my $v = ($y1*$x2 - $x1*$y2)/$d;
418 return (ref $z1)->make($u, $v);
420 _divbyzero "$z1/0" if $z2 == 0;
421 return (ref $z1)->make($x1/$z2, $y1/$z2);
430 # Computes z1**z2 = exp(z2 * log z1)).
433 my ($z1, $z2, $inverted) = @_;
435 return 1 if $z1 == 0 || $z2 == 1;
436 return 0 if $z2 == 0 && Re($z1) > 0;
438 return 1 if $z2 == 0 || $z1 == 1;
439 return 0 if $z1 == 0 && Re($z2) > 0;
441 my $w = $inverted ? &exp($z1 * &log($z2))
442 : &exp($z2 * &log($z1));
443 # If both arguments cartesian, return cartesian, else polar.
444 return $z1->{c_dirty} == 0 &&
445 (not ref $z2 or $z2->{c_dirty} == 0) ?
446 cplx(@{$w->cartesian}) : $w;
452 # Computes z1 <=> z2.
453 # Sorts on the real part first, then on the imaginary part. Thus 2-4i < 3+8i.
456 my ($z1, $z2, $inverted) = @_;
457 my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
458 my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
459 my $sgn = $inverted ? -1 : 1;
460 return $sgn * ($re1 <=> $re2) if $re1 != $re2;
461 return $sgn * ($im1 <=> $im2);
469 # (Required in addition to spaceship() because of NaNs.)
471 my ($z1, $z2, $inverted) = @_;
472 my ($re1, $im1) = ref $z1 ? @{$z1->cartesian} : ($z1, 0);
473 my ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
474 return $re1 == $re2 && $im1 == $im2 ? 1 : 0;
485 my ($r, $t) = @{$z->polar};
486 $t = ($t <= 0) ? $t + pi : $t - pi;
487 return (ref $z)->emake($r, $t);
489 my ($re, $im) = @{$z->cartesian};
490 return (ref $z)->make(-$re, -$im);
496 # Compute complex's conjugate.
501 my ($r, $t) = @{$z->polar};
502 return (ref $z)->emake($r, -$t);
504 my ($re, $im) = @{$z->cartesian};
505 return (ref $z)->make($re, -$im);
511 # Compute or set complex's norm (rho).
519 return CORE::abs($z);
523 $z->{'polar'} = [ $rho, ${$z->polar}[1] ];
528 return ${$z->polar}[0];
535 if ($$theta > pi()) { $$theta -= pit2 }
536 elsif ($$theta <= -pi()) { $$theta += pit2 }
542 # Compute or set complex's argument (theta).
545 my ($z, $theta) = @_;
546 return $z unless ref $z;
547 if (defined $theta) {
549 $z->{'polar'} = [ ${$z->polar}[0], $theta ];
553 $theta = ${$z->polar}[1];
564 # It is quite tempting to use wantarray here so that in list context
565 # sqrt() would return the two solutions. This, however, would
568 # print "sqrt(z) = ", sqrt($z), "\n";
570 # The two values would be printed side by side without no intervening
571 # whitespace, quite confusing.
572 # Therefore if you want the two solutions use the root().
576 my ($re, $im) = ref $z ? @{$z->cartesian} : ($z, 0);
577 return $re < 0 ? cplx(0, CORE::sqrt(-$re)) : CORE::sqrt($re)
579 my ($r, $t) = @{$z->polar};
580 return (ref $z)->emake(CORE::sqrt($r), $t/2);
586 # Compute cbrt(z) (cubic root).
588 # Why are we not returning three values? The same answer as for sqrt().
593 -CORE::exp(CORE::log(-$z)/3) :
594 ($z > 0 ? CORE::exp(CORE::log($z)/3): 0)
596 my ($r, $t) = @{$z->polar};
598 return (ref $z)->emake(CORE::exp(CORE::log($r)/3), $t/3);
607 my $mess = "Root $_[0] illegal, root rank must be positive integer.\n";
611 $mess .= "Died at $up[1] line $up[2].\n";
619 # Computes all nth root for z, returning an array whose size is n.
620 # `n' must be a positive integer.
622 # The roots are given by (for k = 0..n-1):
624 # z^(1/n) = r^(1/n) (cos ((t+2 k pi)/n) + i sin ((t+2 k pi)/n))
628 _rootbad($n) if ($n < 1 or int($n) != $n);
629 my ($r, $t) = ref $z ?
630 @{$z->polar} : (CORE::abs($z), $z >= 0 ? 0 : pi);
633 my $theta_inc = pit2 / $n;
634 my $rho = $r ** (1/$n);
636 my $cartesian = ref $z && $z->{c_dirty} == 0;
637 for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) {
638 my $w = cplxe($rho, $theta);
639 # Yes, $cartesian is loop invariant.
640 push @root, $cartesian ? cplx(@{$w->cartesian}) : $w;
648 # Return or set Re(z).
652 return $z unless ref $z;
654 $z->{'cartesian'} = [ $Re, ${$z->cartesian}[1] ];
658 return ${$z->cartesian}[0];
665 # Return or set Im(z).
669 return $z unless ref $z;
671 $z->{'cartesian'} = [ ${$z->cartesian}[0], $Im ];
675 return ${$z->cartesian}[1];
682 # Return or set rho(w).
685 Math::Complex::abs(@_);
691 # Return or set theta(w).
694 Math::Complex::arg(@_);
704 my ($x, $y) = @{$z->cartesian};
705 return (ref $z)->emake(CORE::exp($x), $y);
711 # Die on logarithm of zero.
714 my $mess = "$_[0]: Logarithm of zero.\n";
717 $mess .= "(Because in the definition of $_[0], the argument ";
718 $mess .= "$_[1] " unless ($_[1] eq '0');
724 $mess .= "Died at $up[1] line $up[2].\n";
737 _logofzero("log") if $z == 0;
738 return $z > 0 ? CORE::log($z) : cplx(CORE::log(-$z), pi);
740 my ($r, $t) = @{$z->polar};
741 _logofzero("log") if $r == 0;
742 if ($t > pi()) { $t -= pit2 }
743 elsif ($t <= -pi()) { $t += pit2 }
744 return (ref $z)->make(CORE::log($r), $t);
752 sub ln { Math::Complex::log(@_) }
761 return Math::Complex::log($_[0]) * uplog10;
767 # Compute logn(z,n) = log(z) / log(n)
771 $z = cplx($z, 0) unless ref $z;
772 my $logn = $LOGN{$n};
773 $logn = $LOGN{$n} = CORE::log($n) unless defined $logn; # Cache log(n)
774 return &log($z) / $logn;
780 # Compute cos(z) = (exp(iz) + exp(-iz))/2.
784 return CORE::cos($z) unless ref $z;
785 my ($x, $y) = @{$z->cartesian};
786 my $ey = CORE::exp($y);
787 my $sx = CORE::sin($x);
788 my $cx = CORE::cos($x);
789 my $ey_1 = $ey ? 1 / $ey : $Inf;
790 return (ref $z)->make($cx * ($ey + $ey_1)/2,
791 $sx * ($ey_1 - $ey)/2);
797 # Compute sin(z) = (exp(iz) - exp(-iz))/2.
801 return CORE::sin($z) unless ref $z;
802 my ($x, $y) = @{$z->cartesian};
803 my $ey = CORE::exp($y);
804 my $sx = CORE::sin($x);
805 my $cx = CORE::cos($x);
806 my $ey_1 = $ey ? 1 / $ey : $Inf;
807 return (ref $z)->make($sx * ($ey + $ey_1)/2,
808 $cx * ($ey - $ey_1)/2);
814 # Compute tan(z) = sin(z) / cos(z).
819 _divbyzero "tan($z)", "cos($z)" if $cz == 0;
820 return &sin($z) / $cz;
826 # Computes the secant sec(z) = 1 / cos(z).
831 _divbyzero "sec($z)", "cos($z)" if ($cz == 0);
838 # Computes the cosecant csc(z) = 1 / sin(z).
843 _divbyzero "csc($z)", "sin($z)" if ($sz == 0);
852 sub cosec { Math::Complex::csc(@_) }
857 # Computes cot(z) = cos(z) / sin(z).
862 _divbyzero "cot($z)", "sin($z)" if ($sz == 0);
863 return &cos($z) / $sz;
871 sub cotan { Math::Complex::cot(@_) }
876 # Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
880 return CORE::atan2(CORE::sqrt(1-$z*$z), $z)
881 if (! ref $z) && CORE::abs($z) <= 1;
882 my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
883 return 0 if $x == 1 && $y == 0;
884 my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
885 my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
886 my $alpha = ($t1 + $t2)/2;
887 my $beta = ($t1 - $t2)/2;
888 $alpha = 1 if $alpha < 1;
889 if ($beta > 1) { $beta = 1 }
890 elsif ($beta < -1) { $beta = -1 }
891 my $u = CORE::atan2(CORE::sqrt(1-$beta*$beta), $beta);
892 my $v = CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
893 $v = -$v if $y > 0 || ($y == 0 && $x < -1);
894 return __PACKAGE__->make($u, $v);
900 # Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)).
904 return CORE::atan2($z, CORE::sqrt(1-$z*$z))
905 if (! ref $z) && CORE::abs($z) <= 1;
906 my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
907 return 0 if $x == 0 && $y == 0;
908 my $t1 = CORE::sqrt(($x+1)*($x+1) + $y*$y);
909 my $t2 = CORE::sqrt(($x-1)*($x-1) + $y*$y);
910 my $alpha = ($t1 + $t2)/2;
911 my $beta = ($t1 - $t2)/2;
912 $alpha = 1 if $alpha < 1;
913 if ($beta > 1) { $beta = 1 }
914 elsif ($beta < -1) { $beta = -1 }
915 my $u = CORE::atan2($beta, CORE::sqrt(1-$beta*$beta));
916 my $v = -CORE::log($alpha + CORE::sqrt($alpha*$alpha-1));
917 $v = -$v if $y > 0 || ($y == 0 && $x < -1);
918 return __PACKAGE__->make($u, $v);
924 # Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)).
928 return CORE::atan2($z, 1) unless ref $z;
929 my ($x, $y) = ref $z ? @{$z->cartesian} : ($z, 0);
930 return 0 if $x == 0 && $y == 0;
931 _divbyzero "atan(i)" if ( $z == i);
932 _logofzero "atan(-i)" if (-$z == i); # -i is a bad file test...
933 my $log = &log((i + $z) / (i - $z));
940 # Computes the arc secant asec(z) = acos(1 / z).
944 _divbyzero "asec($z)", $z if ($z == 0);
951 # Computes the arc cosecant acsc(z) = asin(1 / z).
955 _divbyzero "acsc($z)", $z if ($z == 0);
964 sub acosec { Math::Complex::acsc(@_) }
969 # Computes the arc cotangent acot(z) = atan(1 / z)
973 _divbyzero "acot(0)" if $z == 0;
974 return ($z >= 0) ? CORE::atan2(1, $z) : CORE::atan2(-1, -$z)
976 _divbyzero "acot(i)" if ($z - i == 0);
977 _logofzero "acot(-i)" if ($z + i == 0);
986 sub acotan { Math::Complex::acot(@_) }
991 # Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/2.
998 return $ex ? ($ex + 1/$ex)/2 : $Inf;
1000 my ($x, $y) = @{$z->cartesian};
1001 $ex = CORE::exp($x);
1002 my $ex_1 = $ex ? 1 / $ex : $Inf;
1003 return (ref $z)->make(CORE::cos($y) * ($ex + $ex_1)/2,
1004 CORE::sin($y) * ($ex - $ex_1)/2);
1010 # Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/2.
1016 return 0 if $z == 0;
1017 $ex = CORE::exp($z);
1018 return $ex ? ($ex - 1/$ex)/2 : "-$Inf";
1020 my ($x, $y) = @{$z->cartesian};
1021 my $cy = CORE::cos($y);
1022 my $sy = CORE::sin($y);
1023 $ex = CORE::exp($x);
1024 my $ex_1 = $ex ? 1 / $ex : $Inf;
1025 return (ref $z)->make(CORE::cos($y) * ($ex - $ex_1)/2,
1026 CORE::sin($y) * ($ex + $ex_1)/2);
1032 # Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z).
1037 _divbyzero "tanh($z)", "cosh($z)" if ($cz == 0);
1038 return sinh($z) / $cz;
1044 # Computes the hyperbolic secant sech(z) = 1 / cosh(z).
1049 _divbyzero "sech($z)", "cosh($z)" if ($cz == 0);
1056 # Computes the hyperbolic cosecant csch(z) = 1 / sinh(z).
1061 _divbyzero "csch($z)", "sinh($z)" if ($sz == 0);
1070 sub cosech { Math::Complex::csch(@_) }
1075 # Computes the hyperbolic cotangent coth(z) = cosh(z) / sinh(z).
1080 _divbyzero "coth($z)", "sinh($z)" if $sz == 0;
1081 return cosh($z) / $sz;
1089 sub cotanh { Math::Complex::coth(@_) }
1094 # Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).
1101 my ($re, $im) = @{$z->cartesian};
1103 return CORE::log($re + CORE::sqrt($re*$re - 1))
1105 return cplx(0, CORE::atan2(CORE::sqrt(1 - $re*$re), $re))
1106 if CORE::abs($re) < 1;
1108 my $t = &sqrt($z * $z - 1) + $z;
1109 # Try MacLaurin if looking bad.
1110 $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
1113 $u->Im(-$u->Im) if $im == 0;
1114 return $re < 0 ? -$u : $u;
1120 # Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1))
1125 my $t = $z + CORE::sqrt($z*$z + 1);
1126 return CORE::log($t) if $t;
1128 my $t = &sqrt($z * $z + 1) + $z;
1129 # Try MacLaurin if looking bad.
1130 $t = 1/(2 * $z) - 1/(8 * $z**3) + 1/(16 * $z**5) - 5/(128 * $z**7)
1138 # Computes the arc hyperbolic tangent atanh(z) = 1/2 log((1+z) / (1-z)).
1143 return CORE::log((1 + $z)/(1 - $z))/2 if CORE::abs($z) < 1;
1146 _divbyzero 'atanh(1)', "1 - $z" if (1 - $z == 0);
1147 _logofzero 'atanh(-1)' if (1 + $z == 0);
1148 return 0.5 * &log((1 + $z) / (1 - $z));
1154 # Computes the hyperbolic arc secant asech(z) = acosh(1 / z).
1158 _divbyzero 'asech(0)', "$z" if ($z == 0);
1159 return acosh(1 / $z);
1165 # Computes the hyperbolic arc cosecant acsch(z) = asinh(1 / z).
1169 _divbyzero 'acsch(0)', $z if ($z == 0);
1170 return asinh(1 / $z);
1176 # Alias for acosh().
1178 sub acosech { Math::Complex::acsch(@_) }
1183 # Computes the arc hyperbolic cotangent acoth(z) = 1/2 log((1+z) / (z-1)).
1187 _divbyzero 'acoth(0)' if ($z == 0);
1189 return CORE::log(($z + 1)/($z - 1))/2 if CORE::abs($z) > 1;
1192 _divbyzero 'acoth(1)', "$z - 1" if ($z - 1 == 0);
1193 _logofzero 'acoth(-1)', "1 + $z" if (1 + $z == 0);
1194 return &log((1 + $z) / ($z - 1)) / 2;
1202 sub acotanh { Math::Complex::acoth(@_) }
1207 # Compute atan(z1/z2).
1210 my ($z1, $z2, $inverted) = @_;
1211 my ($re1, $im1, $re2, $im2);
1213 ($re1, $im1) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
1214 ($re2, $im2) = @{$z1->cartesian};
1216 ($re1, $im1) = @{$z1->cartesian};
1217 ($re2, $im2) = ref $z2 ? @{$z2->cartesian} : ($z2, 0);
1220 return CORE::atan2($re1, $re2) if $im1 == 0;
1221 return ($im1<=>0) * pip2 if $re2 == 0;
1223 my $w = atan($z1/$z2);
1224 my ($u, $v) = ref $w ? @{$w->cartesian} : ($w, 0);
1225 $u += pi if $re2 < 0;
1226 $u -= pit2 if $u > pi;
1227 return cplx($u, $v);
1234 # Set (get if no argument) the display format for all complex numbers that
1235 # don't happen to have overridden it via ->display_format
1237 # When called as an object method, this actually sets the display format for
1238 # the current object.
1240 # Valid object formats are 'c' and 'p' for cartesian and polar. The first
1241 # letter is used actually, so the type can be fully spelled out for clarity.
1243 sub display_format {
1245 my %display_format = %DISPLAY_FORMAT;
1247 if (ref $self) { # Called as an object method
1248 if (exists $self->{display_format}) {
1249 my %obj = %{$self->{display_format}};
1250 @display_format{keys %obj} = values %obj;
1253 $display_format{style} = shift;
1256 @display_format{keys %new} = values %new;
1258 } else { # Called as a class method
1260 $display_format{style} = $self;
1263 @display_format{keys %new} = values %new;
1268 if (defined $self) {
1269 $self->{display_format} = { %display_format };
1272 %{$self->{display_format}} :
1273 $self->{display_format}->{style};
1276 %DISPLAY_FORMAT = %display_format;
1280 $DISPLAY_FORMAT{style};
1286 # Show nicely formatted complex number under its cartesian or polar form,
1287 # depending on the current display format:
1289 # . If a specific display format has been recorded for this object, use it.
1290 # . Otherwise, use the generic current default for all complex numbers,
1291 # which is a package global variable.
1296 my $style = $z->display_format;
1298 $style = $DISPLAY_FORMAT{style} unless defined $style;
1300 return $z->stringify_polar if $style =~ /^p/i;
1301 return $z->stringify_cartesian;
1305 # ->stringify_cartesian
1307 # Stringify as a cartesian representation 'a+bi'.
1309 sub stringify_cartesian {
1311 my ($x, $y) = @{$z->cartesian};
1314 my %format = $z->display_format;
1315 my $format = $format{format};
1318 if ($x =~ /^NaN[QS]?$/i) {
1321 if ($x =~ /^-?$Inf$/oi) {
1324 $re = defined $format ? sprintf($format, $x) : $x;
1332 if ($y == 1) { $im = "" }
1333 elsif ($y == -1) { $im = "-" }
1334 elsif ($y =~ /^(NaN[QS]?)$/i) {
1337 if ($y =~ /^-?$Inf$/oi) {
1340 $im = defined $format ? sprintf($format, $y) : $y;
1353 } elsif ($y > 0 || $im =~ /^NaN[QS]?i$/i) {
1354 $str .= "+" if defined $re;
1357 } elsif (!defined $re) {
1368 # Stringify as a polar representation '[r,t]'.
1370 sub stringify_polar {
1372 my ($r, $t) = @{$z->polar};
1375 my %format = $z->display_format;
1376 my $format = $format{format};
1378 if ($t =~ /^NaN[QS]?$/i || $t =~ /^-?$Inf$/oi) {
1380 } elsif ($t == pi) {
1382 } elsif ($r == 0 || $t == 0) {
1383 $theta = defined $format ? sprintf($format, $t) : $t;
1386 return "[$r,$theta]" if defined $theta;
1389 # Try to identify pi/n and friends.
1392 $t -= int(CORE::abs($t) / pit2) * pit2;
1394 if ($format{polar_pretty_print}) {
1398 if (int($b) == $b) {
1399 $b = $b < 0 ? "-" : "" if CORE::abs($b) == 1;
1400 $theta = "${b}pi/$a";
1406 if (defined $format) {
1407 $r = sprintf($format, $r);
1408 $theta = sprintf($format, $theta) unless defined $theta;
1410 $theta = $t unless defined $theta;
1413 return "[$r,$theta]";
1421 Math::Complex - complex numbers and associated mathematical functions
1427 $z = Math::Complex->make(5, 6);
1429 $j = cplxe(1, 2*pi/3);
1433 This package lets you create and manipulate complex numbers. By default,
1434 I<Perl> limits itself to real numbers, but an extra C<use> statement brings
1435 full complex support, along with a full set of mathematical functions
1436 typically associated with and/or extended to complex numbers.
1438 If you wonder what complex numbers are, they were invented to be able to solve
1439 the following equation:
1443 and by definition, the solution is noted I<i> (engineers use I<j> instead since
1444 I<i> usually denotes an intensity, but the name does not matter). The number
1445 I<i> is a pure I<imaginary> number.
1447 The arithmetics with pure imaginary numbers works just like you would expect
1448 it with real numbers... you just have to remember that
1454 5i + 7i = i * (5 + 7) = 12i
1455 4i - 3i = i * (4 - 3) = i
1460 Complex numbers are numbers that have both a real part and an imaginary
1461 part, and are usually noted:
1465 where C<a> is the I<real> part and C<b> is the I<imaginary> part. The
1466 arithmetic with complex numbers is straightforward. You have to
1467 keep track of the real and the imaginary parts, but otherwise the
1468 rules used for real numbers just apply:
1470 (4 + 3i) + (5 - 2i) = (4 + 5) + i(3 - 2) = 9 + i
1471 (2 + i) * (4 - i) = 2*4 + 4i -2i -i*i = 8 + 2i + 1 = 9 + 2i
1473 A graphical representation of complex numbers is possible in a plane
1474 (also called the I<complex plane>, but it's really a 2D plane).
1479 is the point whose coordinates are (a, b). Actually, it would
1480 be the vector originating from (0, 0) to (a, b). It follows that the addition
1481 of two complex numbers is a vectorial addition.
1483 Since there is a bijection between a point in the 2D plane and a complex
1484 number (i.e. the mapping is unique and reciprocal), a complex number
1485 can also be uniquely identified with polar coordinates:
1489 where C<rho> is the distance to the origin, and C<theta> the angle between
1490 the vector and the I<x> axis. There is a notation for this using the
1491 exponential form, which is:
1493 rho * exp(i * theta)
1495 where I<i> is the famous imaginary number introduced above. Conversion
1496 between this form and the cartesian form C<a + bi> is immediate:
1498 a = rho * cos(theta)
1499 b = rho * sin(theta)
1501 which is also expressed by this formula:
1503 z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
1505 In other words, it's the projection of the vector onto the I<x> and I<y>
1506 axes. Mathematicians call I<rho> the I<norm> or I<modulus> and I<theta>
1507 the I<argument> of the complex number. The I<norm> of C<z> will be
1510 The polar notation (also known as the trigonometric
1511 representation) is much more handy for performing multiplications and
1512 divisions of complex numbers, whilst the cartesian notation is better
1513 suited for additions and subtractions. Real numbers are on the I<x>
1514 axis, and therefore I<theta> is zero or I<pi>.
1516 All the common operations that can be performed on a real number have
1517 been defined to work on complex numbers as well, and are merely
1518 I<extensions> of the operations defined on real numbers. This means
1519 they keep their natural meaning when there is no imaginary part, provided
1520 the number is within their definition set.
1522 For instance, the C<sqrt> routine which computes the square root of
1523 its argument is only defined for non-negative real numbers and yields a
1524 non-negative real number (it is an application from B<R+> to B<R+>).
1525 If we allow it to return a complex number, then it can be extended to
1526 negative real numbers to become an application from B<R> to B<C> (the
1527 set of complex numbers):
1529 sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
1531 It can also be extended to be an application from B<C> to B<C>,
1532 whilst its restriction to B<R> behaves as defined above by using
1533 the following definition:
1535 sqrt(z = [r,t]) = sqrt(r) * exp(i * t/2)
1537 Indeed, a negative real number can be noted C<[x,pi]> (the modulus
1538 I<x> is always non-negative, so C<[x,pi]> is really C<-x>, a negative
1539 number) and the above definition states that
1541 sqrt([x,pi]) = sqrt(x) * exp(i*pi/2) = [sqrt(x),pi/2] = sqrt(x)*i
1543 which is exactly what we had defined for negative real numbers above.
1544 The C<sqrt> returns only one of the solutions: if you want the both,
1545 use the C<root> function.
1547 All the common mathematical functions defined on real numbers that
1548 are extended to complex numbers share that same property of working
1549 I<as usual> when the imaginary part is zero (otherwise, it would not
1550 be called an extension, would it?).
1552 A I<new> operation possible on a complex number that is
1553 the identity for real numbers is called the I<conjugate>, and is noted
1554 with an horizontal bar above the number, or C<~z> here.
1561 z * ~z = (a + bi) * (a - bi) = a*a + b*b
1563 We saw that the norm of C<z> was noted C<abs(z)> and was defined as the
1564 distance to the origin, also known as:
1566 rho = abs(z) = sqrt(a*a + b*b)
1570 z * ~z = abs(z) ** 2
1572 If z is a pure real number (i.e. C<b == 0>), then the above yields:
1576 which is true (C<abs> has the regular meaning for real number, i.e. stands
1577 for the absolute value). This example explains why the norm of C<z> is
1578 noted C<abs(z)>: it extends the C<abs> function to complex numbers, yet
1579 is the regular C<abs> we know when the complex number actually has no
1580 imaginary part... This justifies I<a posteriori> our use of the C<abs>
1581 notation for the norm.
1585 Given the following notations:
1587 z1 = a + bi = r1 * exp(i * t1)
1588 z2 = c + di = r2 * exp(i * t2)
1589 z = <any complex or real number>
1591 the following (overloaded) operations are supported on complex numbers:
1593 z1 + z2 = (a + c) + i(b + d)
1594 z1 - z2 = (a - c) + i(b - d)
1595 z1 * z2 = (r1 * r2) * exp(i * (t1 + t2))
1596 z1 / z2 = (r1 / r2) * exp(i * (t1 - t2))
1597 z1 ** z2 = exp(z2 * log z1)
1599 abs(z) = r1 = sqrt(a*a + b*b)
1600 sqrt(z) = sqrt(r1) * exp(i * t/2)
1601 exp(z) = exp(a) * exp(i * b)
1602 log(z) = log(r1) + i*t
1603 sin(z) = 1/2i (exp(i * z1) - exp(-i * z))
1604 cos(z) = 1/2 (exp(i * z1) + exp(-i * z))
1605 atan2(z1, z2) = atan(z1/z2)
1607 The following extra operations are supported on both real and complex
1615 cbrt(z) = z ** (1/3)
1616 log10(z) = log(z) / log(10)
1617 logn(z, n) = log(z) / log(n)
1619 tan(z) = sin(z) / cos(z)
1625 asin(z) = -i * log(i*z + sqrt(1-z*z))
1626 acos(z) = -i * log(z + i*sqrt(1-z*z))
1627 atan(z) = i/2 * log((i+z) / (i-z))
1629 acsc(z) = asin(1 / z)
1630 asec(z) = acos(1 / z)
1631 acot(z) = atan(1 / z) = -i/2 * log((i+z) / (z-i))
1633 sinh(z) = 1/2 (exp(z) - exp(-z))
1634 cosh(z) = 1/2 (exp(z) + exp(-z))
1635 tanh(z) = sinh(z) / cosh(z) = (exp(z) - exp(-z)) / (exp(z) + exp(-z))
1637 csch(z) = 1 / sinh(z)
1638 sech(z) = 1 / cosh(z)
1639 coth(z) = 1 / tanh(z)
1641 asinh(z) = log(z + sqrt(z*z+1))
1642 acosh(z) = log(z + sqrt(z*z-1))
1643 atanh(z) = 1/2 * log((1+z) / (1-z))
1645 acsch(z) = asinh(1 / z)
1646 asech(z) = acosh(1 / z)
1647 acoth(z) = atanh(1 / z) = 1/2 * log((1+z) / (z-1))
1649 I<arg>, I<abs>, I<log>, I<csc>, I<cot>, I<acsc>, I<acot>, I<csch>,
1650 I<coth>, I<acosech>, I<acotanh>, have aliases I<rho>, I<theta>, I<ln>,
1651 I<cosec>, I<cotan>, I<acosec>, I<acotan>, I<cosech>, I<cotanh>,
1652 I<acosech>, I<acotanh>, respectively. C<Re>, C<Im>, C<arg>, C<abs>,
1653 C<rho>, and C<theta> can be used also also mutators. The C<cbrt>
1654 returns only one of the solutions: if you want all three, use the
1657 The I<root> function is available to compute all the I<n>
1658 roots of some complex, where I<n> is a strictly positive integer.
1659 There are exactly I<n> such roots, returned as a list. Getting the
1660 number mathematicians call C<j> such that:
1664 is a simple matter of writing:
1666 $j = ((root(1, 3))[1];
1668 The I<k>th root for C<z = [r,t]> is given by:
1670 (root(z, n))[k] = r**(1/n) * exp(i * (t + 2*k*pi)/n)
1672 The I<spaceship> comparison operator, E<lt>=E<gt>, is also defined. In
1673 order to ensure its restriction to real numbers is conform to what you
1674 would expect, the comparison is run on the real part of the complex
1675 number first, and imaginary parts are compared only when the real
1680 To create a complex number, use either:
1682 $z = Math::Complex->make(3, 4);
1685 if you know the cartesian form of the number, or
1689 if you like. To create a number using the polar form, use either:
1691 $z = Math::Complex->emake(5, pi/3);
1692 $x = cplxe(5, pi/3);
1694 instead. The first argument is the modulus, the second is the angle
1695 (in radians, the full circle is 2*pi). (Mnemonic: C<e> is used as a
1696 notation for complex numbers in the polar form).
1698 It is possible to write:
1700 $x = cplxe(-3, pi/4);
1702 but that will be silently converted into C<[3,-3pi/4]>, since the
1703 modulus must be non-negative (it represents the distance to the origin
1704 in the complex plane).
1706 It is also possible to have a complex number as either argument of
1707 either the C<make> or C<emake>: the appropriate component of
1708 the argument will be used.
1713 =head1 STRINGIFICATION
1715 When printed, a complex number is usually shown under its cartesian
1716 style I<a+bi>, but there are legitimate cases where the polar style
1717 I<[r,t]> is more appropriate.
1719 By calling the class method C<Math::Complex::display_format> and
1720 supplying either C<"polar"> or C<"cartesian"> as an argument, you
1721 override the default display style, which is C<"cartesian">. Not
1722 supplying any argument returns the current settings.
1724 This default can be overridden on a per-number basis by calling the
1725 C<display_format> method instead. As before, not supplying any argument
1726 returns the current display style for this number. Otherwise whatever you
1727 specify will be the new display style for I<this> particular number.
1733 Math::Complex::display_format('polar');
1734 $j = (root(1, 3))[1];
1735 print "j = $j\n"; # Prints "j = [1,2pi/3]"
1736 $j->display_format('cartesian');
1737 print "j = $j\n"; # Prints "j = -0.5+0.866025403784439i"
1739 The polar style attempts to emphasize arguments like I<k*pi/n>
1740 (where I<n> is a positive integer and I<k> an integer within [-9, +9]),
1741 this is called I<polar pretty-printing>.
1743 =head2 CHANGED IN PERL 5.6
1745 The C<display_format> class method and the corresponding
1746 C<display_format> object method can now be called using
1747 a parameter hash instead of just a one parameter.
1749 The old display format style, which can have values C<"cartesian"> or
1750 C<"polar">, can be changed using the C<"style"> parameter. (The one
1751 parameter calling convention also still works.)
1753 There are two new display parameters.
1755 The first one is C<"format">, which is a sprintf()-style format
1756 string to be used for both parts of the complex number(s). The
1757 default is C<undef>, which corresponds usually (this is somewhat
1758 system-dependent) to C<"%.15g">. You can revert to the default by
1759 setting the format string to C<undef>.
1761 # the $j from the above example
1763 $j->display_format('format' => '%.5f');
1764 print "j = $j\n"; # Prints "j = -0.50000+0.86603i"
1765 $j->display_format('format' => '%.6f');
1766 print "j = $j\n"; # Prints "j = -0.5+0.86603i"
1768 Notice that this affects also the return values of the
1769 C<display_format> methods: in list context the whole parameter hash
1770 will be returned, as opposed to only the style parameter value. If
1771 you want to know the whole truth for a complex number, you must call
1772 both the class method and the object method:
1774 The second new display parameter is C<"polar_pretty_print">, which can
1775 be set to true or false, the default being true. See the previous
1776 section for what this means.
1780 Thanks to overloading, the handling of arithmetics with complex numbers
1781 is simple and almost transparent.
1783 Here are some examples:
1787 $j = cplxe(1, 2*pi/3); # $j ** 3 == 1
1788 print "j = $j, j**3 = ", $j ** 3, "\n";
1789 print "1 + j + j**2 = ", 1 + $j + $j**2, "\n";
1791 $z = -16 + 0*i; # Force it to be a complex
1792 print "sqrt($z) = ", sqrt($z), "\n";
1794 $k = exp(i * 2*pi/3);
1795 print "$j - $k = ", $j - $k, "\n";
1797 $z->Re(3); # Re, Im, arg, abs,
1798 $j->arg(2); # (the last two aka rho, theta)
1799 # can be used also as mutators.
1801 =head1 ERRORS DUE TO DIVISION BY ZERO OR LOGARITHM OF ZERO
1803 The division (/) and the following functions
1809 atanh asech acsch acoth
1811 cannot be computed for all arguments because that would mean dividing
1812 by zero or taking logarithm of zero. These situations cause fatal
1813 runtime errors looking like this
1815 cot(0): Division by zero.
1816 (Because in the definition of cot(0), the divisor sin(0) is 0)
1821 atanh(-1): Logarithm of zero.
1824 For the C<csc>, C<cot>, C<asec>, C<acsc>, C<acot>, C<csch>, C<coth>,
1825 C<asech>, C<acsch>, the argument cannot be C<0> (zero). For the the
1826 logarithmic functions and the C<atanh>, C<acoth>, the argument cannot
1827 be C<1> (one). For the C<atanh>, C<acoth>, the argument cannot be
1828 C<-1> (minus one). For the C<atan>, C<acot>, the argument cannot be
1829 C<i> (the imaginary unit). For the C<atan>, C<acoth>, the argument
1830 cannot be C<-i> (the negative imaginary unit). For the C<tan>,
1831 C<sec>, C<tanh>, the argument cannot be I<pi/2 + k * pi>, where I<k>
1834 Note that because we are operating on approximations of real numbers,
1835 these errors can happen when merely `too close' to the singularities
1836 listed above. For example C<tan(2*atan2(1,1)+1e-15)> will die of
1839 =head1 ERRORS DUE TO INDIGESTIBLE ARGUMENTS
1841 The C<make> and C<emake> accept both real and complex arguments.
1842 When they cannot recognize the arguments they will die with error
1843 messages like the following
1845 Math::Complex::make: Cannot take real part of ...
1846 Math::Complex::make: Cannot take real part of ...
1847 Math::Complex::emake: Cannot take rho of ...
1848 Math::Complex::emake: Cannot take theta of ...
1852 Saying C<use Math::Complex;> exports many mathematical routines in the
1853 caller environment and even overrides some (C<sqrt>, C<log>).
1854 This is construed as a feature by the Authors, actually... ;-)
1856 All routines expect to be given real or complex numbers. Don't attempt to
1857 use BigFloat, since Perl has currently no rule to disambiguate a '+'
1858 operation (for instance) between two overloaded entities.
1860 In Cray UNICOS there is some strange numerical instability that results
1861 in root(), cos(), sin(), cosh(), sinh(), losing accuracy fast. Beware.
1862 The bug may be in UNICOS math libs, in UNICOS C compiler, in Math::Complex.
1863 Whatever it is, it does not manifest itself anywhere else where Perl runs.
1867 Raphael Manfredi <F<Raphael_Manfredi@pobox.com>> and
1868 Jarkko Hietaniemi <F<jhi@iki.fi>>.
1870 Extensive patches by Daniel S. Lewart <F<d-lewart@uiuc.edu>>.