NAME


            
              
              Math::Cephes::Complex - Perl interface to the cephes complex number routines

SYNOPSIS


            
              
              use Math::Cephes::Complex qw(cmplx);
my $z1 = cmplx(2,3);          # $z1 = 2 + 3 i
my $z2 = cmplx(3,4);          # $z2 = 3 + 4 i
my $z3 = $z1->radd($z2);      # $z3 = $z1 + $z2

DESCRIPTION

This module is a layer on top of the basic routines in the cephes math library to handle complex numbers. A complex number is created via any of the following syntaxes:


            
              
              my $f = Math::Cephes::Complex->new(3, 2);   # $f = 3 + 2 i
my $g = new Math::Cephes::Complex(5, 3);    # $g = 5 + 3 i
my $h = cmplx(7, 5);                        # $h = 7 + 5 i

the last one being available by importing cmplx. If no arguments are specified, as in


            
              
              my $h = cmplx();

then the defaults $z = 0 + 0 i are assumed. The real and imaginary part of a complex number are represented respectively by


            
              
              $f->{r}; $f->{i};

or, as methods,


            
              
              $f->r;  $f->i;

and can be set according to


            
              
              $f->{r} = 4; $f->{i} = 9;

or, again, as methods,


            
              
              $f->r(4);   $f->i(9);

The complex number can be printed out as


            
              
              print $f->as_string;

A summary of the usage is as follows.

csin: Complex circular sine


            
              
              SYNOPSIS:
# void csin();
# cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->csin;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
DESCRIPTION:
If
    z = x + iy,
then
    w = sin x  cosh y  +  i cos x sinh y.

ccos: Complex circular cosine


            
              
              SYNOPSIS:
# void ccos();
# cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->ccos;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
DESCRIPTION:
If
    z = x + iy,
then
    w = cos x  cosh y  -  i sin x sinh y.

ctan: Complex circular tangent


            
              
              SYNOPSIS:
# void ctan();
# cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->ctan;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
DESCRIPTION:
If
    z = x + iy,
then
          sin 2x  +  i sinh 2y
    w  =  --------------------.
           cos 2x  +  cosh 2y
On the real axis the denominator is zero at odd multiples
of PI/2.  The denominator is evaluated by its Taylor
series near these points.

ccot: Complex circular cotangent


            
              
              SYNOPSIS:
# void ccot();
# cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->ccot;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
DESCRIPTION:
If
    z = x + iy,
then
          sin 2x  -  i sinh 2y
    w  =  --------------------.
           cosh 2y  -  cos 2x
On the real axis, the denominator has zeros at even
multiples of PI/2.  Near these points it is evaluated
by a Taylor series.

casin: Complex circular arc sine


            
              
              SYNOPSIS:
# void casin();
# cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->casin;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
DESCRIPTION:
Inverse complex sine:
                              2
w = -i clog( iz + csqrt( 1 - z ) ).

cacos: Complex circular arc cosine


            
              
              SYNOPSIS:
# void cacos();
# cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->cacos;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
DESCRIPTION:
w = arccos z  =  PI/2 - arcsin z.

catan: Complex circular arc tangent


            
              
              SYNOPSIS:
# void catan();
# cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->catan;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
DESCRIPTION:
If
    z = x + iy,
then
         1       (    2x     )
Re w  =  - arctan(-----------)  +  k PI
         2       (     2    2)
                 (1 - x  - y )
              ( 2         2)
         1    (x  +  (y+1) )
Im w  =  - log(------------)
         4    ( 2         2)
              (x  +  (y-1) )
Where k is an arbitrary integer.

csinh: Complex hyperbolic sine


            
              
               SYNOPSIS:
 # void csinh();
 # cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->csinh;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
 DESCRIPTION:
 csinh z = (cexp(z) - cexp(-z))/2
         = sinh x * cos y  +  i cosh x * sin y .

casinh: Complex inverse hyperbolic sine


            
              
               SYNOPSIS:
 # void casinh();
 # cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->casinh;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
 DESCRIPTION:
 casinh z = -i casin iz .

ccosh: Complex hyperbolic cosine


            
              
               SYNOPSIS:
 # void ccosh();
 # cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->ccosh;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
 DESCRIPTION:
 ccosh(z) = cosh x  cos y + i sinh x sin y .

cacosh: Complex inverse hyperbolic cosine


            
              
               SYNOPSIS:
 # void cacosh();
 # cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->cacosh;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
 DESCRIPTION:
 acosh z = i acos z .

ctanh: Complex hyperbolic tangent


            
              
              SYNOPSIS:
# void ctanh();
# cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->ctanh;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
DESCRIPTION:
tanh z = (sinh 2x  +  i sin 2y) / (cosh 2x + cos 2y) .

catanh: Complex inverse hyperbolic tangent


            
              
               SYNOPSIS:
 # void catanh();
 # cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->catanh;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
 DESCRIPTION:
 Inverse tanh, equal to  -i catan (iz);

cpow: Complex power function


            
              
               SYNOPSIS:
 # void cpow();
 # cmplx a, z, w;
$a = cmplx(5, 6);    # $z = 5 + 6 i
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $a->cpow($z);
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
 DESCRIPTION:
 Raises complex A to the complex Zth power.
 Definition is per AMS55 # 4.2.8,
 analytically equivalent to cpow(a,z) = cexp(z clog(a)).

cmplx: Complex number arithmetic


            
              
              SYNOPSIS:
# typedef struct {
#     double r;     real part
#     double i;     imaginary part
#    }cmplx;
# cmplx *a, *b, *c;
$a = cmplx(3, 5);   # $a = 3 + 5 i
$b = cmplx(2, 3);   # $b = 2 + 3 i
$c = $a->cadd( $b );  #   c = a + b
$c = $a->csub( $b );  #   c = a - b
$c = $a->cmul( $b );  #   c = a * b
$c = $a->cdiv( $b );  #   c = a / b
$c = $a->cneg;        #   c = -a
$c = $a->cmov;        #   c = a
print $c->{r}, '  ', $c->{i};   # prints real and imaginary parts of $c
print $c->as_string;           # prints $c as Re($c) + i Im($c)
DESCRIPTION:
Addition:
   c.r  =  b.r + a.r
   c.i  =  b.i + a.i
Subtraction:
   c.r  =  b.r - a.r
   c.i  =  b.i - a.i
Multiplication:
   c.r  =  b.r * a.r  -  b.i * a.i
   c.i  =  b.r * a.i  +  b.i * a.r
Division:
   d    =  a.r * a.r  +  a.i * a.i
   c.r  = (b.r * a.r  + b.i * a.i)/d
   c.i  = (b.i * a.r  -  b.r * a.i)/d

cabs: Complex absolute value


            
              
              SYNOPSIS:
# double a, cabs();
# cmplx z;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$a = cabs( $z );
DESCRIPTION:
If z = x + iy
then
      a = sqrt( x**2 + y**2 ).
Overflow and underflow are avoided by testing the magnitudes
of x and y before squaring.  If either is outside half of
the floating point full scale range, both are rescaled.

csqrt: Complex square root


            
              
              SYNOPSIS:
# void csqrt();
# cmplx z, w;
$z = cmplx(2, 3);    # $z = 2 + 3 i
$w = $z->csqrt;
print $w->{r}, '  ', $w->{i};  # prints real and imaginary parts of $w
print $w->as_string;           # prints $w as Re($w) + i Im($w)
DESCRIPTION:
If z = x + iy,  r = |z|, then
                      1/2
Im w  =  [ (r - x)/2 ]   ,
Re w  =  y / 2 Im w.
Note that -w is also a square root of z.  The root chosen
is always in the upper half plane.
Because of the potential for cancellation error in r - x,
the result is sharpened by doing a Heron iteration
(see sqrt.c) in complex arithmetic.

BUGS


            
              
              Please report any to Randy Kobes <randy@theoryx5.uwinnipeg.ca>

COPYRIGHT

The C code for the Cephes Math Library is Copyright 1984, 1987, 1989, 2002 by Stephen L. Moshier, and is available at http://www.netlib.org/cephes/. Direct inquiries to 30 Frost Street, Cambridge, MA 02140.

To install Math::Cephes, copy and paste the appropriate command in to your terminal.

cpanm

cpanm Math::Cephes

CPAN shell

perl -MCPAN -e shell
install Math::Cephes

For more information on module installation, please visit the detailed CPAN module installation guide.

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