Math::Cephes::Complex(3pm)

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 <[email protected]>

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.