---
name: "Math::Cephes::Complex"
section: "3pm"
platform: "linux"
package: "libmath-cephes-perl"
version: "0.5307-1"
release: "Debian 13 (Trixie)"
platforms: ["linux"]
url: "https://new.carta.tech/Math::Cephes::Complex"
---

# 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 <randy@theoryx5.uwinnipeg.ca>
```

## SEE ALSO

For the basic interface to the cephes complex number routines, see Math::Cephes. See also Math::Complex for a more extensive interface to complex number routines.

## 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.

The perl interface is copyright 2000, 2002 by Randy Kobes. This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself.