# 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:

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>``