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.\" ========================================================================
.\"
.IX Title "Math::Cephes::Complex 3pm"
.TH Math::Cephes::Complex 3pm "2021-06-22" "perl v5.32.1" "User Contributed Perl Documentation"
.\" For nroff, turn off justification.  Always turn off hyphenation; it makes
.\" way too many mistakes in technical documents.
.if n .ad l
.nh
.SH "NAME"
.Vb 1
\&  Math::Cephes::Complex \- Perl interface to the cephes complex number routines
.Ve
.SH "SYNOPSIS"
.IX Header "SYNOPSIS"
.Vb 4
\&  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
.Ve
.SH "DESCRIPTION"
.IX Header "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:
.PP
.Vb 3
\&  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
.Ve
.PP
the last one being available by importing \fIcmplx\fR. If no arguments
are specified, as in
.PP
.Vb 1
\& my $h = cmplx();
.Ve
.PP
then the defaults \f(CW$z\fR = 0 + 0 i are assumed. The real and imaginary
part of a complex number are represented respectively by
.PP
.Vb 1
\&   $f\->{r}; $f\->{i};
.Ve
.PP
or, as methods,
.PP
.Vb 1
\&   $f\->r;  $f\->i;
.Ve
.PP
and can be set according to
.PP
.Vb 1
\&  $f\->{r} = 4; $f\->{i} = 9;
.Ve
.PP
or, again, as methods,
.PP
.Vb 1
\&  $f\->r(4);   $f\->i(9);
.Ve
.PP
The complex number can be printed out as
.PP
.Vb 1
\&  print $f\->as_string;
.Ve
.PP
A summary of the usage is as follows.
.IP "\fIcsin\fR: Complex circular sine" 4
.IX Item "csin: Complex circular sine"
.Vb 1
\& SYNOPSIS:
\&
\& # void csin();
\& # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->csin;
\& print $w\->{r}, \*(Aq  \*(Aq, $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.
.Ve
.IP "\fIccos\fR: Complex circular cosine" 4
.IX Item "ccos: Complex circular cosine"
.Vb 1
\& SYNOPSIS:
\&
\& # void ccos();
\& # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->ccos;
\& print $w\->{r}, \*(Aq  \*(Aq, $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.
.Ve
.IP "\fIctan\fR: Complex circular tangent" 4
.IX Item "ctan: Complex circular tangent"
.Vb 1
\& SYNOPSIS:
\&
\& # void ctan();
\& # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->ctan;
\& print $w\->{r}, \*(Aq  \*(Aq, $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.
.Ve
.IP "\fIccot\fR: Complex circular cotangent" 4
.IX Item "ccot: Complex circular cotangent"
.Vb 1
\& SYNOPSIS:
\&
\& # void ccot();
\& # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->ccot;
\& print $w\->{r}, \*(Aq  \*(Aq, $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.
.Ve
.IP "\fIcasin\fR: Complex circular arc sine" 4
.IX Item "casin: Complex circular arc sine"
.Vb 1
\& SYNOPSIS:
\&
\& # void casin();
\& # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->casin;
\& print $w\->{r}, \*(Aq  \*(Aq, $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 ) ).
.Ve
.IP "\fIcacos\fR: Complex circular arc cosine" 4
.IX Item "cacos: Complex circular arc cosine"
.Vb 1
\& SYNOPSIS:
\&
\& # void cacos();
\& # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->cacos;
\& print $w\->{r}, \*(Aq  \*(Aq, $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.
.Ve
.IP "\fIcatan\fR: Complex circular arc tangent" 4
.IX Item "catan: Complex circular arc tangent"
.Vb 1
\& SYNOPSIS:
\&
\& # void catan();
\& # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->catan;
\& print $w\->{r}, \*(Aq  \*(Aq, $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.
.Ve
.IP "\fIcsinh\fR: Complex hyperbolic sine" 4
.IX Item "csinh: Complex hyperbolic sine"
.Vb 1
\&  SYNOPSIS:
\&
\&  # void csinh();
\&  # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->csinh;
\& print $w\->{r}, \*(Aq  \*(Aq, $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 .
.Ve
.IP "\fIcasinh\fR: Complex inverse hyperbolic sine" 4
.IX Item "casinh: Complex inverse hyperbolic sine"
.Vb 1
\&  SYNOPSIS:
\&
\&  # void casinh();
\&  # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->casinh;
\& print $w\->{r}, \*(Aq  \*(Aq, $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 .
.Ve
.IP "\fIccosh\fR: Complex hyperbolic cosine" 4
.IX Item "ccosh: Complex hyperbolic cosine"
.Vb 1
\&  SYNOPSIS:
\&
\&  # void ccosh();
\&  # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->ccosh;
\& print $w\->{r}, \*(Aq  \*(Aq, $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 .
.Ve
.IP "\fIcacosh\fR: Complex inverse hyperbolic cosine" 4
.IX Item "cacosh: Complex inverse hyperbolic cosine"
.Vb 1
\&  SYNOPSIS:
\&
\&  # void cacosh();
\&  # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->cacosh;
\& print $w\->{r}, \*(Aq  \*(Aq, $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 .
.Ve
.IP "\fIctanh\fR: Complex hyperbolic tangent" 4
.IX Item "ctanh: Complex hyperbolic tangent"
.Vb 1
\& SYNOPSIS:
\&
\& # void ctanh();
\& # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->ctanh;
\& print $w\->{r}, \*(Aq  \*(Aq, $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) .
.Ve
.IP "\fIcatanh\fR: Complex inverse hyperbolic tangent" 4
.IX Item "catanh: Complex inverse hyperbolic tangent"
.Vb 1
\&  SYNOPSIS:
\&
\&  # void catanh();
\&  # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->catanh;
\& print $w\->{r}, \*(Aq  \*(Aq, $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);
.Ve
.IP "\fIcpow\fR: Complex power function" 4
.IX Item "cpow: Complex power function"
.Vb 1
\&  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}, \*(Aq  \*(Aq, $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)).
.Ve
.IP "\fIcmplx\fR: Complex number arithmetic" 4
.IX Item "cmplx: Complex number arithmetic"
.Vb 1
\& 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}, \*(Aq  \*(Aq, $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
.Ve
.IP "\fIcabs\fR: Complex absolute value" 4
.IX Item "cabs: Complex absolute value"
.Vb 1
\& 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.
.Ve
.IP "\fIcsqrt\fR: Complex square root" 4
.IX Item "csqrt: Complex square root"
.Vb 1
\& SYNOPSIS:
\&
\& # void csqrt();
\& # cmplx z, w;
\&
\& $z = cmplx(2, 3);    # $z = 2 + 3 i
\& $w = $z\->csqrt;
\& print $w\->{r}, \*(Aq  \*(Aq, $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.
.Ve
.SH "BUGS"
.IX Header "BUGS"
.Vb 1
\& Please report any to Randy Kobes <randy@theoryx5.uwinnipeg.ca>
.Ve
.SH "SEE ALSO"
.IX Header "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.
.SH "COPYRIGHT"
.IX Header "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, \s-1MA 02140.\s0
.PP
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.