=encoding utf8

=head1 NAME

Math::Symbolic::AuxFunctions - Auxiliary functions for Math::Symbolic hierarchy

=head1 SYNOPSIS

  use Math::Symbolic::AuxFunctions;
  
  Math::Symbolic::AuxFunctions::acos($x);
  # etc

=head1 DESCRIPTION

This module contains implementations of some auxiliary functions that are
used within the Math::Symbolic hierarchy of modules. In particular, this
module holds all trigonometric functions used for numeric evaluation of
trees by Math::Symbolic::Operator.

=head2 EXPORT

None. On purpose. If I wished this module would pollute others' namespaces,
I'd have put the functions right where they're used.

=cut

package Math::Symbolic::AuxFunctions;

use 5.006;
use strict;
use warnings;

use Carp;

use Math::Symbolic::ExportConstants qw/:all/;
use Memoize;

our $VERSION = '0.612';

=head1 TRIGONOMETRIC FUNCTIONS

=head2 tan

Computes the tangent sin(x) / cos(x).

=cut

sub tan { sin( $_[0] ) / cos( $_[0] ) }

=head2 cot

Computes the cotangent cos(x) / sin(x).

=cut

sub cot { cos( $_[0] ) / sin( $_[0] ) }

=head2 asin

Computes the arc sine asin(z) = -i log(iz + sqrt(1-z*z)).
Above formula is for complex numbers.

=cut

sub asin { atan2( $_[0], sqrt( 1 - $_[0] * $_[0] ) ) }

=head2 acos

Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-1)).
Above formula is for complex numbers.

=cut

sub acos { atan2( sqrt( 1 - $_[0] * $_[0] ), $_[0] ) }

=head2 atan

Computes the arc tangent atan(z) = i/2 log((i+z) / (i-z)).
Above formula is for complex numbers.

=cut

sub atan { atan2( $_[0], 1 ) }

=head2 acot

Computes the arc cotangent ( atan( 1 / x ) ).

=cut

sub acot { atan2( 1 / $_[0], 1 ) }

=head2 asinh

Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z+1))

=cut

sub asinh { log( $_[0] + sqrt( $_[0] * $_[0] + 1 ) ) }

=head2 acosh

Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-1)).

=cut

sub acosh { log( $_[0] + sqrt( $_[0] * $_[0] - 1 ) ) }

=head1 OTHER FUNCTIONS

=cut

=head2 binomial_coeff

Calculates the binomial coefficient n over k of its first two
arguments (n, k).

Code taken from Orwant et al, "Mastering Algorithms with Perl"

=cut

memoize('binomial_coeff');

sub binomial_coeff {
    my ( $n,   $k ) = @_;
    my ( $res, $j ) = ( 1, 1 );

    return 0 if $k > $n || $k < 0;
    $k = ( $n - $k ) if ( $n - $k ) < $k;

    while ( $j <= $k ) {
        $res *= $n--;
        $res /= $j++;
    }
    return $res;
}

=head2 bell_number

The Bell numbers are defined as follows:

  B_0   = 1
  B_n+1 = sum_k=0_to_n( B_k * binomial_coeff(n, k) )

This function uses memoization.

=cut

memoize('bell_number');

sub bell_number {
    my $n = shift;
    return undef if $n < 0;
    return 1     if $n == 0;
    my $bell = 0;
    $bell += bell_number($_) * binomial_coeff( $n - 1, $_ ) for 0 .. $n - 1;
    return $bell;
}

1;
__END__

=head1 AUTHOR

Please send feedback, bug reports, and support requests to the Math::Symbolic
support mailing list:
math-symbolic-support at lists dot sourceforge dot net. Please
consider letting us know how you use Math::Symbolic. Thank you.

If you're interested in helping with the development or extending the
module's functionality, please contact the developers' mailing list:
math-symbolic-develop at lists dot sourceforge dot net.

List of contributors:

  Steffen Müller, symbolic-module at steffen-mueller dot net
  Stray Toaster, mwk at users dot sourceforge dot net
  Oliver Ebenhöh

=head1 SEE ALSO

New versions of this module can be found on
http://steffen-mueller.net or CPAN. The module development takes place on
Sourceforge at http://sourceforge.net/projects/math-symbolic/

L<Math::Symbolic>

=cut