complex.texi

This library is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY without ev

negative real axis. 
@end deftypefun

@deftypefun gsl_complex gsl_complex_log10 (gsl_complex @var{z})
This function returns the complex base-10 logarithm of
the complex number @var{z}, @c{$\log_{10}(z)$}
@math{\log_10 (z)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_log_b (gsl_complex @var{z}, gsl_complex @var{b})
This function returns the complex base-@var{b} logarithm of the complex
number @var{z}, @math{\log_b(z)}. This quantity is computed as the ratio
@math{\log(z)/\log(b)}.
@end deftypefun


@node Complex Trigonometric Functions
@section Complex Trigonometric Functions
@cindex trigonometric functions of complex numbers

@deftypefun gsl_complex gsl_complex_sin (gsl_complex @var{z})
@cindex sin, of complex number
This function returns the complex sine of the complex number @var{z},
@math{\sin(z) = (\exp(iz) - \exp(-iz))/(2i)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_cos (gsl_complex @var{z})
@cindex cosine of complex number
This function returns the complex cosine of the complex number @var{z},
@math{\cos(z) = (\exp(iz) + \exp(-iz))/2}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_tan (gsl_complex @var{z})
@cindex tangent of complex number
This function returns the complex tangent of the complex number @var{z},
@math{\tan(z) = \sin(z)/\cos(z)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_sec (gsl_complex @var{z})
This function returns the complex secant of the complex number @var{z},
@math{\sec(z) = 1/\cos(z)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_csc (gsl_complex @var{z})
This function returns the complex cosecant of the complex number @var{z},
@math{\csc(z) = 1/\sin(z)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_cot (gsl_complex @var{z})
This function returns the complex cotangent of the complex number @var{z},
@math{\cot(z) = 1/\tan(z)}.
@end deftypefun


@node Inverse Complex Trigonometric Functions
@section Inverse Complex Trigonometric Functions
@cindex inverse complex trigonometric functions

@deftypefun gsl_complex gsl_complex_arcsin (gsl_complex @var{z})
This function returns the complex arcsine of the complex number @var{z},
@math{\arcsin(z)}. The branch cuts are on the real axis, less than @math{-1}
and greater than @math{1}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_arcsin_real (double @var{z})
This function returns the complex arcsine of the real number @var{z},
@math{\arcsin(z)}. For @math{z} between @math{-1} and @math{1}, the
function returns a real value in the range @math{[-\pi/2,\pi/2]}. For
@math{z} less than @math{-1} the result has a real part of @math{-\pi/2}
and a positive imaginary part.  For @math{z} greater than @math{1} the
result has a real part of @math{\pi/2} and a negative imaginary part.
@end deftypefun

@deftypefun gsl_complex gsl_complex_arccos (gsl_complex @var{z})
This function returns the complex arccosine of the complex number @var{z},
@math{\arccos(z)}. The branch cuts are on the real axis, less than @math{-1}
and greater than @math{1}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_arccos_real (double @var{z})
This function returns the complex arccosine of the real number @var{z},
@math{\arccos(z)}. For @math{z} between @math{-1} and @math{1}, the
function returns a real value in the range @math{[0,\pi]}. For @math{z}
less than @math{-1} the result has a real part of @math{\pi} and a
negative imaginary part.  For @math{z} greater than @math{1} the result
is purely imaginary and positive.
@end deftypefun

@deftypefun gsl_complex gsl_complex_arctan (gsl_complex @var{z})
This function returns the complex arctangent of the complex number
@var{z}, @math{\arctan(z)}. The branch cuts are on the imaginary axis,
below @math{-i} and above @math{i}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_arcsec (gsl_complex @var{z})
This function returns the complex arcsecant of the complex number @var{z},
@math{\arcsec(z) = \arccos(1/z)}. 
@end deftypefun

@deftypefun gsl_complex gsl_complex_arcsec_real (double @var{z})
This function returns the complex arcsecant of the real number @var{z},
@math{\arcsec(z) = \arccos(1/z)}. 
@end deftypefun

@deftypefun gsl_complex gsl_complex_arccsc (gsl_complex @var{z})
This function returns the complex arccosecant of the complex number @var{z},
@math{\arccsc(z) = \arcsin(1/z)}. 
@end deftypefun

@deftypefun gsl_complex gsl_complex_arccsc_real (double @var{z})
This function returns the complex arccosecant of the real number @var{z},
@math{\arccsc(z) = \arcsin(1/z)}. 
@end deftypefun

@deftypefun gsl_complex gsl_complex_arccot (gsl_complex @var{z})
This function returns the complex arccotangent of the complex number @var{z},
@math{\arccot(z) = \arctan(1/z)}. 
@end deftypefun


@node Complex Hyperbolic Functions
@section Complex Hyperbolic Functions
@cindex hyperbolic functions, complex numbers

@deftypefun gsl_complex gsl_complex_sinh (gsl_complex @var{z})
This function returns the complex hyperbolic sine of the complex number
@var{z}, @math{\sinh(z) = (\exp(z) - \exp(-z))/2}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_cosh (gsl_complex @var{z})
This function returns the complex hyperbolic cosine of the complex number
@var{z}, @math{\cosh(z) = (\exp(z) + \exp(-z))/2}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_tanh (gsl_complex @var{z})
This function returns the complex hyperbolic tangent of the complex number
@var{z}, @math{\tanh(z) = \sinh(z)/\cosh(z)}.
@end deftypefun


@deftypefun gsl_complex gsl_complex_sech (gsl_complex @var{z})
This function returns the complex hyperbolic secant of the complex
number @var{z}, @math{\sech(z) = 1/\cosh(z)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_csch (gsl_complex @var{z})
This function returns the complex hyperbolic cosecant of the complex
number @var{z}, @math{\csch(z) = 1/\sinh(z)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_coth (gsl_complex @var{z})
This function returns the complex hyperbolic cotangent of the complex
number @var{z}, @math{\coth(z) = 1/\tanh(z)}.
@end deftypefun


@node Inverse Complex Hyperbolic Functions
@section Inverse Complex Hyperbolic Functions
@cindex inverse hyperbolic functions, complex numbers

@deftypefun gsl_complex gsl_complex_arcsinh (gsl_complex @var{z})
This function returns the complex hyperbolic arcsine of the
complex number @var{z}, @math{\arcsinh(z)}.  The branch cuts are on the
imaginary axis, below @math{-i} and above @math{i}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_arccosh (gsl_complex @var{z})
This function returns the complex hyperbolic arccosine of the complex
number @var{z}, @math{\arccosh(z)}.  The branch cut is on the real axis,
less than @math{1}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_arccosh_real (double @var{z})
This function returns the complex hyperbolic arccosine of
the real number @var{z}, @math{\arccosh(z)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_arctanh (gsl_complex @var{z})
This function returns the complex hyperbolic arctangent of the complex
number @var{z}, @math{\arctanh(z)}.  The branch cuts are on the real
axis, less than @math{-1} and greater than @math{1}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_arctanh_real (double @var{z})
This function returns the complex hyperbolic arctangent of the real
number @var{z}, @math{\arctanh(z)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_arcsech (gsl_complex @var{z})
This function returns the complex hyperbolic arcsecant of the complex
number @var{z}, @math{\arcsech(z) = \arccosh(1/z)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_arccsch (gsl_complex @var{z})
This function returns the complex hyperbolic arccosecant of the complex
number @var{z}, @math{\arccsch(z) = \arcsin(1/z)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_arccoth (gsl_complex @var{z})
This function returns the complex hyperbolic arccotangent of the complex
number @var{z}, @math{\arccoth(z) = \arctanh(1/z)}.
@end deftypefun

@node Complex Number References and Further Reading
@section References and Further Reading

The implementations of the elementary and trigonometric functions are
based on the following papers,

@itemize @asis
@item
T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
``Implementing Complex Elementary Functions Using Exception
Handling'', @cite{ACM Transactions on Mathematical Software}, Volume 20
(1994), pp 215--244, Corrigenda, p553

@item
T. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,
``Implementing the complex arcsin and arccosine functions using exception
handling'', @cite{ACM Transactions on Mathematical Software}, Volume 23
(1997) pp 299--335
@end itemize

@noindent
The general formulas and details of branch cuts can be found in the
following books,

@itemize @asis
@item
Abramowitz and Stegun, @cite{Handbook of Mathematical Functions},
``Circular Functions in Terms of Real and Imaginary Parts'', Formulas
4.3.55--58,
``Inverse Circular Functions in Terms of Real and Imaginary Parts'',
Formulas 4.4.37--39,
``Hyperbolic Functions in Terms of Real and Imaginary Parts'',
Formulas 4.5.49--52,
``Inverse Hyperbolic Functions---relation to Inverse Circular Functions'',
Formulas 4.6.14--19.

@item
Dave Gillespie, @cite{Calc Manual}, Free Software Foundation, ISBN
1-882114-18-3
@end itemize