@end deftypefun@deftypefun gsl_complex gsl_complex_log10 (gsl_complex @var{z})This function returns the complex base-10 logarithm ofthe 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 complexnumber @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 numberThis 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 numberThis 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 numberThis 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}, thefunction returns a real value in the range @math{(-\pi,\pi]}. 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} theresult 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}, thefunction 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/2} and anegative imaginary part.  For @math{z} greater than @math{1} the resultis 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 complexnumber @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 complexnumber @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 complexnumber @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 thecomplex number @var{z}, @math{\arcsinh(z)}.  The branch cuts are on theimaginary 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 complexnumber @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 ofthe 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 complexnumber @var{z}, @math{\arctanh(z)}.  The branch cuts are on the realaxis, 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 realnumber @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 complexnumber @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 complexnumber @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 complexnumber @var{z}, @math{\arccoth(z) = \arctanh(1/z)}.@end deftypefun@node Complex Number References and Further Reading@section References and Further Reading@noindentThe implementations of the elementary and trigonometric functions arebased on the following papers,@itemize @asis@itemT. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,"Implementing Complex Elementary Functions Using ExceptionHandling", @cite{ACM Transactions on Mathematical Software}, Volume 20(1994), pp 215-244, Corrigenda, p553@itemT. E. Hull, Thomas F. Fairgrieve, Ping Tak Peter Tang,"Implementing the complex arcsin and arccosine functions using exceptionhandling", @cite{ACM Transactions on Mathematical Software}, Volume 23(1997) pp 299-335@end itemize@noindentThe general formulas and details of branch cuts can be found in thefollowing books,@itemize @asis@itemAbramowitz and Stegun, @cite{Handbook of Mathematical Functions},"Circular Functions in Terms of Real and Imaginary Parts", Formulas4.3.55--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.@itemDave Gillespie, @cite{Calc Manual}, Free Software Foundation, ISBN1-882114-18-3@end itemize