complex.texi

用于VC.net的gsl的lib库文件包

@cindex complex numbers

The functions described in this chapter provide support for complex
numbers.  The algorithms take care to avoid unnecessary intermediate
underflows and overflows, allowing the functions to be evaluated over 
as much of the complex plane as possible. 

@comment FIXME: this still needs to be
@comment done for the csc,sec,cot,csch,sech,coth functions

For multiple-valued functions the branch cuts have been chosen to follow
the conventions of Abramowitz and Stegun in the @cite{Handbook of
Mathematical Functions}. The functions return principal values which are
the same as those in GNU Calc, which in turn are the same as those in
@cite{Common Lisp, The Language (Second Edition)} (n.b. The second
edition uses different definitions from the first edition) and the
HP-28/48 series of calculators.

The complex types are defined in the header file @file{gsl_complex.h},
while the corresponding complex functions and arithmetic operations are
defined in @file{gsl_complex_math.h}.

@menu
* Complex numbers::             
* Properties of complex numbers::  
* Complex arithmetic operators::  
* Elementary Complex Functions::  
* Complex Trigonometric Functions::  
* Inverse Complex Trigonometric Functions::  
* Complex Hyperbolic Functions::  
* Inverse Complex Hyperbolic Functions::  
* Complex Number References and Further Reading::  
@end menu

@node Complex numbers
@section Complex numbers
@cindex representations of complex numbers
@cindex polar form of complex numbers

Complex numbers are represented using the type @code{gsl_complex}. The
internal representation of this type may vary across platforms and
should not be accessed directly. The functions and macros described
below allow complex numbers to be manipulated in a portable way.

For reference, the default form of the @code{gsl_complex} type is
given by the following struct,

@example
typedef struct
@{
  double dat[2];
@} gsl_complex;
@end example
@noindent
The real and imaginary part are stored in contiguous elements of a two
element array. This eliminates any padding between the real and
imaginary parts, @code{dat[0]} and @code{dat[1]}, allowing the struct to
be mapped correctly onto packed complex arrays.

@deftypefun gsl_complex gsl_complex_rect (double @var{x}, double @var{y})
This function uses the rectangular cartesian components
(@var{x},@var{y}) to return the complex number @math{z = x + i y}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_polar (double @var{r}, double @var{theta})
This function returns the complex number @math{z = r \exp(i \theta) = r
(\cos(\theta) + i \sin(\theta))} from the polar representation
(@var{r},@var{theta}).
@end deftypefun

@defmac GSL_REAL (@var{z})
@defmacx GSL_IMAG (@var{z})
These macros return the real and imaginary parts of the complex number
@var{z}.
@end defmac

@defmac GSL_SET_COMPLEX (@var{zp}, @var{x}, @var{y})
This macro uses the cartesian components (@var{x},@var{y}) to set the
real and imaginary parts of the complex number pointed to by @var{zp}.
For example,

@example
GSL_SET_COMPLEX(&z, 3, 4)
@end example
@noindent
sets @var{z} to be @math{3 + 4i}.
@end defmac

@defmac GSL_SET_REAL (@var{zp},@var{x})
@defmacx GSL_SET_IMAG (@var{zp},@var{y})
These macros allow the real and imaginary parts of the complex number
pointed to by @var{zp} to be set independently.
@end defmac

@node Properties of complex numbers
@section Properties of complex numbers

@deftypefun double gsl_complex_arg (gsl_complex @var{z})
@cindex argument of complex number 
This function returns the argument of the complex number @var{z},
@math{\arg(z)}, where @c{$-\pi < \arg(z) \leq \pi$}
@math{-\pi < \arg(z) <= \pi}.
@end deftypefun

@deftypefun double gsl_complex_abs (gsl_complex @var{z})
@cindex magnitude of complex number 
This function returns the magnitude of the complex number @var{z}, @math{|z|}.
@end deftypefun

@deftypefun double gsl_complex_abs2 (gsl_complex @var{z})
This function returns the squared magnitude of the complex number
@var{z}, @math{|z|^2}.
@end deftypefun

@deftypefun double gsl_complex_logabs (gsl_complex @var{z})
This function returns the natural logarithm of the magnitude of the
complex number @var{z}, @math{\log|z|}.  It allows an accurate
evaluation of @math{\log|z|} when @math{|z|} is close to one. The direct
evaluation of @code{log(gsl_complex_abs(z))} would lead to a loss of
precision in this case.
@end deftypefun


@node Complex arithmetic operators
@section Complex arithmetic operators
@cindex complex arithmetic

@deftypefun gsl_complex gsl_complex_add (gsl_complex @var{a}, gsl_complex @var{b})
This function returns the sum of the complex numbers @var{a} and
@var{b}, @math{z=a+b}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_sub (gsl_complex @var{a}, gsl_complex @var{b})
This function returns the difference of the complex numbers @var{a} and
@var{b}, @math{z=a-b}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_mul (gsl_complex @var{a}, gsl_complex @var{b})
This function returns the product of the complex numbers @var{a} and
@var{b}, @math{z=ab}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_div (gsl_complex @var{a}, gsl_complex @var{b})
This function returns the quotient of the complex numbers @var{a} and
@var{b}, @math{z=a/b}.
@end deftypefun


@deftypefun gsl_complex gsl_complex_add_real (gsl_complex @var{a}, double @var{x})
This function returns the sum of the complex number @var{a} and the
real number @var{x}, @math{z=a+x}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_sub_real (gsl_complex @var{a}, double @var{x})
This function returns the difference of the complex number @var{a} and the
real number @var{x}, @math{z=a-x}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_mul_real (gsl_complex @var{a}, double @var{x})
This function returns the product of the complex number @var{a} and the
real number @var{x}, @math{z=ax}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_div_real (gsl_complex @var{a}, double @var{x})
This function returns the quotient of the complex number @var{a} and the
real number @var{x}, @math{z=a/x}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_add_imag (gsl_complex @var{a}, double @var{y})
This function returns the sum of the complex number @var{a} and the
imaginary number @math{i}@var{y}, @math{z=a+iy}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_sub_imag (gsl_complex @var{a}, double @var{y})
This function returns the difference of the complex number @var{a} and the
imaginary number @math{i}@var{y}, @math{z=a-iy}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_mul_imag (gsl_complex @var{a}, double @var{y})
This function returns the product of the complex number @var{a} and the
imaginary number @math{i}@var{y}, @math{z=a*(iy)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_div_imag (gsl_complex @var{a}, double @var{y})
This function returns the quotient of the complex number @var{a} and the
imaginary number @math{i}@var{y}, @math{z=a/(iy)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_conjugate (gsl_complex @var{z})
@cindex conjugate of complex number
This function returns the complex conjugate of the complex number
@var{z}, @math{z^* = x - i y}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_inverse (gsl_complex @var{z})
This function returns the inverse, or reciprocal, of the complex number
@var{z}, @math{1/z = (x - i y)/(x^2 + y^2)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_negative (gsl_complex @var{z})
This function returns the negative of the complex number
@var{z}, @math{-z = (-x) + i(-y)}.
@end deftypefun


@node Elementary Complex Functions
@section Elementary Complex Functions

@deftypefun gsl_complex gsl_complex_sqrt (gsl_complex @var{z})
@cindex square root of complex number
This function returns the square root of the complex number @var{z},
@math{\sqrt z}. The branch cut is the negative real axis. The result
always lies in the right half of the complex plane.
@end deftypefun

@deftypefun gsl_complex gsl_complex_sqrt_real (double @var{x})
This function returns the complex square root of the real number
@var{x}, where @var{x} may be negative.
@end deftypefun


@deftypefun gsl_complex gsl_complex_pow (gsl_complex @var{z}, gsl_complex @var{a})
@cindex power of complex number
@cindex exponentiation of complex number
The function returns the complex number @var{z} raised to the complex
power @var{a}, @math{z^a}. This is computed as @math{\exp(\log(z)*a)}
using complex logarithms and complex exponentials.
@end deftypefun

@deftypefun gsl_complex gsl_complex_pow_real (gsl_complex @var{z}, double @var{x})
This function returns the complex number @var{z} raised to the real
power @var{x}, @math{z^x}.
@end deftypefun


@deftypefun gsl_complex gsl_complex_exp (gsl_complex @var{z})
This function returns the complex exponential of the complex number
@var{z}, @math{\exp(z)}.
@end deftypefun

@deftypefun gsl_complex gsl_complex_log (gsl_complex @var{z})
@cindex logarithm of complex number
This function returns the complex natural logarithm (base @math{e}) of
the complex number @var{z}, @math{\log(z)}.  The branch cut is the
negative real axis.