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Diffstat (limited to 'gsl-1.9/doc/specfunc-psi.texi')
-rw-r--r-- | gsl-1.9/doc/specfunc-psi.texi | 90 |
1 files changed, 90 insertions, 0 deletions
diff --git a/gsl-1.9/doc/specfunc-psi.texi b/gsl-1.9/doc/specfunc-psi.texi new file mode 100644 index 0000000..8bbef3b --- /dev/null +++ b/gsl-1.9/doc/specfunc-psi.texi @@ -0,0 +1,90 @@ +@cindex psi function +@cindex digamma function +@cindex polygamma functions + +The polygamma functions of order @math{n} are defined by +@tex +\beforedisplay +$$ +\psi^{(n)}(x) = \left(d \over dx\right)^n \psi(x) = \left(d \over dx\right)^{n+1} \log(\Gamma(x)) +$$ +\afterdisplay +@end tex +@ifinfo + +@example +\psi^@{(n)@}(x) = (d/dx)^n \psi(x) = (d/dx)^@{n+1@} \log(\Gamma(x)) +@end example + +@end ifinfo +@noindent +where @math{\psi(x) = \Gamma'(x)/\Gamma(x)} is known as the digamma function. +These functions are declared in the header file @file{gsl_sf_psi.h}. + +@menu +* Digamma Function:: +* Trigamma Function:: +* Polygamma Function:: +@end menu + +@node Digamma Function +@subsection Digamma Function + +@deftypefun double gsl_sf_psi_int (int @var{n}) +@deftypefunx int gsl_sf_psi_int_e (int @var{n}, gsl_sf_result * @var{result}) +These routines compute the digamma function @math{\psi(n)} for positive +integer @var{n}. The digamma function is also called the Psi function. +@comment Domain: n integer, n > 0 +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun + + +@deftypefun double gsl_sf_psi (double @var{x}) +@deftypefunx int gsl_sf_psi_e (double @var{x}, gsl_sf_result * @var{result}) +These routines compute the digamma function @math{\psi(x)} for general +@math{x}, @math{x \ne 0}. +@comment Domain: x != 0.0, -1.0, -2.0, ... +@comment Exceptional Return Values: GSL_EDOM, GSL_ELOSS +@end deftypefun + + +@deftypefun double gsl_sf_psi_1piy (double @var{y}) +@deftypefunx int gsl_sf_psi_1piy_e (double @var{y}, gsl_sf_result * @var{result}) +These routines compute the real part of the digamma function on the line +@math{1+i y}, @math{\Re[\psi(1 + i y)]}. +@comment exceptions: none +@comment Exceptional Return Values: none +@end deftypefun + + +@node Trigamma Function +@subsection Trigamma Function + +@deftypefun double gsl_sf_psi_1_int (int @var{n}) +@deftypefunx int gsl_sf_psi_1_int_e (int @var{n}, gsl_sf_result * @var{result}) +These routines compute the Trigamma function @math{\psi'(n)} for +positive integer @math{n}. +@comment Domain: n integer, n > 0 +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun + +@deftypefun double gsl_sf_psi_1 (double @var{x}) +@deftypefunx int gsl_sf_psi_1_e (double @var{x}, gsl_sf_result * @var{result}) +These routines compute the Trigamma function @math{\psi'(x)} for +general @math{x}. +@comment Domain: x != 0.0, -1.0, -2.0, ... +@comment Exceptional Return Values: GSL_EDOM, GSL_ELOSS +@end deftypefun + +@node Polygamma Function +@subsection Polygamma Function + +@deftypefun double gsl_sf_psi_n (int @var{n}, double @var{x}) +@deftypefunx int gsl_sf_psi_n_e (int @var{n}, double @var{x}, gsl_sf_result * @var{result}) +These routines compute the polygamma function @c{$\psi^{(n)}(x)$} +@math{\psi^@{(n)@}(x)} for +@c{$n \ge 0$} +@math{n >= 0}, @math{x > 0}. +@comment Domain: n >= 0, x > 0.0 +@comment Exceptional Return Values: GSL_EDOM +@end deftypefun |