.. index::

   single: coupling coefficients

   single: 3-j symbols

   single: 6-j symbols

   single: 9-j symbols

   single: Wigner coefficients

   single: Racah coefficients

The Wigner 3-j, 6-j and 9-j symbols give the coupling coefficients for

combined angular momentum vectors.  Since the arguments of the standard

coupling coefficient functions are integer or half-integer, the

arguments of the following functions are, by convention, integers equal

to twice the actual spin value.  For information on the 3-j coefficients

see Abramowitz & Stegun, Section 27.9.  The functions described in this

section are declared in the header file :file:`gsl_sf_coupling.h`.

3-j Symbols

-----------

.. function:: double gsl_sf_coupling_3j (int two_ja, int two_jb, int two_jc, int two_ma, int two_mb, int two_mc)

              int gsl_sf_coupling_3j_e (int two_ja, int two_jb, int two_jc, int two_ma, int two_mb, int two_mc, gsl_sf_result * result)

   These routines compute the Wigner 3-j coefficient, 

   .. only:: not texinfo

      .. math::

         \left(

         \begin{array}{ccc}

           ja & jb & jc \\

           ma & mb & mc

         \end{array}

         \right)

   .. only:: texinfo

      | ( ja jb jc )

      | ( ma mb mc )

   where the arguments are given in half-integer units, :math:`ja` =

   :data:`two_ja`/2, :math:`ma` = :data:`two_ma`/2, etc.

.. Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW

6-j Symbols

-----------

.. function:: double gsl_sf_coupling_6j (int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf)

              int gsl_sf_coupling_6j_e (int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, gsl_sf_result * result) 

   These routines compute the Wigner 6-j coefficient, 

   .. only:: not texinfo

      .. math::

         \left\{

         \begin{array}{ccc}

           ja & jb & jc \\

           jd & je & jf

         \end{array}

         \right\}

   .. only:: texinfo

      | { ja jb jc }

      | { jd je jf }

   where the arguments are given in half-integer units, :math:`ja` =

   :data:`two_ja`/2, :math:`ma` = :data:`two_ma`/2, etc.

.. Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW

9-j Symbols

-----------

.. function:: double gsl_sf_coupling_9j (int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, int two_jg, int two_jh, int two_ji)

              int gsl_sf_coupling_9j_e (int two_ja, int two_jb, int two_jc, int two_jd, int two_je, int two_jf, int two_jg, int two_jh, int two_ji, gsl_sf_result * result) 

   These routines compute the Wigner 9-j coefficient, 

   .. only:: not texinfo

      .. math::

         \left\{

         \begin{array}{ccc}

           ja & jb & jc \\

           jd & je & jf \\

           jg & jh & ji

         \end{array}

         \right\}

   .. only:: texinfo

      | { ja jb jc }

      | { jd je jf }

      | { jg jh ji }

   where the arguments are given in half-integer units, :math:`ja` =

   :data:`two_ja`/2, :math:`ma` = :data:`two_ma`/2, etc.

.. Exceptional Return Values: GSL_EDOM, GSL_EOVRFLW