# Coupling 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.

gsl_sf_coupling_3j(two_ja, two_jb, two_jc, two_ma, two_mb, two_mc)

These routines compute the Wigner 3-j coefficient,

$\begin{split}\left( \begin{array}{ccc} ja & jb & jc \\ ma & mb & mc \end{array} \right)\end{split}$

where the arguments are given in half-integer units, ja = two_ja / 2, ma = two_ma / 2, etc.

gsl_sf_coupling_6j(two_ja, two_jb, two_jc, two_jd, two_je, two_jf)

These routines compute the Wigner 6-j coefficient,

$\begin{split}\left( \begin{array}{ccc} ja & jb & jc \\ jd & je & jf \end{array} \right)\end{split}$

where the arguments are given in half-integer units, ja = two_ja / 2, jb = two_jb / 2, etc.

gsl_sf_coupling_9j(two_ja, two_jb, two_jc, two_jd, two_je, two_jf, two_jg, two_jh, two_ji)

These routines compute the Wigner 9-j coefficient,

$\begin{split}\left( \begin{array}{ccc} ja & jb & jc \\ jd & je & jf \\ jg & jh & ji \end{array} \right)\end{split}$

where the arguments are given in half-integer units, ja = two_ja / 2, jb = two_jb / 2, etc.