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where j l = J1 /2 , j2 = J2/2, j = J /2 , ml = M1/2, m2 = M2/2, m = M/2. [Reference fo rmula 3.6.11, p. 45 of EDMONDS, Alan R. Angu la r m o m e n t m n in q u a n t u m mechan ics . In Investigations in Physics, ~, P r i n c e t o n U. Press , 1957.]. T h e p a r a m e t e r s of t he procedure , J1, J2, J, M1, M2 and M, are i n t e r p r e t e d as be ing twice the i r phys ica l value, so t h a t ac tua l p a r a m e t e r s m a y be in tegers . T h u s to call t he p rocedure to ca lcula te (1⁄2 0 1⁄2 0 ] 1⁄2 1⁄2 0 0) t he call would be VCC(1, 1, O, O, O, O, factorial). The procedure checks t h a t t he t r iangle cond i t ions for t he ex is tence of a coefficient are sa t is f ied and t h a t j l -4j2 4j is in tegra l . If t he condi t ions are no t sa t is f ied the value of t he p rocedure is zero. The p a r a m e t e r factorial is an a r r ay con ta in ing the fac tor ia l s f rom 0 up to j l + j2 4j + 1. Since in ac tua l ca lcu la t ions t he p rocedure VCC will be cal led m a n y t imes i t is more economica l to have the fac tor ia l s in a global a r r ay r a t h e r t h a n compu t e t h e m on every e n t r y to t he p rocedure ; b e g i n i n t e g e r z, zmin, zmax; r e a l cc; i f M1 + M2 # M V abs(M1) > abs(J1) V abs(M2) > abs(J2) V abs(M) > abs(J) V d > J1 + J2 V d < a b s ( J 1 J 2 ) V J1 + J2 4d # 2 M ( (d l+J24-d )+2) t h e n VCC := 0 e l s e b e g i n zmin := 0; i f J J2 + M1 < 0 t h e n zmin := J + J2 -M1; i f J J1 M2 + zmin < 0 t h e n zmin := J + d l + M2; zmax: = J1 + J2 J; i f J 2 + M2 -zmax < 0 t h e n zmax := J 2 + M2; i f J1 M1 zmax < 0 t h e n zmax := J1 -M1; ce := 0; f o r z := zmin s t e p 2 u n t i l zmax do cc := cc + ( i f z = d X (z+4) t h e n 1 e l s e --1)/(faclorial[z+2] X f a c t o r i a l [ ( J l + J 2 J z ) +2] X fac tor ia l [ (d l -Mlz ) +2] X factor ial[(J24-M2-z) +2] X factorial[(J--J2+M14-z) +2] X faetorial[(J--J1--M24-z) +2]) ; VCC := sqr t ( (J+l ) X f a c t o r i a l [ ( J l + J 2 J ) + 2 ] X fac tor ia l [ (J1 -J2+J) +2] X f a c t o r i a l [ ( d l + J 2 + J ) +2] X factorial[(Jl+M1) +2] X factorial[(orl--M1)+2] X factorial[(J2+M2)+2] X faetor ia l[ (J2-M2)+2] X factorial[(J+M)+2] X fac tor ia l[ (d--M)+2]/ fac tor ia l[ (J l+J2+d+2)+2]) X cc e n d e n d VCC

Certification of algorithm 21 [S17]: Bessel function for a set of integer orders