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1: 19.2 Definitions
where p j is a polynomial in t while ρ and σ are rational functions of t . … Here a , b , p are real parameters, and k c and x are real or complex variables, with p 0 , k c 0 . …
§19.2(iv) A Related Function: R C ( x , y )
Formulas involving Π ( ϕ , α 2 , k ) that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using R C ( x , y ) . … For the special cases of R C ( x , x ) and R C ( 0 , y ) see (19.6.15). …
2: 34.6 Definition: 9 j Symbol
34.6.1 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = all  m r s ( j 11 j 12 j 13 m 11 m 12 m 13 ) ( j 21 j 22 j 23 m 21 m 22 m 23 ) ( j 31 j 32 j 33 m 31 m 32 m 33 ) ( j 11 j 21 j 31 m 11 m 21 m 31 ) ( j 12 j 22 j 32 m 12 m 22 m 32 ) ( j 13 j 23 j 33 m 13 m 23 m 33 ) ,
34.6.2 { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = j ( 1 ) 2 j ( 2 j + 1 ) { j 11 j 21 j 31 j 32 j 33 j } { j 12 j 22 j 32 j 21 j j 23 } { j 13 j 23 j 33 j j 11 j 12 } .
3: 34.7 Basic Properties: 9 j Symbol
34.7.1 { j 11 j 12 j 13 j 21 j 22 j 13 j 31 j 31 0 } = ( 1 ) j 12 + j 21 + j 13 + j 31 ( ( 2 j 13 + 1 ) ( 2 j 31 + 1 ) ) 1 2 { j 11 j 12 j 13 j 22 j 21 j 31 } .
34.7.2 j 12 j 34 ( 2 j 12 + 1 ) ( 2 j 34 + 1 ) ( 2 j 13 + 1 ) ( 2 j 24 + 1 ) { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } = δ j 13 , j 13 δ j 24 , j 24 .
34.7.3 j 13 j 24 ( 1 ) 2 j 2 + j 24 + j 23 j 34 ( 2 j 13 + 1 ) ( 2 j 24 + 1 ) { j 1 j 2 j 12 j 3 j 4 j 34 j 13 j 24 j } { j 1 j 3 j 13 j 4 j 2 j 24 j 14 j 23 j } = { j 1 j 2 j 12 j 4 j 3 j 34 j 14 j 23 j } .
34.7.4 ( j 13 j 23 j 33 m 13 m 23 m 33 ) { j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } = m r 1 , m r 2 , r = 1 , 2 , 3 ( j 11 j 12 j 13 m 11 m 12 m 13 ) ( j 21 j 22 j 23 m 21 m 22 m 23 ) ( j 31 j 32 j 33 m 31 m 32 m 33 ) ( j 11 j 21 j 31 m 11 m 21 m 31 ) ( j 12 j 22 j 32 m 12 m 22 m 32 ) .
34.7.5 j ( 2 j + 1 ) { j 11 j 12 j j 21 j 22 j 23 j 31 j 32 j 33 } { j 11 j 12 j j 23 j 33 j } = ( 1 ) 2 j { j 21 j 22 j 23 j 12 j j 32 } { j 31 j 32 j 33 j j 11 j 21 } .
4: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
These are given by the following equations in which a 1 , a 2 , , a n are nonnegative integers such that … M 2 is the number of permutations of { 1 , 2 , , n } with a 1 cycles of length 1, a 2 cycles of length 2, , and a n cycles of length n : … M 3 is the number of set partitions of { 1 , 2 , , n } with a 1 subsets of size 1, a 2 subsets of size 2, , and a n subsets of size n : …For each n all possible values of a 1 , a 2 , , a n are covered. … where the summation is over all nonnegative integers n 1 , n 2 , , n k such that n 1 + n 2 + + n k = n . …
5: 16.12 Products
16.12.3 ( F 1 2 ( a , b c ; z ) ) 2 = k = 0 ( 2 a ) k ( 2 b ) k ( c 1 2 ) k ( c ) k ( 2 c 1 ) k k ! F 3 4 ( 1 2 k , 1 2 ( 1 k ) , a + b c + 1 2 , 1 2 a + 1 2 , b + 1 2 , 3 2 k c ; 1 ) z k , | z | < 1 .
6: 10.48 Graphs
See accompanying text
Figure 10.48.1: 𝗃 n ( x ) , n = 0 ( 1 ) 4 , 0 x 12 . Magnify
See accompanying text
Figure 10.48.2: 𝗒 n ( x ) , n = 0 ( 1 ) 4 , 0 < x 12 . Magnify
See accompanying text
Figure 10.48.3: 𝗃 5 ( x ) , 𝗒 5 ( x ) , 𝗃 5 2 ( x ) + 𝗒 5 2 ( x ) , 0 x 12 . Magnify
See accompanying text
Figure 10.48.4: 𝗃 5 ( x ) , 𝗒 5 ( x ) , 𝗃 5 2 ( x ) + 𝗒 5 2 ( x ) , 0 x 12 . Magnify
See accompanying text
Figure 10.48.5: 𝗂 0 ( 1 ) ( x ) , 𝗂 0 ( 2 ) ( x ) , 𝗄 0 ( x ) , 0 x 4 . Magnify
7: 34.1 Special Notation
( j 1 j 2 j 3 m 1 m 2 m 3 ) ,
{ j 1 j 2 j 3 l 1 l 2 l 3 } ,
{ j 11 j 12 j 13 j 21 j 22 j 23 j 31 j 32 j 33 } .
An often used alternative to the 3 j symbol is the Clebsch–Gordan coefficient
34.1.1 ( j 1 m 1 j 2 m 2 | j 1 j 2 j 3 m 3 ) = ( 1 ) j 1 j 2 + m 3 ( 2 j 3 + 1 ) 1 2 ( j 1 j 2 j 3 m 1 m 2 m 3 ) ;
8: 32.3 Graphics
Plots of solutions w k ( x ) of P I  with w k ( 0 ) = 0 and w k ( 0 ) = k for various values of k , and the parabola 6 w 2 + x = 0 . … Here w k ( x ) is the solution of P II  with α = 0 and such that … Here u = u k ( x ; ν ) is the solution of …The corresponding solution of P IV  is given by …with β = 0 , α = 2 ν + 1 , and …
9: 4.17 Special Values and Limits
Table 4.17.1: Trigonometric functions: values at multiples of 1 12 π .
θ sin θ cos θ tan θ csc θ sec θ cot θ
π / 12 1 4 2 ( 3 1 ) 1 4 2 ( 3 + 1 ) 2 3 2 ( 3 + 1 ) 2 ( 3 1 ) 2 + 3
4.17.1 lim z 0 sin z z = 1 ,
4.17.2 lim z 0 tan z z = 1 .
4.17.3 lim z 0 1 cos z z 2 = 1 2 .
10: 34.5 Basic Properties: 6 j Symbol
If any lower argument in a 6 j symbol is 0 , 1 2 , or 1 , then the 6 j symbol has a simple algebraic form. …
34.5.9 { j 1 j 2 j 3 l 1 l 2 l 3 } = { j 1 1 2 ( j 2 + l 2 + j 3 l 3 ) 1 2 ( j 2 l 2 + j 3 + l 3 ) l 1 1 2 ( j 2 + l 2 j 3 + l 3 ) 1 2 ( j 2 + l 2 + j 3 + l 3 ) } ,
34.5.11 ( 2 j 1 + 1 ) ( ( J 3 + J 2 J 1 ) ( L 3 + L 2 J 1 ) 2 ( J 3 L 3 + J 2 L 2 J 1 L 1 ) ) { j 1 j 2 j 3 l 1 l 2 l 3 } = j 1 E ( j 1 + 1 ) { j 1 + 1 j 2 j 3 l 1 l 2 l 3 } + ( j 1 + 1 ) E ( j 1 ) { j 1 1 j 2 j 3 l 1 l 2 l 3 } ,
34.5.16 ( 1 ) j 1 + j 2 + j 3 + j 1 + j 2 + l 1 + l 2 { j 1 j 2 j 3 l 1 l 2 l 3 } { j 1 j 2 j 3 l 1 l 2 l 3 } = j ( 1 ) l 3 + l 3 + j ( 2 j + 1 ) { j 1 j 1 j j 2 j 2 j 3 } { l 3 l 3 j j 1 j 1 l 2 } { l 3 l 3 j j 2 j 2 l 1 } .
34.5.23 ( j 1 j 2 j 3 m 1 m 2 m 3 ) { j 1 j 2 j 3 l 1 l 2 l 3 } = m 1 m 2 m 3 ( 1 ) l 1 + l 2 + l 3 + m 1 + m 2 + m 3 ( j 1 l 2 l 3 m 1 m 2 m 3 ) ( l 1 j 2 l 3 m 1 m 2 m 3 ) ( l 1 l 2 j 3 m 1 m 2 m 3 ) .