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1: 19.2 Definitions
Assume 1 sin 2 ϕ ( , 0 ] and 1 k 2 sin 2 ϕ ( , 0 ] , except that one of them may be 0, and 1 α 2 sin 2 ϕ { 0 } . …The integral for E ( ϕ , k ) is well defined if k 2 = sin 2 ϕ = 1 , and the Cauchy principal value (§1.4(v)) of Π ( ϕ , α 2 , k ) is taken if 1 α 2 sin 2 ϕ vanishes at an interior point of the integration path. …The circular and hyperbolic cases alternate in the four intervals of the real line separated by the points α 2 = 0 , k 2 , 1 . … If 1 < k 1 / sin ϕ , then k c is pure imaginary. …
§19.2(iv) A Related Function: R C ( x , y )
2: 10.55 Continued Fractions
For continued fractions for 𝗃 n + 1 ( z ) / 𝗃 n ( z ) and 𝗂 n + 1 ( 1 ) ( z ) / 𝗂 n ( 1 ) ( z ) see Cuyt et al. (2008, pp. 350, 353, 362, 363, 367–369).
3: 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
0 0 1 0 1
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 .
4: 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. …
34.5.4 { j 1 j 2 j 3 1 j 3 1 j 2 1 } = ( 1 ) J ( J ( J + 1 ) ( J 2 j 1 ) ( J 2 j 1 1 ) ( 2 j 2 1 ) 2 j 2 ( 2 j 2 + 1 ) ( 2 j 3 1 ) 2 j 3 ( 2 j 3 + 1 ) ) 1 2 ,
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 } ,
34.5.18 j ( 1 ) j 1 + j 2 + j ( 2 j + 1 ) { j 1 j 2 j j 2 j 1 j } = ( 2 j 1 + 1 ) ( 2 j 2 + 1 ) δ j , 0 ,
34.5.22 l ( 1 ) l + j + j 1 + j 2 1 l ( l + 1 ) { j 1 j 2 l j 2 j 1 j } = 1 j 1 ( j 1 + 1 ) j 2 ( j 2 + 1 ) ( ( 2 j 1 j ) ! ( 2 j 2 + j + 1 ) ! ( 2 j 2 j ) ! ( 2 j 1 + j + 1 ) ! ) 1 2 , j 2 < j 1 .
5: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions
M 1 is the multinominal coefficient (26.4.2): … 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 . …
6: 18.31 Bernstein–Szegő Polynomials
Let ρ ( x ) be a polynomial of degree and positive when 1 x 1 . The Bernstein–Szegő polynomials { p n ( x ) } , n = 0 , 1 , , are orthogonal on ( 1 , 1 ) with respect to three types of weight function: ( 1 x 2 ) 1 2 ( ρ ( x ) ) 1 , ( 1 x 2 ) 1 2 ( ρ ( x ) ) 1 , ( 1 x ) 1 2 ( 1 + x ) 1 2 ( ρ ( x ) ) 1 . In consequence, p n ( cos θ ) can be given explicitly in terms of ρ ( cos θ ) and sines and cosines, provided that < 2 n in the first case, < 2 n + 2 in the second case, and < 2 n + 1 in the third case. …
7: 24.20 Tables
Abramowitz and Stegun (1964, Chapter 23) includes exact values of k = 1 m k n , m = 1 ( 1 ) 100 , n = 1 ( 1 ) 10 ; k = 1 k n , k = 1 ( 1 ) k 1 k n , k = 0 ( 2 k + 1 ) n , n = 1 , 2 , , 20D; k = 0 ( 1 ) k ( 2 k + 1 ) n , n = 1 , 2 , , 18D. Wagstaff (1978) gives complete prime factorizations of N n and E n for n = 20 ( 2 ) 60 and n = 8 ( 2 ) 42 , respectively. … For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).
8: 10.10 Continued Fractions
Assume J ν 1 ( z ) 0 . …
10.10.1 J ν ( z ) J ν 1 ( z ) = 1 2 ν z 1 1 2 ( ν + 1 ) z 1 1 2 ( ν + 2 ) z 1 , z 0 ,
10.10.2 J ν ( z ) J ν 1 ( z ) = 1 2 z / ν 1 1 4 z 2 / ( ν ( ν + 1 ) ) 1 1 4 z 2 / ( ( ν + 1 ) ( ν + 2 ) ) 1 , ν 0 , 1 , 2 , .
9: 10.33 Continued Fractions
Assume I ν 1 ( z ) 0 . …
10.33.1 I ν ( z ) I ν 1 ( z ) = 1 2 ν z 1 + 1 2 ( ν + 1 ) z 1 + 1 2 ( ν + 2 ) z 1 + , z 0 ,
10.33.2 I ν ( z ) I ν 1 ( z ) = 1 2 z / ν 1 + 1 4 z 2 / ( ν ( ν + 1 ) ) 1 + 1 4 z 2 / ( ( ν + 1 ) ( ν + 2 ) ) 1 + , ν 0 , 1 , 2 , .
10: 34.8 Approximations for Large Parameters
34.8.1 { j 1 j 2 j 3 j 2 j 1 l 3 } = ( 1 ) j 1 + j 2 + j 3 + l 3 ( 4 π ( 2 j 1 + 1 ) ( 2 j 2 + 1 ) ( 2 l 3 + 1 ) sin θ ) 1 2 ( cos ( ( l 3 + 1 2 ) θ 1 4 π ) + o ( 1 ) ) , j 1 , j 2 , j 3 l 3 1 ,
34.8.2 cos θ = j 1 ( j 1 + 1 ) + j 2 ( j 2 + 1 ) j 3 ( j 3 + 1 ) 2 j 1 ( j 1 + 1 ) j 2 ( j 2 + 1 ) ,
and the symbol o ( 1 ) denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i). …