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1: 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
π / 4 1 2 2 1 2 2 1 2 2 1
2 π / 3 1 2 3 1 2 3 2 3 3 2 1 3 3
3 π / 4 1 2 2 1 2 2 1 2 2 1
4.17.3 lim z 0 1 cos z z 2 = 1 2 .
2: 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. …
34.5.5 { j 1 j 2 j 3 1 j 3 1 j 2 } = ( 1 ) J ( 2 ( J + 1 ) ( J 2 j 1 ) ( J 2 j 2 ) ( J 2 j 3 + 1 ) 2 j 2 ( 2 j 2 + 1 ) ( 2 j 2 + 2 ) ( 2 j 3 1 ) 2 j 3 ( 2 j 3 + 1 ) ) 1 2 ,
34.5.6 { j 1 j 2 j 3 1 j 3 1 j 2 + 1 } = ( 1 ) J ( ( J 2 j 2 1 ) ( J 2 j 2 ) ( J 2 j 3 + 1 ) ( J 2 j 3 + 2 ) ( 2 j 2 + 1 ) ( 2 j 2 + 2 ) ( 2 j 2 + 3 ) ( 2 j 3 1 ) 2 j 3 ( 2 j 3 + 1 ) ) 1 2 ,
34.5.7 { j 1 j 2 j 3 1 j 3 j 2 } = ( 1 ) J + 1 2 ( j 2 ( j 2 + 1 ) + j 3 ( j 3 + 1 ) j 1 ( j 1 + 1 ) ) ( 2 j 2 ( 2 j 2 + 1 ) ( 2 j 2 + 2 ) 2 j 3 ( 2 j 3 + 1 ) ( 2 j 3 + 2 ) ) 1 2 .
34.5.13 E ( j ) = ( ( j 2 ( j 2 j 3 ) 2 ) ( ( j 2 + j 3 + 1 ) 2 j 2 ) ( j 2 ( l 2 l 3 ) 2 ) ( ( l 2 + l 3 + 1 ) 2 j 2 ) ) 1 2 .
3: 29.1 Special Notation
The main functions treated in this chapter are the eigenvalues a ν 2 m ( k 2 ) , a ν 2 m + 1 ( k 2 ) , b ν 2 m + 1 ( k 2 ) , b ν 2 m + 2 ( k 2 ) , the Lamé functions Ec ν 2 m ( z , k 2 ) , Ec ν 2 m + 1 ( z , k 2 ) , Es ν 2 m + 1 ( z , k 2 ) , Es ν 2 m + 2 ( z , k 2 ) , and the Lamé polynomials uE 2 n m ( z , k 2 ) , sE 2 n + 1 m ( z , k 2 ) , cE 2 n + 1 m ( z , k 2 ) , dE 2 n + 1 m ( z , k 2 ) , scE 2 n + 2 m ( z , k 2 ) , sdE 2 n + 2 m ( z , k 2 ) , cdE 2 n + 2 m ( z , k 2 ) , scdE 2 n + 3 m ( z , k 2 ) . … Other notations that have been used are as follows: Ince (1940a) interchanges a ν 2 m + 1 ( k 2 ) with b ν 2 m + 1 ( k 2 ) . The relation to the Lamé functions L c ν ( m ) , L s ν ( m ) of Jansen (1977) is given by …
Es ν 2 m + 2 ( z , k 2 ) = s ν 2 m + 2 ( k 2 ) Es ν 2 m + 2 ( z , k 2 ) ,
where the positive factors c ν m ( k 2 ) and s ν m ( k 2 ) are determined by …
4: 4.30 Elementary Properties
Table 4.30.1: Hyperbolic functions: interrelations. …
sinh θ = a cosh θ = a tanh θ = a csch θ = a sech θ = a coth θ = a
sinh θ a ( a 2 1 ) 1 / 2 a ( 1 a 2 ) 1 / 2 a 1 a 1 ( 1 a 2 ) 1 / 2 ( a 2 1 ) 1 / 2
cosh θ ( 1 + a 2 ) 1 / 2 a ( 1 a 2 ) 1 / 2 a 1 ( 1 + a 2 ) 1 / 2 a 1 a ( a 2 1 ) 1 / 2
tanh θ a ( 1 + a 2 ) 1 / 2 a 1 ( a 2 1 ) 1 / 2 a ( 1 + a 2 ) 1 / 2 ( 1 a 2 ) 1 / 2 a 1
csch θ a 1 ( a 2 1 ) 1 / 2 a 1 ( 1 a 2 ) 1 / 2 a a ( 1 a 2 ) 1 / 2 ( a 2 1 ) 1 / 2
sech θ ( 1 + a 2 ) 1 / 2 a 1 ( 1 a 2 ) 1 / 2 a ( 1 + a 2 ) 1 / 2 a a 1 ( a 2 1 ) 1 / 2
5: 29.15 Fourier Series and Chebyshev Series
Polynomial uE 2 n m ( z , k 2 )
Polynomial sE 2 n + 1 m ( z , k 2 )
Polynomial scE 2 n + 2 m ( z , k 2 )
Polynomial sdE 2 n + 2 m ( z , k 2 )
Polynomial cdE 2 n + 2 m ( z , k 2 )
6: 22.9 Cyclic Identities
§22.9(ii) Typical Identities of Rank 2
These identities are cyclic in the sense that each of the indices m , n in the first product of, for example, the form s m , p ( 4 ) s n , p ( 4 ) are simultaneously permuted in the cyclic order: m m + 1 m + 2 p 1 2 m 1 ; n n + 1 n + 2 p 1 2 n 1 . …
22.9.11 ( d 1 , 2 ( 2 ) ) 2 d 2 , 2 ( 2 ) ± ( d 2 , 2 ( 2 ) ) 2 d 1 , 2 ( 2 ) = k ( d 1 , 2 ( 2 ) ± d 2 , 2 ( 2 ) ) ,
22.9.12 c 1 , 2 ( 2 ) s 1 , 2 ( 2 ) d 2 , 2 ( 2 ) + c 2 , 2 ( 2 ) s 2 , 2 ( 2 ) d 1 , 2 ( 2 ) = 0 .
22.9.21 k 2 c 1 , 2 ( 2 ) s 1 , 2 ( 2 ) c 2 , 2 ( 2 ) s 2 , 2 ( 2 ) = k ( 1 ( s 1 , 2 ( 2 ) ) 2 ( s 2 , 2 ( 2 ) ) 2 ) .
7: 30.3 Eigenvalues
With μ = m = 0 , 1 , 2 , , the spheroidal wave functions Ps n m ( x , γ 2 ) are solutions of Equation (30.2.1) which are bounded on ( 1 , 1 ) , or equivalently, which are of the form ( 1 x 2 ) 1 2 m g ( x ) where g ( z ) is an entire function of z . These solutions exist only for eigenvalues λ n m ( γ 2 ) , n = m , m + 1 , m + 2 , , of the parameter λ . … The eigenvalues λ n m ( γ 2 ) are analytic functions of the real variable γ 2 and satisfy … has the solutions λ = λ m + 2 j m ( γ 2 ) , j = 0 , 1 , 2 , . If p is an odd positive integer, then Equation (30.3.5) has the solutions λ = λ m + 2 j + 1 m ( γ 2 ) , j = 0 , 1 , 2 , . …
8: 14.29 Generalizations
14.29.1 ( 1 z 2 ) d 2 w d z 2 2 z d w d z + ( ν ( ν + 1 ) μ 1 2 2 ( 1 z ) μ 2 2 2 ( 1 + z ) ) w = 0
As in the case of (14.21.1), the solutions are hypergeometric functions, and (14.29.1) reduces to (14.21.1) when μ 1 = μ 2 = μ . …
9: 29.6 Fourier Series
§29.6(i) Function Ec ν 2 m ( z , k 2 )
In the special case ν = 2 n , m = 0 , 1 , , n , there is a unique nontrivial solution with the property A 2 p = 0 , p = n + 1 , n + 2 , . …
§29.6(ii) Function Ec ν 2 m + 1 ( z , k 2 )
§29.6(iii) Function Es ν 2 m + 1 ( z , k 2 )
§29.6(iv) Function Es ν 2 m + 2 ( z , k 2 )
10: 29.12 Definitions
where n = 0 , 1 , 2 , , m = 0 , 1 , 2 , , n . … In the fourth column the variable z and modulus k of the Jacobian elliptic functions have been suppressed, and P ( sn 2 ) denotes a polynomial of degree n in sn 2 ( z , k ) (different for each type). … where ρ , σ , τ are either 0 or 1 2 . The polynomial P ( ξ ) is of degree n and has m zeros (all simple) in ( 0 , 1 ) and n m zeros (all simple) in ( 1 , k 2 ) . … defined for ( t 1 , t 2 , , t n ) with …