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鉄鹿山学院大学学士成绩单【仿证微CXFK69】2Efze

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1: 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 …
2: 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
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
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
5 π / 12 1 4 2 ( 3 + 1 ) 1 4 2 ( 3 - 1 ) 2 + 3 2 ( 3 - 1 ) 2 ( 3 + 1 ) 2 - 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 .
4: 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 ) .
5: 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 .
6: 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 , . …
7: 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 )
8: 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 …
9: 34.3 Basic Properties: 3 j Symbol
When any one of j 1 , j 2 , j 3 is equal to 0 , 1 2 , or 1 , the 3 j symbol has a simple algebraic form. …For these and other results, and also cases in which any one of j 1 , j 2 , j 3 is 3 2 or 2 , see Edmonds (1974, pp. 125–127). … Even permutations of columns of a 3 j symbol leave it unchanged; odd permutations of columns produce a phase factor ( - 1 ) j 1 + j 2 + j 3 , for example, …
34.3.13 ( ( j 1 + j 2 + j 3 + 1 ) ( - j 1 + j 2 + j 3 ) ) 1 2 ( j 1 j 2 j 3 m 1 m 2 m 3 ) = ( ( j 2 + m 2 ) ( j 3 - m 3 ) ) 1 2 ( j 1 j 2 - 1 2 j 3 - 1 2 m 1 m 2 - 1 2 m 3 + 1 2 ) - ( ( j 2 - m 2 ) ( j 3 + m 3 ) ) 1 2 ( j 1 j 2 - 1 2 j 3 - 1 2 m 1 m 2 + 1 2 m 3 - 1 2 ) ,
34.3.15 ( 2 j 1 + 1 ) ( ( j 2 ( j 2 + 1 ) - j 3 ( j 3 + 1 ) ) m 1 - j 1 ( j 1 + 1 ) ( m 3 - m 2 ) ) ( j 1 j 2 j 3 m 1 m 2 m 3 ) = ( j 1 + 1 ) ( j 1 2 - ( j 2 - j 3 ) 2 ) 1 2 ( ( j 2 + j 3 + 1 ) 2 - j 1 2 ) 1 2 ( j 1 2 - m 1 2 ) 1 2 ( j 1 - 1 j 2 j 3 m 1 m 2 m 3 ) + j 1 ( ( j 1 + 1 ) 2 - ( j 2 - j 3 ) 2 ) 1 2 ( ( j 2 + j 3 + 1 ) 2 - ( j 1 + 1 ) 2 ) 1 2 ( ( j 1 + 1 ) 2 - m 1 2 ) 1 2 ( j 1 + 1 j 2 j 3 m 1 m 2 m 3 ) .
10: 28.14 Fourier Series
The coefficients satisfy
28.14.4 q c 2 m + 2 - ( a - ( ν + 2 m ) 2 ) c 2 m + q c 2 m - 2 = 0 , a = λ ν ( q ) , c 2 m = c 2 m ν ( q ) ,
28.14.5 m = - ( c 2 m ν ( q ) ) 2 = 1 ;
28.14.7 c - 2 m - ν ( q ) = c 2 m ν ( q ) ,
28.14.8 c 2 m ν ( - q ) = ( - 1 ) m c 2 m ν ( q ) .