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1: 35.10 Methods of Computation
β–ΊSee Yan (1992) for the F 1 1 and F 1 2 functions of matrix argument in the case m = 2 , and Bingham et al. (1992) for Monte Carlo simulation on 𝐎 ⁑ ( m ) applied to a generalization of the integral (35.5.8). …
2: 35.7 Gaussian Hypergeometric Function of Matrix Argument
β–Ί
Case m = 2
3: 16.17 Definition
β–ΊThere are three possible choices for L , illustrated in Figure 16.17.1 in the case m = 1 , n = 2 : …
4: 18.32 OP’s with Respect to Freud Weights
β–ΊOf special interest are the cases Q ⁑ ( x ) = x 2 ⁒ m , m = 1 , 2 , , and the case Q ⁑ ( x ) = 1 4 ⁒ x 4 t ⁒ x 2 ( t ℝ ), see §32.15. …
5: 29.3 Definitions and Basic Properties
β–ΊThey are denoted by a Ξ½ 2 ⁒ m ⁑ ( k 2 ) , a Ξ½ 2 ⁒ m + 1 ⁑ ( k 2 ) , b Ξ½ 2 ⁒ m + 1 ⁑ ( k 2 ) , b Ξ½ 2 ⁒ m + 2 ⁑ ( k 2 ) , where m = 0 , 1 , 2 , ; see Table 29.3.1. … β–ΊFor the special case k = k = 1 / 2 see Erdélyi et al. (1955, §15.5.2). … β–ΊThe quantity H = 2 ⁒ a Ξ½ 2 ⁒ m + 1 ⁑ ( k 2 ) Ξ½ ⁒ ( Ξ½ + 1 ) ⁒ k 2 satisfies equation (29.3.10) with … β–ΊThe quantity H = 2 ⁒ b Ξ½ 2 ⁒ m + 2 ⁑ ( k 2 ) Ξ½ ⁒ ( Ξ½ + 1 ) ⁒ k 2 satisfies equation (29.3.10) with … β–ΊFor m p , …
6: 15.8 Transformations of Variable
β–ΊWith m = 0 , 1 , 2 , , polynomial cases of (15.8.2)–(15.8.5) are given by …
7: 25.11 Hurwitz Zeta Function
β–ΊFor the more general case ΞΆ ⁑ ( m , a ) , m = 1 , 2 , , see Elizalde (1986). …
8: 22.21 Tables
β–ΊSpenceley and Spenceley (1947) tabulates sn ⁑ ( K ⁑ ⁒ x , k ) , cn ⁑ ( K ⁑ ⁒ x , k ) , dn ⁑ ( K ⁑ ⁒ x , k ) , am ⁑ ( K ⁑ ⁒ x , k ) , β„° ⁑ ( K ⁑ ⁒ x , k ) for arcsin ⁑ k = 1 ∘ ⁒ ( 1 ∘ ) ⁒ 89 ∘ and x = 0 ⁒ ( 1 90 ) ⁒ 1 to 12D, or 12 decimals of a radian in the case of am ⁑ ( K ⁑ ⁒ x , k ) . β–ΊCurtis (1964b) tabulates sn ⁑ ( m ⁒ K ⁑ / n , k ) , cn ⁑ ( m ⁒ K ⁑ / n , k ) , dn ⁑ ( m ⁒ K ⁑ / n , k ) for n = 2 ⁒ ( 1 ) ⁒ 15 , m = 1 ⁒ ( 1 ) ⁒ n 1 , and q (not k ) = 0 ⁒ ( .005 ) ⁒ 0.35 to 20D. … β–Ί5 to 2. 2. … β–ΊZhang and Jin (1996, p. 678) tabulates sn ⁑ ( K ⁑ ⁒ x , k ) , cn ⁑ ( K ⁑ ⁒ x , k ) , dn ⁑ ( K ⁑ ⁒ x , k ) for k = 1 4 , 1 2 and x = 0 ⁒ ( .1 ) ⁒ 4 to 7D. …
9: 29.6 Fourier Series
β–ΊIn the special case Ξ½ = 2 ⁒ n , m = 0 , 1 , , n , there is a unique nontrivial solution with the property A 2 ⁒ p = 0 , p = n + 1 , n + 2 , . …
10: 10.21 Zeros
β–ΊHere a m , b m , a m , b m are the m th negative zeros of Ai ⁑ ( x ) , Bi ⁑ ( x ) , Ai ⁑ ( x ) , Bi ⁑ ( x ) , respectively (§9.9), Ξ± k , Ξ² k , Ξ± k , Ξ² k are given by (10.21.25), (10.21.26), (10.21.30), and (10.21.31), with Ξ± = 2 1 3 ⁒ a m in the case of j Ξ½ , m and J Ξ½ ⁑ ( j Ξ½ , m ) , Ξ± = 2 1 3 ⁒ b m in the case of y Ξ½ , m and Y Ξ½ ⁑ ( y Ξ½ , m ) , Ξ± = 2 1 3 ⁒ a m in the case of j Ξ½ , m and J Ξ½ ⁑ ( j Ξ½ , m ) , Ξ± = 2 1 3 ⁒ b m in the case of y Ξ½ , m and Y Ξ½ ⁑ ( y Ξ½ , m ) . …