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1: 34.5 Basic Properties: 6 ⁒ j Symbol
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34.5.8 { j 1 j 2 j 3 l 1 l 2 l 3 } = { j 2 j 1 j 3 l 2 l 1 l 3 } = { j 1 l 2 l 3 l 1 j 2 j 3 } .
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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 ) } ,
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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 .
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34.5.19 l { j 1 j 2 l j 2 j 1 j } = 0 , 2 ⁒ μ j odd, μ = min ⁑ ( j 1 , j 2 ) ,
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34.5.20 l ( 1 ) l + j ⁒ { j 1 j 2 l j 1 j 2 j } = ( 1 ) 2 ⁒ μ 2 ⁒ j + 1 , μ = min ⁑ ( j 1 , j 2 ) ,
2: Bibliography
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  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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  • V. S. Adamchik (1998) Polygamma functions of negative order. J. Comput. Appl. Math. 100 (2), pp. 191–199.
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  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
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  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
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  • A. Apelblat (1983) Table of Definite and Infinite Integrals. Physical Sciences Data, Vol. 13, Elsevier Scientific Publishing Co., Amsterdam.
  • 3: Bibliography G
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  • W. Gautschi (1994) Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.
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  • A. Gil, D. Ruiz-Antolín, J. Segura, and N. M. Temme (2016) Algorithm 969: computation of the incomplete gamma function for negative values of the argument. ACM Trans. Math. Softw. 43 (3), pp. 26:1–26:9.
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  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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  • M. L. Glasser (1976) Definite integrals of the complete elliptic integral K . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.
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  • Ya. I. GranovskiΔ­, I. M. Lutzenko, and A. S. Zhedanov (1992) Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.
  • 4: 10.3 Graphics
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    See accompanying text
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    Figure 10.3.10: H 0 ( 1 ) ⁑ ( x + i ⁒ y ) , 10 x 5 , 2.8 y 4 . …There is a cut along the negative real axis. Magnify 3D Help
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    See accompanying text
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    Figure 10.3.12: H 1 ( 1 ) ⁑ ( x + i ⁒ y ) , 10 x 5 , 2.8 y 4 . …There is a cut along the negative real axis. Magnify 3D Help
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    See accompanying text
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    Figure 10.3.14: H 5 ( 1 ) ⁑ ( x + i ⁒ y ) , 20 x 10 , 4 y 4 . …There is a cut along the negative real axis. Magnify 3D Help
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    See accompanying text
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    Figure 10.3.15: J 5.5 ⁑ ( x + i ⁒ y ) , 10 x 10 , 4 y 4 . …There is a cut along the negative real axis. Magnify 3D Help
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    See accompanying text
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    Figure 10.3.16: H 5.5 ( 1 ) ⁑ ( x + i ⁒ y ) , 20 x 10 , 4 y 4 . …There is a cut along the negative real axis. Magnify 3D Help
    5: 25.7 Integrals
    β–ΊFor definite integrals of the Riemann zeta function see Prudnikov et al. (1986b, §2.4), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).
    6: 34.8 Approximations for Large Parameters
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    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 Ο€ ⁒ ( 2 ⁒ j 1 + 1 ) ⁒ ( 2 ⁒ j 2 + 1 ) ⁒ ( 2 ⁒ l 3 + 1 ) ⁒ sin ⁑ ΞΈ ) 1 2 ⁒ ( cos ⁑ ( ( l 3 + 1 2 ) ⁒ ΞΈ 1 4 ⁒ Ο€ ) + o ⁑ ( 1 ) ) , j 1 , j 2 , j 3 ≫ l 3 ≫ 1 ,
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    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 ) ,
    7: Bibliography N
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  • D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.
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  • W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.
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  • E. W. Ng and M. Geller (1969) A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.
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  • C. J. Noble and I. J. Thompson (1984) COULN, a program for evaluating negative energy Coulomb functions. Comput. Phys. Comm. 33 (4), pp. 413–419.
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  • C. J. Noble (2004) Evaluation of negative energy Coulomb (Whittaker) functions. Comput. Phys. Comm. 159 (1), pp. 55–62.
  • 8: Sidebar 5.SB1: Gamma & Digamma Phase Plots
    β–ΊThe color encoded phases of Ξ“ ⁑ ( z ) (above) and ψ ⁑ ( z ) (below), are constrasted in the negative half of the complex plane. β–ΊIn the upper half of the image, the poles of Ξ“ ⁑ ( z ) are clearly visible at negative integer values of z : the phase changes by 2 ⁒ Ο€ around each pole, showing a full revolution of the color wheel. …
    9: 12.11 Zeros
    β–ΊFor large negative values of a the real zeros of U ⁑ ( a , x ) , U ⁑ ( a , x ) , V ⁑ ( a , x ) , and V ⁑ ( a , x ) can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …Here Ξ± = ΞΌ 4 3 ⁒ a s , a s denoting the s th negative zero of the function Ai (see §9.9(i)). … β–Ίwhere Ξ² = ΞΌ 4 3 ⁒ a s , a s denoting the s th negative zero of the function Ai and … β–Ί
    12.11.9 u a , 1 2 1 2 ⁒ ΞΌ ⁒ ( 1 1.85575 708 ⁒ ΞΌ 4 / 3 0.34438 34 ⁒ ΞΌ 8 / 3 0.16871 5 ⁒ ΞΌ 4 0.11414 ⁒ ΞΌ 16 / 3 0.0808 ⁒ ΞΌ 20 / 3 β‹― ) ,
    10: 20 Theta Functions
    Chapter 20 Theta Functions