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11: 2.4 Contour Integrals
β–ΊAssume that p ⁑ ( t ) and q ⁑ ( t ) are analytic on an open domain 𝐓 that contains 𝒫 , with the possible exceptions of t = a and t = b . … β–ΊThe coefficients b s are determined as in §2.3(iii), the branch of ph ⁑ p 0 being chosen to satisfy … β–ΊAdditionally, it may be advantageous to arrange that ⁑ ( z ⁒ p ⁑ ( t ) ) is constant on the path: this will usually lead to greater regions of validity and sharper error bounds. … β–ΊHigher coefficients b 2 ⁒ s in (2.4.15) can be found from (2.3.18) with Ξ» = 1 , ΞΌ = 2 , and s replaced by 2 ⁒ s . … β–ΊFor a symbolic method for evaluating the coefficients in the asymptotic expansions see VidΕ«nas and Temme (2002). …
12: 29.7 Asymptotic Expansions
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29.7.1 a Ξ½ m ⁑ ( k 2 ) p ⁒ ΞΊ Ο„ 0 Ο„ 1 ⁒ ΞΊ 1 Ο„ 2 ⁒ ΞΊ 2 β‹― ,
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29.7.3 Ο„ 0 = 1 2 3 ⁒ ( 1 + k 2 ) ⁒ ( 1 + p 2 ) ,
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29.7.4 Ο„ 1 = p 2 6 ⁒ ( ( 1 + k 2 ) 2 ⁒ ( p 2 + 3 ) 4 ⁒ k 2 ⁒ ( p 2 + 5 ) ) .
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29.7.6 Ο„ 2 = 1 2 10 ⁒ ( 1 + k 2 ) ⁒ ( 1 k 2 ) 2 ⁒ ( 5 ⁒ p 4 + 34 ⁒ p 2 + 9 ) ,
β–ΊIn Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions 𝐸𝑐 Ξ½ m ⁑ ( z , k 2 ) and 𝐸𝑠 Ξ½ m ⁑ ( z , k 2 ) . …
13: 30.8 Expansions in Series of Ferrers Functions
β–ΊThen the set of coefficients a n , k m ⁑ ( Ξ³ 2 ) , k = R , R + 1 , R + 2 , is the solution of the difference equation … β–ΊThe coefficients a n , k m ⁑ ( Ξ³ 2 ) satisfy (30.8.4) for all k when we set a n , k m ⁑ ( Ξ³ 2 ) = 0 for k < N . For k R they agree with the coefficients defined in §30.8(i). …The set of coefficients a n , k m ⁑ ( Ξ³ 2 ) , k = N 1 , N 2 , , is the recessive solution of (30.8.4) as k that is normalized by …It should be noted that if the forward recursion (30.8.4) beginning with f N 1 = 0 , f N = 1 leads to f R = 0 , then a n , k m ⁑ ( Ξ³ 2 ) is undefined for n < R and π–°π—Œ n m ⁑ ( x , Ξ³ 2 ) does not exist. …
14: 28.6 Expansions for Small q
β–ΊLeading terms of the power series for a m ⁑ ( q ) and b m ⁑ ( q ) for m 6 are: … β–ΊLeading terms of the of the power series for m = 7 , 8 , 9 , are: … β–ΊFor more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2). … β–ΊHigher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations: … β–ΊLeading terms of the power series for the normalized functions are: …
15: 3.3 Interpolation
β–Ίand A k n are the Lagrangian interpolation coefficients defined by β–Ί
3.3.10 A k n = ( 1 ) n 1 + k ( k n 0 ) ! ⁒ ( n 1 k ) ! ⁒ ( t k ) ⁒ m = n 0 n 1 ( t m ) .
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3.3.12 c n = 1 ( n + 1 ) ! ⁒ max ⁒ k = n 0 n 1 | t k | ,
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3.3.15 c 1 = 1 8 , 0 < t < 1 .
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3.3.18 c 2 = 1 / ( 9 ⁒ 3 ) = 0.0641 ⁒ , | t | < 1 .
16: 2.10 Sums and Sequences
β–ΊIn the present example it leads to … β–Ίwhere … β–Ί
§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method
β–ΊLet f ⁑ ( z ) be analytic on the annulus 0 < | z | < r , with Laurent expansion … β–ΊWe need a “comparison function” g ⁑ ( z ) with the properties: …
17: 16.23 Mathematical Applications
β–ΊA variety of problems in classical mechanics and mathematical physics lead to Picard–Fuchs equations. … β–ΊMany combinatorial identities, especially ones involving binomial and related coefficients, are special cases of hypergeometric identities. …
18: 18.33 Polynomials Orthogonal on the Unit Circle
β–Ίβ–ΊAskey (1982a) and Sri Ranga (2010) give more general results leading to what seem to be the right analogues of Jacobi polynomials on the unit circle. … β–Ίwith complex coefficients c k and of a certain degree n define the reversed polynomial p ⁒ ( z ) by …The Verblunsky coefficients (also called Schur parameters or reflection coefficients) are the coefficients Ξ± n in the SzegΕ‘ recurrence relationsβ–ΊFor w ⁑ ( z ) as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure ΞΌ in (18.33.17)) and with Ξ± n the Verblunsky coefficients in (18.33.23), (18.33.24), SzegΕ‘’s theorem states that …
19: 8.11 Asymptotic Approximations and Expansions
β–Ίwhere … β–Ί
8.11.9 b k ⁑ ( λ ) = λ ⁒ ( 1 λ ) ⁒ b k 1 ⁑ ( λ ) + ( 2 ⁒ k 1 ) ⁒ λ ⁒ b k 1 ⁑ ( λ ) .
β–ΊThis reference also contains explicit formulas for the coefficients in terms of Stirling numbers. … β–Ί
8.11.18 S n ⁑ ( x ) k = 0 d k ⁑ ( x ) ⁒ n k , n ,
β–Ίuniformly for x ( , 1 Ξ΄ ] , with …
20: Errata
β–Ί
  • Section 16.11(i)

    A sentence indicating that explicit representations for the coefficients c k are given in Volkmer (2023) was inserted just below (16.11.5).

  • β–Ί
  • Additions

    Equations: (3.3.3_1), (3.3.3_2), (5.15.9) (suggested by Calvin Khor on 2021-09-04), (8.15.2), Pochhammer symbol representation in (10.17.1) for a k ⁒ ( Ξ½ ) coefficient, Pochhammer symbol representation in (11.9.4) for a k ⁒ ( ΞΌ , Ξ½ ) coefficient, and (12.14.4_5).

  • β–Ί
  • Equation (3.3.34)

    In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

    Reported by Nico Temme on 2021-06-01

  • β–Ί
  • Section 34.1

    The reference for Clebsch-Gordan coefficients, Condon and Shortley (1935), was replaced by Edmonds (1974) and Rotenberg et al. (1959). The references for 3 ⁒ j , 6 ⁒ j , 9 ⁒ j symbols were made more precise.

  • β–Ί
  • Equation (10.20.14)
    10.20.14 B 3 ⁒ ( 0 ) = 959 71711 84603 25 47666 37125 00000 ⁒ 2 1 3

    Originally this coefficient was given incorrectly as B 3 ⁒ ( 0 ) = 430 99056 39368 59253 5 68167 34399 42500 00000 ⁒ 2 1 3 . The other coefficients in this equation have not been changed.

    Reported 2012-05-11 by Antony Lee.