leading coefficients
(0.002 seconds)
11—20 of 21 matching pages
11: 2.4 Contour Integrals
…
βΊAssume that and are analytic on an open domain that contains , with the possible exceptions of and .
…
βΊThe coefficients
are determined as in §2.3(iii), the branch of being chosen to satisfy
…
βΊAdditionally, it may be advantageous to arrange that is constant on the path: this will usually lead to greater regions of validity and sharper error bounds.
…
βΊHigher coefficients
in (2.4.15) can be found from (2.3.18) with , , and replaced by .
…
βΊFor a symbolic method for evaluating the coefficients in the asymptotic expansions see VidΕ«nas and Temme (2002).
…
12: 29.7 Asymptotic Expansions
…
βΊ
29.7.1
…
βΊ
29.7.3
βΊ
29.7.4
…
βΊ
29.7.6
…
βΊIn Müller (1966c) it is shown how these expansions lead to asymptotic expansions for the Lamé functions and .
…
13: 30.8 Expansions in Series of Ferrers Functions
…
βΊThen the set of coefficients
, is the solution of the difference equation
…
βΊThe coefficients
satisfy (30.8.4) for all when we set for .
For they agree with the coefficients defined in §30.8(i).
…The set of coefficients
, , is the recessive solution of (30.8.4) as that is normalized by
…It should be noted that if the forward recursion (30.8.4) beginning with ,
leads to , then is undefined for and does not exist.
…
14: 28.6 Expansions for Small
…
βΊLeading terms of the power series for and for are:
…
βΊLeading terms of the of the power series for are:
…
βΊFor more details on these expansions and recurrence relations for the coefficients see Frenkel and Portugal (2001, §2).
…
βΊHigher coefficients in the foregoing series can be found by equating coefficients in the following continued-fraction equations:
…
βΊLeading terms of the power series for the normalized functions are:
…
15: 3.3 Interpolation
…
βΊand are the Lagrangian interpolation coefficients defined by
βΊ
3.3.10
…
βΊ
3.3.12
…
βΊ
3.3.15
.
…
βΊ
3.3.18
.
…
16: 2.10 Sums and Sequences
…
βΊIn the present example it leads to
…
βΊwhere
…
βΊ
§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method
βΊLet be analytic on the annulus , with Laurent expansion … βΊWe need a “comparison function” with the properties: …17: 16.23 Mathematical Applications
…
βΊA variety of problems in classical mechanics and mathematical physics lead to Picard–Fuchs equations.
…
βΊMany combinatorial identities, especially ones involving binomial and related coefficients, are special cases of hypergeometric identities.
…
18: 18.33 Polynomials Orthogonal on the Unit Circle
…
βΊ
…
βΊAskey (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.
…
βΊwith complex coefficients
and of a certain degree define the reversed polynomial
by
…The Verblunsky coefficients (also called Schur parameters or reflection coefficients) are the coefficients
in the SzegΕ recurrence relations
…
βΊFor as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure in (18.33.17)) and with the Verblunsky coefficients in (18.33.23), (18.33.24), SzegΕ’s theorem states that
…
19: 8.11 Asymptotic Approximations and Expansions
…
βΊwhere
…
βΊ
8.11.9
…
βΊThis reference also contains explicit formulas for the coefficients in terms of Stirling numbers.
…
βΊ
8.11.18
,
βΊuniformly for , with
…
20: Errata
…
βΊ
Section 16.11(i)
…
βΊ
Additions
…
βΊ
Equation (3.3.34)
…
βΊ
Section 34.1
…
βΊ
Equation (10.20.14)
…
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
10.20.14
Originally this coefficient was given incorrectly as . The other coefficients in this equation have not been changed.
Reported 2012-05-11 by Antony Lee.