Lagrange formula for reversion of series
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21: 14.28 Sums
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§14.28(ii) Heine’s Formula
… ►The series converges uniformly for outside or on , and within or on . …22: 6.18 Methods of Computation
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►For small or moderate values of and , the expansion in power series (§6.6) or in series of spherical Bessel functions (§6.10(ii)) can be used.
For large or these series suffer from slow convergence or cancellation (or both).
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►Also, other ranges of can be covered by use of the continuation formulas of §6.4.
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►For example, the Gauss–Laguerre formula (§3.5(v)) can be applied to (6.2.2); see Todd (1954) and Tseng and Lee (1998).
For an application of the Gauss–Legendre formula (§3.5(v)) see Tooper and Mark (1968).
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23: 25.13 Periodic Zeta Function
24: 25.10 Zeros
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§25.10(ii) Riemann–Siegel Formula
… ►Sign changes of are determined by multiplying (25.9.3) by to obtain the Riemann–Siegel formula: ►
25.10.3
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►The error term can be expressed as an asymptotic series that begins
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►Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of in the critical strip are on the critical line (van de Lune et al. (1986)).
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25: 25.2 Definition and Expansions
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§25.2(ii) Other Infinite Series
… ►For further expansions of functions similar to (25.2.1) (Dirichlet series) see §27.4. … ►§25.2(iii) Representations by the Euler–Maclaurin Formula
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25.2.8
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25.2.10
, .
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26: 24.17 Mathematical Applications
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Euler–Maclaurin Summation Formula
… ►Boole Summation Formula
… ►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and -series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); -adic analysis (Koblitz (1984, Chapter 2)). …27: 18.18 Sums
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§18.18(v) Linearization Formulas
… ►Ultraspherical
… ►Hermite
… ►Formula (18.18.27) is known as the Hille–Hardy formula. … ►Hermite
…28: 5.19 Mathematical Applications
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►As shown in Temme (1996b, §3.4), the results given in §5.7(ii) can be used to sum infinite series of rational functions.
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29: 15.6 Integral Representations
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15.6.1
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15.6.2
; , .
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15.6.3
; , .
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15.6.8
; .
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►However, this reverses the direction of the integration contour, and in consequence (15.6.5) would need to be multiplied by .
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30: Bibliography W
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Reduction formulae for products of theta functions.
J. Res. Nat. Inst. Standards and Technology 117, pp. 297–303.
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Recursion formulae for hypergeometric functions.
Math. Comp. 22 (102), pp. 363–373.
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Explicit formulas for the associated Jacobi polynomials and some applications.
Canad. J. Math. 39 (4), pp. 983–1000.
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On the connection formulas of the fourth Painlevé transcendent.
Anal. Appl. (Singap.) 7 (4), pp. 419–448.
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Quadrature formulas for oscillatory integral transforms.
Numer. Math. 39 (3), pp. 351–360.
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