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11: 28.16 Asymptotic Expansions for Large $q$

§28.16 Asymptotic Expansions for Large $q$

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28.16.1 $${\lambda }_{\nu }\left(h^2\right)\sim -2h^2+2sh-\frac{1}{8}(s^2+1)-\frac{1}{2^{7}h}(s^3+3s)-\frac{1}{2^{12}{h}^2}(5s^4+34s^2+9)-\frac{1}{2^{17}{h}^3}(33s^5+410s^3+405s)-\frac{1}{2^{20}{h}^4}(63s^6+1260s^4+2943s^2+486)-\frac{1}{2^{25}{h}^5}(527s^7+15617s^5+69001s^3+41607s)+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $\sim$: Poincaré asymptotic expansion, $h$: parameter and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.16.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.16 and Ch.28

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12: 6.16 Mathematical Applications

… ►The $n$th partial sum is given by …uniformly for $x\in [-\pi ,\pi ]$. … ►Compare Figure 6.16.1. … ►It occurs with Fourier-series expansions of all piecewise continuous functions. … …

13: 25.20 Approximations

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Cody et al. (1971) gives rational approximations for $\zeta \left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\le s\le 5$, $5\le s\le 11$, $11\le s\le 25$, $25\le s\le 55$. Precision is varied, with a maximum of 20S.

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Piessens and Branders (1972) gives the coefficients of the Chebyshev-series expansions of $s\zeta \left(s+1\right)$ and $\zeta \left(s+k\right)$, $k=2,3,4,5,8$, for $0\le s\le 1$ (23D).

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Luke (1969b, p. 306) gives coefficients in Chebyshev-series expansions that cover $\zeta \left(s\right)$ for $0\le s\le 1$ (15D), $\zeta \left(s+1\right)$ for $0\le s\le 1$ (20D), and $\ln \xi \left(\frac{1}{2}+\mathrm{i}x\right)$ (§25.4) for $-1\le x\le 1$ (20D). For errata see Piessens and Branders (1972).

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14: 8 Incomplete Gamma and Related
Functions

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15: 28 Mathieu Functions and Hill’s Equation

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16: 25.12 Polylogarithms

… ►The notation ${\mathrm{Li}}_2\left(z\right)$ was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828): … ►The remainder of the equations in this subsection apply to principal branches. … ►The cosine series in (25.12.7) has the elementary sum … ►For real or complex $s$ and $z$ the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ is defined by … ►(In the latter case (25.12.11) becomes (25.5.1)). …

17: Gergő Nemes

… ► 1988 in Szeged, Hungary) is a Research Fellow at the Alfréd Rényi Institute of Mathematics in Budapest, Hungary. … ► in mathematics (with distinction) and a M. …in mathematics (with honours) from Loránd Eötvös University, Budapest, Hungary and a Ph. … in mathematics from Central European University in Budapest, Hungary. … ►As of September 20, 2021, Nemes performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 25 Zeta and Related Functions. …

18: 16.22 Asymptotic Expansions

§16.22 Asymptotic Expansions

►Asymptotic expansions of $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for large $z$ are given in Luke (1969a, §§5.7 and 5.10) and Luke (1975, §5.9). For asymptotic expansions of Meijer $G$-functions with large parameters see Fields (1973, 1983).

19: 23 Weierstrass Elliptic and Modular
Functions

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20: Wolter Groenevelt

… ► 1976 in Leidschendam, the Netherlands) is an Associate Professor at the Delft University of Technology in Delft, The Netherlands. … ► in mathematics at the Delft University of Technology in 2004. ►Groenevelt’s research interests is in special functions and orthogonal polynomials and their relations with representation theory and interacting particle systems. ►As of September 20, 2022, Groenevelt performed a complete analysis and acted as main consultant for the update of the source citation and proof metadata for every formula in Chapter 18 Orthogonal Polynomials. ►In July 2023, Groenevelt was named Contributing Developer of the NIST Digital Library of Mathematical Functions.