asymptotic expansions for large q

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11: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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§10.19(i) Asymptotic Forms

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§10.19(ii) Debye’s Expansions

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§10.19(iii) Transition Region

… ►See also §10.20(i).

12: 16.11 Asymptotic Expansions

§16.11 Asymptotic Expansions

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§16.11(i) Formal Series

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§16.11(ii) Expansions for Large Variable

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§16.11(iii) Expansions for Large Parameters

… ►Asymptotic expansions for the polynomials ${}_{p+2}{}^{}F_q^{}(-r,r+a_0,𝐚;𝐛;z)$ as $r\to \mathrm{\infty }$ through integer values are given in Fields and Luke (1963b, a) and Fields (1965).

13: Bibliography J

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Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions ${\overline{ps}}_n^r(\eta ,h)$ and ${\overline{qs}}_n^r(\eta ,h)$ for large $h$ . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

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Referenced by:

§30.9(ii).

Encodings:

BibTeX

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14: 28.4 Fourier Series

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§28.4(iv) Case $q=0$

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§28.4(v) Change of Sign of $q$

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§28.4(vi) Behavior for Small $q$

… ►For further terms and expansions see Meixner and Schäfke (1954, p. 122) and McLachlan (1947, §3.33). ►

§28.4(vii) Asymptotic Forms for Large $m$

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15: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(i) Numerical Use of Asymptotic Expansions

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§2.11(iii) Exponentially-Improved Expansions

… ► … ►For large $|z|$, with $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$ (), the Whittaker function of the second kind has the asymptotic expansion (§13.19) …

16: 8.25 Methods of Computation

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§8.25(i) Series Expansions

►Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. For large $|z|$ the corresponding asymptotic expansions (generally divergent) are used instead. … ►

§8.25(iii) Asymptotic Expansions

►DiDonato and Morris (1986) describes an algorithm for computing $P(a,x)$ and $Q(a,x)$ for $a\ge 0$, $x\ge 0$, and $a+x\ne 0$ from the uniform expansions in §8.12. …

17: 2.10 Sums and Sequences

… ►for large $n$. … ►This identity can be used to find asymptotic approximations for large $n$ when the factor $v_j$ changes slowly with $j$, and $u_j$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). … ►As a first estimate for large $n$ … ►

§2.10(iii) Asymptotic Expansions of Entire Functions

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Example

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18: 15.12 Asymptotic Approximations

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§15.12(i) Large Variable

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§15.12(ii) Large $c$

… ►As $\lambda \to \mathrm{\infty }$, … ►For this result and an extension to an asymptotic expansion with error bounds see Jones (2001). … ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

19: 8.18 Asymptotic Expansions of $I_x(a,b)$

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§8.18(ii) Large Parameters: Uniform Asymptotic Expansions

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Symmetric Case

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General Case

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Inverse Function

►For asymptotic expansions for large values of $a$ and/or $b$ of the $x$-solution of the equation …

20: 13.8 Asymptotic Approximations for Large Parameters

§13.8 Asymptotic Approximations for Large Parameters

… ►For other asymptotic expansions for large $b$ and $z$ see López and Pagola (2010). … ►

§13.8(iii) Large $a$

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§13.8(iv) Large $a$ and $b$

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