asymptotic expansions for large zeros

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1: 12.11 Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

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§12.11(iii) Asymptotic Expansions for Large Parameter

►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …

2: 10.21 Zeros

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§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros

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§10.21(vii) Asymptotic Expansions for Large Order

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§10.21(viii) Uniform Asymptotic Approximations for Large Order

… ►The asymptotic expansion of the large positive zeros (not necessarily the $m$th) of the function …

3: 12.14 The Function $W(a,x)$

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§12.14(xi) Zeros of $W(a,x)$, $W^{\prime }(a,x)$

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4: 9.9 Zeros

§9.9 Zeros

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§9.9(iv) Asymptotic Expansions

►For large $k$ … ►For error bounds for the asymptotic expansions of $a_k$, $b_k$, $a_k^{\prime }$, and $b_k^{\prime }$ see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999). ►

§9.9(v) Tables

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5: 10.24 Functions of Imaginary Order

… ►For mathematical properties and applications of ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$, including zeros and uniform asymptotic expansions for large $\nu$, see Dunster (1990a). …

6: 10.45 Functions of Imaginary Order

… ►For properties of ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$, including uniform asymptotic expansions for large $\nu$ and zeros, see Dunster (1990a). …

7: Bibliography O

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A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Proc. Cambridge Philos. Soc. 47, pp. 699–712.

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External Links:

Document, MathReview (J. Todd), ZentralBlatt

Referenced by:

§10.21(vii).

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8: 10.72 Mathematical Applications

… ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). …

9: 10.1 Special Notation

… ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

10: 18.26 Wilson Class: Continued

… ►For asymptotic expansions of Wilson polynomials of large degree see Wilson (1991), and for asymptotic approximations to their largest zeros see Chen and Ismail (1998). …