limiting forms for large ℓ

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1: 33.18 Limiting Forms for Large $\mathrm{\ell }$

§33.18 Limiting Forms for Large $\mathrm{\ell }$

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2: 33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

§33.10 Limiting Forms for Large $\rho$ or Large $\left|\eta \right|$

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§33.10(i) Large $\rho$

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§33.10(ii) Large Positive $\eta$

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§33.10(iii) Large Negative $\eta$

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3: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

§33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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§33.5(iv) Large $\mathrm{\ell }$

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4: 33.11 Asymptotic Expansions for Large $\rho$

§33.11 Asymptotic Expansions for Large $\rho$

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5: 11.13 Methods of Computation

… ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that ${𝐇}_{\nu }\left(x\right)$ and ${𝐋}_{\nu }\left(x\right)$ can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. …

6: 33.21 Asymptotic Approximations for Large $|r|$

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§33.21(i) Limiting Forms

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7: 10.45 Functions of Imaginary Order

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$${\tilde{K}}_{\nu }\left(x\right)={(\pi /(2x))}^{\frac{1}{2}}{\mathrm{e}}^{-x}\left(1+O\left(x^{-1}\right)\right).$$

… ►In consequence of (10.45.5)–(10.45.7), ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.45.1) when $x$ is large, and either ${\tilde{I}}_{\nu }\left(x\right)$ and $(1/\pi )\sinh \left(\pi \nu \right){\tilde{K}}_{\nu }\left(x\right)$, or ${\tilde{I}}_{\nu }\left(x\right)$ and ${\tilde{K}}_{\nu }\left(x\right)$, comprise a numerically satisfactory pair when $x$ is small, depending whether $\nu \ne 0$ or $\nu =0$. … ►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). …

8: 10.24 Functions of Imaginary Order

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$${\tilde{Y}}_{\nu }\left(x\right)=\sqrt{2/(\pi x)}\sin \left(x-\frac{1}{4}\pi \right)+O\left(x^{-\frac{3}{2}}\right).$$

… ►In consequence of (10.24.6), when $x$ is large ${\tilde{J}}_{\nu }\left(x\right)$ and ${\tilde{Y}}_{\nu }\left(x\right)$ comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). … … ►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). …

9: 13.29 Methods of Computation

… ►Although the Maclaurin series expansion (13.2.2) converges for all finite values of $z$, it is cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. For large $|z|$ the asymptotic expansions of §13.7 should be used instead. Accuracy is limited by the magnitude of $|z|$. However, this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied by the combination of (13.7.10) and (13.7.11), or by use of the hyperasymptotic expansions given in Olde Daalhuis and Olver (1995a). For large values of the parameters $a$ and $b$ the approximations in §13.8 are available. …

10: 9.17 Methods of Computation

… ►Although the Maclaurin-series expansions of §§9.4 and 9.12(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 asymptotic expansions of §§9.7 and 9.12(viii) should be used instead. Since these expansions diverge, the accuracy they yield is limited by the magnitude of $|z|$. However, in the case of $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$ this accuracy can be increased considerably by use of the exponentially-improved forms of expansion supplied in §9.7(v). … ►For details, including the application of a generalized form of Gaussian quadrature, see Gordon (1969, Appendix A) and Schulten et al. (1979). …