in terms of Bessel functions of fixed order

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1: 2.8 Differential Equations with a Parameter

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§2.8(iv) Case III: Simple Pole

… ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). …

2: 33.12 Asymptotic Expansions for Large $\eta$

… ►The second set is in terms of Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$, and they are uniform for fixed $\mathrm{\ell }$ and $0\le z\le 1-\delta$, where $\delta$ again denotes an arbitrary small positive constant. …

3: 10.68 Modulus and Phase Functions

… ►where $M_{\nu }\left(x\right)(>0)$, $N_{\nu }\left(x\right)(>0)$, ${\theta }_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $x$ and $\nu$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to \mathrm{\infty }$. … ►Equations (10.68.8)–(10.68.14) also hold with the symbols $\mathrm{ber}$, $\mathrm{bei}$, $M$, and $\theta$ replaced throughout by $\ker$, $\mathrm{kei}$, $N$, and $\varphi$, respectively. In place of (10.68.7), … ►When $\nu$ is fixed, $\mu =4{\nu }^2$, and $x\to \mathrm{\infty }$ … ►Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). …

4: 18.15 Asymptotic Approximations

… ►The latter expansions are in terms of Bessel functions, and are uniform in complex $z$-domains not containing neighborhoods of 1. … ►These expansions are in terms of Bessel functions and modified Bessel functions, respectively. … ►

In Terms of Elementary Functions

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In Terms of Bessel Functions

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In Terms of Airy Functions

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5: 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).

6: 10.40 Asymptotic Expansions for Large Argument

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$\nu$-Derivative

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§10.40(ii) Error Bounds for Real Argument and Order

… ►For the error term in (10.40.1) see §10.40(iii). ►

§10.40(iii) Error Bounds for Complex Argument and Order

… ►For higher re-expansions of the remainder term see Olde Daalhuis and Olver (1995a), Olde Daalhuis (1995, 1996), and Paris (2001a, b).

7: 14.15 Uniform Asymptotic Approximations

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§14.15(i) Large $\mu$, Fixed $\nu$

… ►Here $I$ and $K$ are the modified Bessel functions (§10.25(ii)). … ►For asymptotic expansions and explicit error bounds, see Dunster (2003b). ►

§14.15(iii) Large $\nu$, Fixed $\mu$

… ►See also Olver (1997b, pp. 311–313) and §18.15(iii) for a generalized asymptotic expansion in terms of elementary functions for Legendre polynomials $P_n\left(\cos \theta \right)$ as $n\to \mathrm{\infty }$ with $\theta$ fixed. …

8: 13.9 Zeros

… ►When and $b>0$ let ${\varphi }_r$, $r=1,2,3,\mathrm{\dots }$, be the positive zeros of $M(a,b,x)$ arranged in increasing order of magnitude, and let $j_{b-1,r}$ be the $r$th positive zero of the Bessel function $J_{b-1}\left(x\right)$ (§10.21(i)). … ►For fixed $a,b\in ℂ$ the large $z$-zeros of $M(a,b,z)$ satisfy … ►For fixed $b$ and $z$ in $ℂ$ the large $a$-zeros of $M(a,b,z)$ are given by … ►For fixed $a$ and $z$ in $ℂ$ the function $M(a,b,z)$ has only a finite number of $b$-zeros. … ►For fixed $b$ and $z$ in $ℂ$ the large $a$-zeros of $U(a,b,z)$ are given by …

9: 10.17 Asymptotic Expansions for Large Argument

§10.17 Asymptotic Expansions for Large Argument

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§10.17(iii) Error Bounds for Real Argument and Order

… ►If these expansions are terminated when $k=\mathrm{\ell }-1$, then the remainder term is bounded in absolute value by the first neglected term, provided that $\mathrm{\ell }\ge \max (\nu -\frac{1}{2},1)$. … ►For higher re-expansions of the remainder terms see Olde Daalhuis and Olver (1995a) and Olde Daalhuis (1995, 1996).

10: 10.41 Asymptotic Expansions for Large Order

§10.41 Asymptotic Expansions for Large Order

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§10.41(i) Asymptotic Forms

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