in terms of Bessel functions of variable order

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1: 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)). … ►Then for large $u$ asymptotic approximations of the solutions $w$ can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on $u$ and $\alpha$). …

2: 2.8 Differential Equations with a Parameter

… ►For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. ►For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii). …

3: 33.20 Expansions for Small $|ϵ|$

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§33.20(i) Case $ϵ=0$

… ►where $A(ϵ,\mathrm{\ell })$ is given by (33.14.11), (33.14.12), and …The functions $Y$ and $K$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by (33.20.6). ►

§33.20(iv) Uniform Asymptotic Expansions

… ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

4: 33.12 Asymptotic Expansions for Large $\eta$

… ►Then as $\eta \to \mathrm{\infty }$, … ►For derivations and additional terms in the expansions in this subsection see Abramowitz and Rabinowitz (1954) and Fröberg (1955). … ►The first set is in terms of Airy functions and the expansions are uniform for fixed $\mathrm{\ell }$ and , where $\delta$ is an arbitrary small positive constant. … ►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. … ►

5: 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), … ►Additional properties of the modulus and phase functions are given in Young and Kirk (1964, pp. xi–xv). …Thus this reference gives ${\varphi }_1\left(0\right)=\frac{5}{4}\pi$ (Eq. …

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: 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).

8: 18.38 Mathematical Applications

… ►However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. … ►The orthogonality relations (34.5.14) for the $6j$ symbols can be rewritten in terms of orthogonality relations for Racah polynomials as given by (18.25.9)–(18.25.12). … ►The Dunkl operator, introduced by Dunkl (1989), is an operator associated with reflection groups or root systems which has terms involving first order partial derivatives and reflection terms. … ►where the Bessel function $J_{\nu }\left(z\right)$ is defined in (10.2.2). …

9: Guide to Searching the DLMF

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Terms, Phrases and Expressions

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a textual word, a number, or a math symbol.

… ►If you do not want a term or a sequence of terms in your query to undergo math processing, you should quote them as a phrase. … ►For example, for the Bessel function $K_n\left(z\right)$, you can write K_n(z), BesselK_n(z), BesselK(n,z), or BesselK[n,z]. … ►

Single-letter terms

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10: 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). … ►For the Bessel functions $J$ and $Y$ see §10.2(ii), and for the $env$ functions associated with $J$ and $Y$ see §2.8(iv). … ►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. …