# 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 $\tfrac{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 $|\epsilon|$
###### §33.20(i) Case $\epsilon=0$
where $A(\epsilon,\ell)$ 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 $2\ell+1$ and $2\ell+2$.
##### 4: 33.12 Asymptotic Expansions for Large $\eta$
Then as $\eta\to\infty$, … 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 $\ell$ and $\delta\leq z<\infty$, where $\delta$ is an arbitrary small positive constant. … The second set is in terms of Bessel functions of orders $2\ell+1$ and $2\ell+2$, and they are uniform for fixed $\ell$ and $0\leq z\leq 1-\delta$, 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 $\phi_{\nu}\left(x\right)$ are continuous real functions of $x$ and $\nu$, with the branches of $\theta_{\nu}\left(x\right)$ and $\phi_{\nu}\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to\infty$. … Equations (10.68.8)–(10.68.14) also hold with the symbols $\operatorname{ber}$, $\operatorname{bei}$, $M$, and $\theta$ replaced throughout by $\operatorname{ker}$, $\operatorname{kei}$, $N$, and $\phi$, 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 $\phi_{1}\left(0\right)=\tfrac{5}{4}\pi$ (Eq. …
##### 6: 10.40 Asymptotic Expansions for Large Argument
###### §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).
##### 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
###### Terms, Phrases and Expressions
• term:

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

• ##### 10: 14.15 Uniform Asymptotic Approximations
###### §14.15(i) Large $\mu$, Fixed $\nu$
Here $I$ and $K$ are the modified Bessel functions10.25(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 $\operatorname{env}$ 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 $n\to\infty$ with $\theta$ fixed. …