# in terms of Bessel functions of variable order

(0.022 seconds)

## 1—10 of 34 matching pages

##### 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 $|\u03f5|$

…
►

###### §33.20(i) Case $\u03f5=0$

… ►where $A(\u03f5,\mathrm{\ell})$ 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 ${M}_{\nu}\left(x\right)\phantom{\rule{0.3888888888888889em}{0ex}}(>0)$, ${N}_{\nu}\left(x\right)\phantom{\rule{0.3888888888888889em}{0ex}}(>0)$, ${\theta}_{\nu}\left(x\right)$, 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 $\mathrm{ker}$, $\mathrm{kei}$, $N$, 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

…
►

###### $\nu $-Derivative

… ►###### §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

►###### §10.19(i) Asymptotic Forms

… ►###### §10.19(ii) Debye’s Expansions

… ►###### §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

…
►
term:
…
►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
•
…

###### Terms, Phrases and Expressions

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

`K_n(z)`

, `BesselK_n(z)`

, `BesselK(n,z)`

, or `BesselK[n,z]`

.
…
►
Single-letter terms

##### 10: 14.15 Uniform Asymptotic Approximations

…
►