of arbitrary order

SearchAdvancedHelp

(0.002 seconds)

21—30 of 49 matching pages

21: 18.15 Asymptotic Approximations

… ►

18.15.7

ⓘ

Defines:

${\epsilon }_M(\rho ,\theta )$ (locally)

Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\delta$: arbitrary small positive constant and $\rho$

Permalink:

http://dlmf.nist.gov/18.15.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.15(i), §18.15 and Ch.18

…

22: 28.1 Special Notation

… ► ►►►►

$m,n$	integers.
…
$\nu$	order of the Mathieu function or modified Mathieu function. (When $\nu$ is an integer it is often replaced by $n$.)
$\delta$	arbitrary small positive number.
…

…

23: 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. …

24: 30.1 Special Notation

… ► ►►►►

$x$	real variable. Except in §§30.7(iv), 30.11(ii), 30.13, and 30.14, .
…
$m$	order, a nonnegative integer.
…
$\delta$	arbitrary small positive constant.

…

25: 14.20 Conical (or Mehler) Functions

… ►uniformly for $\theta \in (0,\pi -\delta ]$, where $I$ and $K$ are the modified Bessel functions (§10.25(ii)) and $\delta$ is an arbitrary constant such that . … ►In this subsection and §14.20(ix), $A$ and $\delta$ denote arbitrary constants such that $A>0$ and . … ►

14.20.19 $$\alpha =\mu /\tau ,$$

ⓘ

Symbols:

$\tau$: real variable, $\mu$: general order and $\alpha$

Permalink:

http://dlmf.nist.gov/14.20.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §14.20(viii), §14.20 and Ch.14

… ►

§14.20(ix) Asymptotic Approximations: Large $\mu$, $0\le \tau \le A\mu$

… ►For the case of purely imaginary order and argument see Dunster (2013). …

26: 9.9 Zeros

… ►They are denoted by $a_k$, $a_k^{\prime }$, $b_k$, $b_k^{\prime }$, respectively, arranged in ascending order of absolute value for $k=1,2,\mathrm{\dots }.$ … ►They lie in the sectors and , and are denoted by ${\beta }_k$, ${\beta }_k^{\prime }$, respectively, in the former sector, and by $\overline{{\beta }_k}$, $\overline{{\beta }_k^{\prime }}$, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for $k=1,2,\mathrm{\dots }.$ See §9.3(ii) for visualizations. ►For the distribution in $ℂ$ of the zeros of ${\mathrm{Ai}}^{\prime }\left(z\right)-\sigma \mathrm{Ai}\left(z\right)$, where $\sigma$ is an arbitrary complex constant, see Muraveĭ (1976) and Gil and Segura (2014). …

27: 32.7 Bäcklund Transformations

… ►

32.7.6 $$({\alpha }_1,{\beta }_1,{\gamma }_1,{\delta }_1)=(-{\alpha }_0,-{\beta }_0,{\gamma }_0,{\delta }_0),$$

ⓘ

Symbols:

$\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\delta$: arbitrary constant

Permalink:

http://dlmf.nist.gov/32.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §32.7(iii), §32.7 and Ch.32

►

32.7.7 $$({\alpha }_2,{\beta }_2,{\gamma }_2,{\delta }_2)=(-{\beta }_0,-{\alpha }_0,-{\delta }_0,-{\gamma }_0).$$

ⓘ

Symbols:

$\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\delta$: arbitrary constant

Permalink:

http://dlmf.nist.gov/32.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §32.7(iii), §32.7 and Ch.32

… ►

32.7.36 $$({\alpha }_1,{\beta }_1,{\gamma }_1,{\delta }_1)=({\alpha }_0,{\beta }_0,-{\delta }_0+\frac{1}{2},-{\gamma }_0+\frac{1}{2}),$$

ⓘ

Symbols:

$\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\delta$: arbitrary constant

Permalink:

http://dlmf.nist.gov/32.7.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §32.7(vii), §32.7 and Ch.32

►

32.7.37 $$({\alpha }_2,{\beta }_2,{\gamma }_2,{\delta }_2)=({\alpha }_0,-{\gamma }_0,-{\beta }_0,{\delta }_0),$$

ⓘ

Symbols:

$\alpha$: arbitrary constant, $\beta$: arbitrary constant, $\gamma$: arbitrary constant and $\delta$: arbitrary constant

Permalink:

http://dlmf.nist.gov/32.7.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §32.7(vii), §32.7 and Ch.32

… ►The transformations ${𝒮}_j$, for $j=1,2,3$, generate a group of order 24. …

28: 28.20 Definitions and Basic Properties

… ►Then from §2.7(ii) it is seen that equation (28.20.2) has independent and unique solutions that are asymptotic to ${\zeta }^{1/2}{\mathrm{e}}^{\pm 2\mathrm{i}h\zeta }$ as $\zeta \to \mathrm{\infty }$ in the respective sectors $|\mathrm{ph}\left(\mp \mathrm{i}\zeta \right)|\le \frac{3}{2}\pi -\delta$, $\delta$ being an arbitrary small positive constant. … ►

28.20.9 $${\mathrm{M}}_{\nu }^{(3)}(z,h)=H_{\nu }^{(1)}\left(2h\cosh z\right)\left(1+O\left(\mathrm{sech}z\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

… ►

28.20.10 $${\mathrm{M}}_{\nu }^{(4)}(z,h)=H_{\nu }^{(2)}\left(2h\cosh z\right)\left(1+O\left(\mathrm{sech}z\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.20.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

… ►

28.20.11 $${\mathrm{M}}_{\nu }^{(1)}(z,h)=J_{\nu }\left(2h\cosh z\right)+{\mathrm{e}}^{|\mathrm{\Im }\left(2h\cosh z\right)|}O\left({\left(\mathrm{sech}z\right)}^{3/2}\right),$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{\Im }$: imaginary part, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.20.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

►

28.20.12 $${\mathrm{M}}_{\nu }^{(2)}(z,h)=Y_{\nu }\left(2h\cosh z\right)+{\mathrm{e}}^{|\mathrm{\Im }\left(2h\cosh z\right)|}O\left({(\mathrm{sech}z)}^{3/2}\right),$$

ⓘ

Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $O\left(x\right)$: order not exceeding, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, $\mathrm{\Im }$: imaginary part, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.33(ii)

Permalink:

http://dlmf.nist.gov/28.20.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

…

29: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

§1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

§1.18(iv) Formally Self-adjoint Linear Second Order Differential Operators

… ► … ► … ►

§1.18(v) Point Spectra and Eigenfunction Expansions

…

30: 2.1 Definitions and Elementary Properties

… ►

§2.1(i) Asymptotic and Order Symbols

… ►As $x\to c$ in $𝐗$ … ►(Here and elsewhere in this chapter $\delta$ is an arbitrary small positive constant.) … ►

§2.1(ii) Integration and Differentiation

►Integration of asymptotic and order relations is permissible, subject to obvious convergence conditions. …