asymptotic expansions for large variable

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31: 10.67 Asymptotic Expansions for Large Argument

§10.67 Asymptotic Expansions for Large Argument

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§10.67(i) ${\mathrm{ber}}_{\nu }x,{\mathrm{bei}}_{\nu }x,{\ker }_{\nu }x,{\mathrm{kei}}_{\nu }x$, and Derivatives

… ►The contributions of the terms in ${\ker }_{\nu }x$, ${\mathrm{kei}}_{\nu }x$, ${\ker }_{\nu }^{\prime }x$, and ${\mathrm{kei}}_{\nu }^{\prime }x$ on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). … ►

§10.67(ii) Cross-Products and Sums of Squares in the Case $\nu =0$

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32: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

§28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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28.25.1 $${\mathrm{M}}_{\nu }^{(3,4)}(z,h)\sim \frac{{\mathrm{e}}^{\pm \mathrm{i}\left(2h\cosh z-\left(\frac{1}{2}\nu +\frac{1}{4}\right)\pi \right)}}{{\left(\pi h(\cosh z+1)\right)}^{\frac{1}{2}}}\sum _{m=0}^{\mathrm{\infty }}\frac{D_m^{\pm }}{{\left(\mp 4\mathrm{i}h(\cosh z+1)\right)}^m},$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $m$: integer, $h$: parameter, $z$: complex variable, $\nu$: complex parameter and $D_m^{\pm }$: coefficients

A&S Ref:

20.9.1 (for 3 and for integer $\nu$) 20.9.3 (for 4 and for integer $\nu$)

Referenced by:

§28.25

Permalink:

http://dlmf.nist.gov/28.25.E1

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See also:

Annotations for §28.25 and Ch.28

… ►The expansion (28.25.1) is valid for ${\mathrm{M}}_{\nu }^{(3)}(z,h)$ when ►

28.25.4 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le 2\pi -\delta$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $h$: parameter, $z$: complex variable and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/28.25.E4

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See also:

Annotations for §28.25 and Ch.28

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28.25.5 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-2\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le \pi -\delta$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $h$: parameter, $z$: complex variable and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/28.25.E5

Encodings:

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See also:

Annotations for §28.25 and Ch.28

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33: 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).

34: Bibliography B

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Asymptotic expansions of the modified Bessel function of the third kind of imaginary order. SIAM J. Appl. Math. 15, pp. 1315–1323.

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External Links:

ISSN 0036-1399, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§10.45.

Encodings:

BibTeX

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Chapter 13 of Ramanujan’s second notebook: Integrals and asymptotic expansions. Expo. Math. 2 (4), pp. 289–347.

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External Links:

ISSN 0723-0869, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.10(iii).

Encodings:

BibTeX

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Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal. 1 (3), pp. 294–313.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Eberhard Wagner), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

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Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation. J. Phys. A 30 (2), pp. 559–571.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§31.13.

Encodings:

BibTeX

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On the asymptotic expansion of some integrals. Arch. Math. (Basel) 42 (3), pp. 253–259.

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External Links:

ISSN 0003-889X, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§2.5(ii).

Encodings:

BibTeX

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35: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

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§30.9(i) Prolate Spheroidal Wave Functions

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). … ►

36: Bibliography O

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Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.

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External Links:

ISSN 0219-5305, Document, MathReview, ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.

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External Links:

ISSN 0219-5305, Document, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii).

Encodings:

BibTeX

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A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Proc. Cambridge Philos. Soc. 47, pp. 699–712.

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External Links:

Document, MathReview (J. Todd), ZentralBlatt

Referenced by:

§10.21(vii).

Encodings:

BibTeX

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Some new asymptotic expansions for Bessel functions of large orders. Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.

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External Links:

Document, MathReview (F. Oberhettinger), ZentralBlatt

Referenced by:

§10.19(iii), §10.21(vii).

Encodings:

BibTeX

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37: 28.26 Asymptotic Approximations for Large $q$

§28.26 Asymptotic Approximations for Large $q$

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§28.26(i) Goldstein’s Expansions

… ►The asymptotic expansions of ${\mathrm{Fs}}_m(z,h)$ and ${\mathrm{Gs}}_m(z,h)$ in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively. … ►

§28.26(ii) Uniform Approximations

… ►For asymptotic approximations for ${\mathrm{M}}_{\nu }^{(3,4)}(z,h)$ see also Naylor (1984, 1987, 1989).

38: 25.11 Hurwitz Zeta Function

… ►The function $\zeta (s,a)$ was introduced in Hurwitz (1882) and defined by the series expansion … ►For other series expansions similar to (25.11.10) see Coffey (2008). … ►

§25.11(xii) $a$-Asymptotic Behavior

… ►As $a\to \mathrm{\infty }$ in the sector , with $s(\ne 1)$ and $\delta$ fixed, we have the asymptotic expansion … ►Similarly, as $a\to \mathrm{\infty }$ in the sector , …

39: 10.68 Modulus and Phase Functions

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10.68.1 $$M_{\nu }\left(x\right){\mathrm{e}}^{\mathrm{i}{\theta }_{\nu }\left(x\right)}={\mathrm{ber}}_{\nu }x+\mathrm{i}{\mathrm{bei}}_{\nu }x,$$

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Symbols:

${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $M_{\nu }\left(x\right)$: modulus of Bessel functions, ${\theta }_{\nu }\left(x\right)$: phase of Bessel functions, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.68.E1

Encodings:

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See also:

Annotations for §10.68(i), §10.68 and Ch.10

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10.68.2 $$N_{\nu }\left(x\right){\mathrm{e}}^{\mathrm{i}{\varphi }_{\nu }\left(x\right)}={\ker }_{\nu }x+\mathrm{i}{\mathrm{kei}}_{\nu }x,$$

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Symbols:

${\mathrm{kei}}_{\nu }\left(x\right)$: Kelvin function, ${\ker }_{\nu }\left(x\right)$: Kelvin function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $N_{\nu }\left(x\right)$: modulus of derivatives of Bessel functions, ${\varphi }_{\nu }\left(x\right)$: phase of derivatives of Bessel functions, $x$: real variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.68.E2

Encodings:

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See also:

Annotations for §10.68(i), §10.68 and Ch.10

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§10.68(iii) Asymptotic Expansions for Large Argument

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10.68.16 $$M_{\nu }\left(x\right)=\frac{{\mathrm{e}}^{x/\sqrt{2}}}{{(2\pi x)}^{\frac{1}{2}}}\left(1-\frac{\mu -1}{8\sqrt{2}}\frac{1}{x}+\frac{{(\mu -1)}^2}{256}\frac{1}{x^2}-\frac{(\mu -1)({\mu }^2+14\mu -399)}{6144\sqrt{2}}\frac{1}{x^3}+O\left(\frac{1}{x^4}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.10.21

Permalink:

http://dlmf.nist.gov/10.68.E16

Encodings:

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See also:

Annotations for §10.68(iii), §10.68 and Ch.10

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10.68.17 $$\ln M_{\nu }\left(x\right)=\frac{x}{\sqrt{2}}-\frac{1}{2}\ln \left(2\pi x\right)-\frac{\mu -1}{8\sqrt{2}}\frac{1}{x}-\frac{(\mu -1)(\mu -25)}{384\sqrt{2}}\frac{1}{x^3}-\frac{(\mu -1)(\mu -13)}{128}\frac{1}{x^4}+O\left(\frac{1}{x^5}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $M_{\nu }\left(x\right)$: modulus of Bessel functions, $\ln z$: principal branch of logarithm function, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.10.22

Permalink:

http://dlmf.nist.gov/10.68.E17

Encodings:

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See also:

Annotations for §10.68(iii), §10.68 and Ch.10

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40: 15.12 Asymptotic Approximations

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§15.12(i) Large Variable

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§15.12(ii) Large $c$

… ►As $\lambda \to \mathrm{\infty }$, … ►For this result and an extension to an asymptotic expansion with error bounds see Jones (2001). … ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).