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51: 2.5 Mellin Transform Methods

… ►where $J_{\nu }$ denotes the Bessel function (§10.2(ii)), and $x$ is a large positive parameter. …

52: 8.13 Zeros

… ►For information on the distribution and computation of zeros of $\gamma (a,\lambda a)$ and $\mathrm{\Gamma }(a,\lambda a)$ in the complex $\lambda$-plane for large values of the positive real parameter $a$ see Temme (1995a). …

53: Bibliography K

… ►

Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters. J. B. Wolters, Groningen.

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External Links:

MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

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54: 10.20 Uniform Asymptotic Expansions for Large Order

… ►

10.20.5 $$Y_{\nu }\left(\nu z\right)\sim -{\left(\frac{4\zeta }{1-z^2}\right)}^{\frac{1}{4}}\left(\frac{\mathrm{Bi}\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{1}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Bi}}^{\prime }\left({\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{5}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{B_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $Y_{\nu }\left(z\right)$: Bessel function of the second kind, $\sim$: Poincaré asymptotic expansion, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $A_k(\zeta )$: coefficients and $B_k(\zeta )$: coefficients

A&S Ref:

9.3.36

Permalink:

http://dlmf.nist.gov/10.20.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

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10.20.9 $$\begin{array}{c}H_{\nu }^{(1)}{}_{}{}^{\prime }\left(\nu z\right)\\ H_{\nu }^{(2)}{}_{}{}^{\prime }\left(\nu z\right)\end{array}\}\sim \frac{4{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}}{z}{\left(\frac{1-z^2}{4\zeta }\right)}^{\frac{1}{4}}\left(\frac{{\mathrm{e}}^{\mp 2\pi \mathrm{i}/3}\mathrm{Ai}\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{4}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{C_k(\zeta )}{{\nu }^{2k}}+\frac{{\mathrm{Ai}}^{\prime }\left({\mathrm{e}}^{\pm 2\pi \mathrm{i}/3}{\nu }^{\frac{2}{3}}\zeta \right)}{{\nu }^{\frac{2}{3}}}\sum _{k=0}^{\mathrm{\infty }}\frac{D_k(\zeta )}{{\nu }^{2k}}\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\zeta (z)$: solution, $C_k(\zeta )$: coefficients and $D_k(\zeta )$: coefficients

A&S Ref:

9.3.45

Referenced by:

§10.20(ii)

Permalink:

http://dlmf.nist.gov/10.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §10.20(i), §10.20 and Ch.10

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55: 28.34 Methods of Computation

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Use of asymptotic expansions and approximations for large $q$ (§§28.8(i), 28.16). See also Zhang and Jin (1996, pp. 482–485).

… ►

Use of asymptotic expansions and approximations for large $q$ (§§28.8(ii)–28.8(iv)).

… ►

Direct numerical integration (§3.7) of the differential equation (28.20.1) for moderate values of the parameters.

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Use of asymptotic expansions for large $z$ or large $q$. See §§28.25 and 28.26.

56: 13.19 Asymptotic Expansions for Large Argument

§13.19 Asymptotic Expansions for Large Argument

… ►

13.19.2 $$M_{\kappa ,\mu }\left(z\right)\sim \frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\mathrm{e}}^{\frac{1}{2}z}{z}^{-\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-\mu +\kappa \right)}_s{\left(\frac{1}{2}+\mu +\kappa \right)}_s}{s!}{z}^{-s}+\frac{\mathrm{\Gamma }\left(1+2\mu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)}{\mathrm{e}}^{-\frac{1}{2}z\pm (\frac{1}{2}+\mu -\kappa )\pi \mathrm{i}}{z}^{\kappa }\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}+\mu -\kappa \right)}_s{\left(\frac{1}{2}-\mu -\kappa \right)}_s}{s!}{(-z)}^{-s},$$ $-\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $s$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant

Referenced by:

§13.14(iv)

Permalink:

http://dlmf.nist.gov/13.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.19 and Ch.13

►provided that both $\mu \mp \kappa \ne -\frac{1}{2},-\frac{3}{2},\mathrm{\dots }$. … ►For an asymptotic expansion of $W_{\kappa ,\mu }\left(z\right)$ as $z\to \mathrm{\infty }$ that is valid in the sector $|\mathrm{ph}z|\le \pi -\delta$ and where the real parameters $\kappa$, $\mu$ are subject to the growth conditions $\kappa =o\left(z\right)$, $\mu =o\left(\sqrt{z}\right)$, see Wong (1973a).

57: 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

… ►

10.67.1 $${\ker }_{\nu }x\sim {\mathrm{e}}^{-x/\sqrt{2}}{\left(\frac{\pi }{2x}\right)}^{\frac{1}{2}}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{x^k}\cos \left(\frac{x}{\sqrt{2}}+\left(\frac{\nu }{2}+\frac{k}{4}+\frac{1}{8}\right)\pi \right),$$

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Symbols:

${\ker }_{\nu }\left(x\right)$: Kelvin function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.10.3, 9.10.5--9.10.7 (modified)

Referenced by:

§10.61(v), §10.67(ii), §10.68(iii)

Permalink:

http://dlmf.nist.gov/10.67.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.67(i), §10.67 and Ch.10

… ► ►

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

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58: 10.70 Zeros

§10.70 Zeros

►Asymptotic approximations for large zeros are as follows. … ►

10.70.1 $$\frac{\mu -1}{16t}+\frac{\mu -1}{32t^2}+\frac{(\mu -1)(5\mu +19)}{1536t^3}+\frac{3{(\mu -1)}^2}{512t^4}+\mathrm{\cdots }.$$

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Defines:

$f(t)$: function (locally)

A&S Ref:

9.10.35

Referenced by:

§10.70

Permalink:

http://dlmf.nist.gov/10.70.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.70 and Ch.10

►If $m$ is a large positive integer, then ►

$$\text{zeros of }{\mathrm{ber}}_{\nu }x\sim \sqrt{2}(t-f(t)),$$ $t=(m-\frac{1}{2}\nu -\frac{3}{8})\pi$,

…

59: 10.19 Asymptotic Expansions for Large Order

§10.19 Asymptotic Expansions for Large Order

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§10.19(i) Asymptotic Forms

… ►

§10.19(ii) Debye’s Expansions

… ►

§10.19(iii) Transition Region

… ►See also §10.20(i).

60: 10.40 Asymptotic Expansions for Large Argument

… ►

10.40.2 $$K_{\nu }\left(z\right)\sim {\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{a_k(\nu )}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $\delta$: small positive constant and $a_k(\nu )$: polynomial coefficient

A&S Ref:

9.7.2

Referenced by:

§10.40(i), §10.40(i), §10.40(ii), §10.40(iii), §10.41(iv), §10.45

Permalink:

http://dlmf.nist.gov/10.40.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.40(i), §10.40 and Ch.10

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