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11: 14.26 Uniform Asymptotic Expansions

… ►The uniform asymptotic approximations given in §14.15 for $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). For an extension of §14.15(iv) to complex argument and imaginary parameters, see Dunster (1990b). …

12: Staff

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Howard S. Cohl, Technical Editor, NIST

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T. Mark Dunster, San Diego State University, Chap. 14

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Roderick S. C. Wong, City University of Hong Kong, Chaps. 1, 2, 18

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T. Mark Dunster, San Diego State University, for Chap. 14

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Roderick S. C. Wong, City University of Hong Kong, for Chaps. 2, 18

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13: 14 Legendre and Related Functions

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14: 14.32 Methods of Computation

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For the computation of conical functions see Gil et al. (2009, 2012), and Dunster (2014).

15: 28.8 Asymptotic Expansions for Large $q$

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§28.8(iii) Goldstein’s Expansions

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Barrett’s Expansions

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Dunster’s Approximations

►Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). … ►

16: 8.22 Mathematical Applications

… ►The function $\mathrm{\Gamma }(a,z)$, with $|\mathrm{ph}a|\le \frac{1}{2}\pi$ and $\mathrm{ph}z=\frac{1}{2}\pi$, has an intimate connection with the Riemann zeta function $\zeta \left(s\right)$ (§25.2(i)) on the critical line $\mathrm{\Re }s=\frac{1}{2}$. … ►If ${\zeta }_x(s)$ denotes the incomplete Riemann zeta function defined by …so that ${lim}_{x\to \mathrm{\infty }}{\zeta }_x(s)=\zeta \left(s\right)$, then ►

8.22.3 $${\zeta }_x(s)=\sum _{k=1}^{\mathrm{\infty }}{k}^{-s}P(s,kx),$$ $\mathrm{\Re }s>1$.

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Symbols:

$P(a,z)$: normalized incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $k$: nonnegative integer and ${\zeta }_x(s)$: incomplete Riemann zeta function

Permalink:

http://dlmf.nist.gov/8.22.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.22(ii), §8.22 and Ch.8

►For further information on ${\zeta }_x(s)$, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …

17: Preface

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T. Mark Dunster, San Diego State University

… ►Boggs, S. … S. … S. … S. …

18: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►where $F\left(x\right)$ is Dawson’s integral; see §7.2(ii). Then as $a\to \mathrm{\infty }$ in the sector , … ►For the asymptotic behavior of $c_k(\eta )$ as $k\to \mathrm{\infty }$ see Dunster et al. (1998) and Olde Daalhuis (1998c). … ►Lastly, a uniform approximation for $\mathrm{\Gamma }(a,ax)$ for large $a$, with error bounds, can be found in Dunster (1996a). ►For other uniform asymptotic approximations of the incomplete gamma functions in terms of the function $\mathrm{erfc}$ see Paris (2002b) and Dunster (1996a). …

19: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

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8.20.1 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z}\left(\sum _{k=0}^{n-1}{(-1)}^k\frac{{\left(p\right)}_k}{z^k}+{(-1)}^n\frac{{\left(p\right)}_n{\mathrm{e}}^z}{z^{n-1}}{E}_{n+p}\left(z\right)\right),$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

5.1.51

Permalink:

http://dlmf.nist.gov/8.20.E1

Encodings:

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See also:

Annotations for §8.20(i), §8.20 and Ch.8

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8.20.2 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{{\left(p\right)}_k}{z^k},$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E2

Encodings:

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See also:

Annotations for §8.20(i), §8.20 and Ch.8

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8.20.3 $$E_p\left(z\right)\sim \pm \frac{2\pi \mathrm{i}}{\mathrm{\Gamma }\left(p\right)}{\mathrm{e}}^{\mp p\pi \mathrm{i}}{z}^{p-1}+\frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\left(p\right)}_k}{z^k},$$ $\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{7}{2}\pi -\delta$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant

Referenced by:

§2.11(ii), §8.20(i)

Permalink:

http://dlmf.nist.gov/8.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.20(i), §8.20 and Ch.8

… ►For further information, including extensions to complex values of $x$ and $p$, see Temme (1994b, §4) and Dunster (1996b, 1997).

20: 2.8 Differential Equations with a Parameter

… ►For another approach to these problems based on convergent inverse factorial series expansions see Dunster et al. (1993) and Dunster (2001a, 2004). … ►For error bounds, more delicate error estimates, extensions to complex $\xi$, $\nu$, and $u$, zeros, and examples see Olver (1997b, Chapter 12), Boyd (1990a), and Dunster (1990a). … ►For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. … ►For a coalescing turning point and double pole see Boyd and Dunster (1986) and Dunster (1990b); in this case the uniform approximants are Bessel functions of variable order. ►For a coalescing turning point and simple pole see Nestor (1984) and Dunster (1994b); in this case the uniform approximants are Whittaker functions (§13.14(i)) with a fixed value of the second parameter. …