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1: 12.9 Asymptotic Expansions for Large Variable

§12.9 Asymptotic Expansions for Large Variable

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12.9.2 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ .

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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, $!$: factorial (as in $n!$), $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

A&S Ref:

19.8.2 (modification of)

Referenced by:

§12.9(i), §12.9(ii)

Permalink:

http://dlmf.nist.gov/12.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.3 $$U(a,z)\sim {\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s}\pm \mathrm{i}\frac{\sqrt{2\pi }}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\mathrm{e}}^{\mp \mathrm{i}\pi a}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s},$$ $\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{5}{4}\pi -\delta$ ,

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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, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/12.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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12.9.4 $$V(a,z)\sim \sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}\frac{{\left(\frac{1}{2}-a\right)}_{2s}}{s!{(2z^2)}^s}\pm \frac{\mathrm{i}}{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(\frac{1}{2}+a\right)}_{2s}}{s!{(2z^2)}^s},$$ $-\frac{1}{4}\pi +\delta \le \pm \mathrm{ph}z\le \frac{3}{4}\pi -\delta$.

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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, $!$: factorial (as in $n!$), $\mathrm{i}$: imaginary unit, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable, $s$: nonnegative integer, $a$: real or complex parameter and $\delta$: arbitrary small positive constant

Referenced by:

§12.9(i)

Permalink:

http://dlmf.nist.gov/12.9.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §12.9(i), §12.9 and Ch.12

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§12.9(ii) Bounds and Re-Expansions for the Remainder Terms

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2: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

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§8.20(i) Large $z$

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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:

pMML, png, TeX

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:

pMML, png, TeX

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

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§8.20(ii) Large $p$

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3: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

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§8.11(i) Large $z$, Fixed $a$

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§8.11(ii) Large $a$, Fixed $z$

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§8.11(iv) Large $a$, Bounded $(x-a)/{(2a)}^{\frac{1}{2}}$

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4: 15.12 Asymptotic Approximations

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§15.12(i) Large Variable

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§15.12(ii) Large $c$

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§15.12(iii) Other Large Parameters

… ►If , then as $\lambda \to \mathrm{\infty }$ with $|\mathrm{ph}\lambda |\le \frac{1}{2}\pi -\delta$, … ►For other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).

5: 12.11 Zeros

… ►If , $n=1,2,\mathrm{\dots }$, then $U(a,x)$ has $n$ positive real zeros. … ►

§12.11(ii) Asymptotic Expansions of Large Zeros

… ►When $a>-\frac{1}{2}$ the zeros are asymptotically given by $z_{a,s}$ and $\overline{z_{a,s}}$, where $s$ is a large positive integer and … ►

§12.11(iii) Asymptotic Expansions for Large Parameter

►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …

6: 16.11 Asymptotic Expansions

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§16.11(ii) Expansions for Large Variable

►In this subsection we assume that none of $a_1,a_2,\mathrm{\dots },a_p$ is a nonpositive integer. … ►The special case $a_1=1$, $p=q=2$ is discussed in Kim (1972). … ►

Case $p\le q-2$

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§16.11(iii) Expansions for Large Parameters

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7: 12.14 The Function $W(a,x)$

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§12.14(viii) Asymptotic Expansions for Large Variable

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§12.14(ix) Uniform Asymptotic Expansions for Large Parameter

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Positive $a$,

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Airy-type Uniform Expansions

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8: Bibliography F

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Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

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External Links:

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

Referenced by:

§15.12(iii), §15.12(iii).

Encodings:

BibTeX

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The third Appell function for one large variable. J. Approx. Theory 165, pp. 60–69.

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External Links:

ISSN 0021-9045, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

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Incomplete gamma functions for large values of their variables. Adv. in Appl. Math. 34 (3), pp. 467–485.

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External Links:

ISSN 0196-8858, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

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Uniform asymptotic expansions of certain classes of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 4 (3), pp. 482–507.

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External Links:

Document, MathReview (C. M. Joshi), ZentralBlatt

Referenced by:

§16.22.

Encodings:

BibTeX

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Uniform asymptotic expansions of a class of Meijer $G$-functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.

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External Links:

ISSN 0036-1410, Document, MathReview (Roop N. Kesarwani), ZentralBlatt

Referenced by:

§16.22.

Encodings:

BibTeX

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9: 8.21 Generalized Sine and Cosine Integrals

… ►Furthermore, $\mathrm{si}(a,z)$ and $\mathrm{ci}(a,z)$ are entire functions of $a$, and $\mathrm{Si}(a,z)$ and $\mathrm{Ci}(a,z)$ are meromorphic functions of $a$ with simple poles at $a=-1,-3,-5,\mathrm{\dots }$ and $a=0,-2,-4,\mathrm{\dots }$, respectively. … ►When $\mathrm{ph}z=0$ (and when $a\ne -1,-3,-5,\mathrm{\dots }$, in the case of $\mathrm{Si}(a,z)$, or $a\ne 0,-2,-4,\mathrm{\dots }$, in the case of $\mathrm{Ci}(a,z)$) the principal values of $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ are defined by (8.21.1) and (8.21.2) with the incomplete gamma functions assuming their principal values (§8.2(i)). … ►When , … ►

§8.21(viii) Asymptotic Expansions

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8.21.26 $$f(a,z)\sim z^{a-1}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\left(1-a\right)}_{2k}}{z^{2k}},$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $z$: complex variable, $a$: parameter, $k$: nonnegative integer and $f(a,z)$: auxiliary function

Permalink:

http://dlmf.nist.gov/8.21.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §8.21(viii), §8.21 and Ch.8

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10: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

… ►in each case uniformly with respect to $\lambda$ in the sector $|\mathrm{ph}\lambda |\le 2\pi -\delta$ (). … ►and ${\alpha }_3,{\alpha }_4,\mathrm{\dots }$ are defined by … ►Lastly, a uniform approximation for $\mathrm{\Gamma }(a,ax)$ for large $a$, with error bounds, can be found in Dunster (1996a). … ►

Inverse Function

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