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11: 8.12 Uniform Asymptotic Expansions for Large Parameter

§8.12 Uniform Asymptotic Expansions for Large Parameter

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Inverse Function

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12: 12.10 Uniform Asymptotic Expansions for Large Parameter

§12.10 Uniform Asymptotic Expansions for Large Parameter

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§12.10(vi) Modifications of Expansions in Elementary Functions

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Modified Expansions

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13: 16.11 Asymptotic Expansions

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§16.11(iii) Expansions for Large Parameters

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16.11.10 $${}_{p+1}{}^{}F_p^{}(\genfrac{}{}{0pt}{}{a_1+r,\mathrm{\dots },a_{k-1}+r,a_k,\mathrm{\dots },a_{p+1}}{b_1+r,\mathrm{\dots },b_k+r,b_{k+1},\mathrm{\dots },b_p};z)=\sum _{n=0}^{m-1}\frac{{\left(a_1+r\right)}_n\mathrm{\cdots }{\left(a_{k-1}+r\right)}_n{\left(a_k\right)}_n\mathrm{\cdots }{\left(a_{p+1}\right)}_n}{{\left(b_1+r\right)}_n\mathrm{\cdots }{\left(b_k+r\right)}_n{\left(b_{k+1}\right)}_n\mathrm{\cdots }{\left(b_p\right)}_n}\frac{z^n}{n!}+O\left(\frac{1}{r^m}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $r$: large parameter

Permalink:

http://dlmf.nist.gov/16.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(iii), §16.11 and Ch.16

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16.11.11 $${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1+r,\mathrm{\dots },a_p+r}{b_1+r,\mathrm{\dots },b_q+r};z)=\sum _{n=0}^{m-1}\frac{{\left(a_1+r\right)}_n\mathrm{\cdots }{\left(a_p+r\right)}_n}{{\left(b_1+r\right)}_n\mathrm{\cdots }{\left(b_q+r\right)}_n}\frac{z^n}{n!}+O\left(\frac{1}{r^{(q-p)m}}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $r$: large parameter

Permalink:

http://dlmf.nist.gov/16.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(iii), §16.11 and Ch.16

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14: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

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§8.20(ii) Large $p$

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15: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

§13.21 Uniform Asymptotic Approximations for Large $\kappa$

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16: 12.14 The Function $W(a,x)$

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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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17: 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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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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18: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

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19: Bibliography L

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The solutions of the Mathieu equation with a complex variable and at least one parameter large. Trans. Amer. Math. Soc. 36 (3), pp. 637–695.

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External Links:

ISSN 0002-9947, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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Asymptotics of the first Appell function $F_1$ with large parameters II. Integral Transforms Spec. Funct. 24 (12), pp. 982–999.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

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Asymptotics of the first Appell function $F_1$ with large parameters. Integral Transforms Spec. Funct. 24 (9), pp. 715–733.

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External Links:

ISSN 1065-2469, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

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Asymptotic expansions of the Whittaker functions for large order parameter. Methods Appl. Anal. 6 (2), pp. 249–256.

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Notes:

Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part II

External Links:

ISSN 1073-2772, Pdf, MathReview, ZentralBlatt

Referenced by:

§13.21(i), §13.24(ii).

Encodings:

BibTeX

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20: Bibliography D

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Uniform asymptotic expansions for prolate spheroidal functions with large parameters. SIAM J. Math. Anal. 17 (6), pp. 1495–1524.

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External Links:

ISSN 0036-1410, Document, MathReview (R. K. Gupta), ZentralBlatt

Referenced by:

§30.9(i).

Encodings:

BibTeX

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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. SIAM J. Math. Anal. 21 (4), pp. 995–1018.

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Notes:

Errata: In eq. (2.8) replace ${(x^2/4)}^2$ by ${(x^2/4)}^s$. In eq. (4.2) $\varphi (\zeta )$ should be replaced by $\psi (\zeta )$. In the second line of eq. (4.7) insert an external factor $e^{-2\pi ij/3}$ and change the upper limit of the sum to $n-1$. In eq. (4.15) the factor $\zeta$ is missing from the arguments of the functions Bi_ν and Bi’_ν.

External Links:

ISSN 0036-1410, Document, MathReview (A. de Castro Brzezicki), ZentralBlatt

Referenced by:

§10.24, §10.45, §2.8(iv).

Encodings:

BibTeX

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Conical functions with one or both parameters large. Proc. Roy. Soc. Edinburgh Sect. A 119 (3-4), pp. 311–327.

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External Links:

ISSN 0308-2105, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.20(i), §14.20(ix), §14.20(ix), §14.20(viii), §14.23.

Encodings:

BibTeX

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