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1: 12.16 Mathematical Applications

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2: 12.9 Asymptotic Expansions for Large Variable

§12.9 Asymptotic Expansions for Large Variable

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

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3: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

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§8.20(i) Large $z$

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4: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

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5: 15.12 Asymptotic Approximations

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§15.12(i) Large Variable

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6: 12.11 Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

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7: 16.11 Asymptotic Expansions

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§16.11(ii) Expansions for Large Variable

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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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8: 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(xi) Zeros of $W(a,x)$, $W^{\prime }(a,x)$

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9: Bibliography F

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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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The second Appell function for one large variable. Mediterr. J. Math. 10 (4), pp. 1853–1865.

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External Links:

ISSN 1660-5446, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

Encodings:

BibTeX

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10: 8.21 Generalized Sine and Cosine Integrals

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§8.21(viii) Asymptotic Expansions

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