Euler–Maclaurin formula

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11: 9.12 Scorer Functions

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§9.12(v) Connection Formulas

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§9.12(vi) Maclaurin Series

… ►where the integration contour separates the poles of $\mathrm{\Gamma }\left(\frac{1}{3}+\frac{1}{3}t\right)$ from those of $\mathrm{\Gamma }\left(-t\right)$. … ►For other phase ranges combine these results with the connection formulas (9.12.11)–(9.12.14) and the asymptotic expansions given in §9.7. … ►where $\gamma$ is Euler’s constant (§5.2(ii)). …

12: 13.2 Definitions and Basic Properties

… ►The first two standard solutions are: … ►Although $M(a,b,z)$ does not exist when $b=-n$, $n=0,1,2,\mathrm{\dots }$, many formulas containing $M(a,b,z)$ continue to apply in their limiting form. … ►

13.2.18 $$U(a,b,z)=\frac{\mathrm{\Gamma }\left(b-1\right)}{\mathrm{\Gamma }\left(a\right)}{z}^{1-b}+\frac{\mathrm{\Gamma }\left(1-b\right)}{\mathrm{\Gamma }\left(a-b+1\right)}+O\left(z^{2-\mathrm{\Re }b}\right),$$ , $b\ne 1$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathrm{\Re }$: real part and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E18

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See also:

Annotations for §13.2(iii), §13.2 and Ch.13

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13.2.19 $$U(a,1,z)=-\frac{1}{\mathrm{\Gamma }\left(a\right)}\left(\ln z+\psi \left(a\right)+2\gamma \right)+O\left(z\ln z\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $U(a,b,z)$: Kummer confluent hypergeometric function, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

13.5.9 (as corrected in later editions)

Referenced by:

§13.2(iii)

Permalink:

http://dlmf.nist.gov/13.2.E19

Encodings:

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See also:

Annotations for §13.2(iii), §13.2 and Ch.13

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§13.2(vii) Connection Formulas

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13: 15.2 Definitions and Analytical Properties

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§15.2(i) Gauss Series

… ►Because of the analytic properties with respect to $a$, $b$, and $c$, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. … ►This formula is also valid when $c=-m-\mathrm{\ell }$, $\mathrm{\ell }=0,1,2,\mathrm{\dots }$, provided that we use the interpretation … ►Formula (15.4.6) reads $F(a,b;a;z)={(1-z)}^{-b}$. …In that case we are using interpretation (15.2.6) since with interpretation (15.2.5) we would obtain that $F(-m,b;-m;z)$ is equal to the first $m+1$ terms of the Maclaurin series for ${(1-z)}^{-b}$.

14: Bibliography F

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Euler sums and contour integral representations. Experiment. Math. 7 (1), pp. 15–35.

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External Links:

ISSN 1058-6458, Link, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

(25.16.13), §25.16(ii), §25.16(ii), §25.16.

Encodings:

BibTeX

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The Edmonds asymptotic formulas for the $3j$ and $6j$ symbols. J. Math. Phys. 39 (7), pp. 3906–3915.

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External Links:

ISSN 0022-2488, Document, MathReview (A. J. Coleman), ZentralBlatt

Referenced by:

§34.8.

Encodings:

BibTeX

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Studies on Divergent Series and Summability & The Asymptotic Developments of Functions Defined by Maclaurin Series. Chelsea Publishing Co., New York.

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Notes:

Reprint with corrections of two books published originally in 1916 and 1936.

External Links:

MathReview

Referenced by:

§2.10(iii).

Encodings:

BibTeX

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On the coefficients in the recursion formulae of orthogonal polynomials. Proc. Roy. Irish Acad. Sect. A 76 (1), pp. 1–6.

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External Links:

MathReview (A. G. Law), ZentralBlatt

Referenced by:

§18.32, §32.15.

Encodings:

BibTeX

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15: 13.29 Methods of Computation

… ►Although the Maclaurin series expansion (13.2.2) converges for all finite values of $z$, it is cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. … ►For $U(a,b,z)$ and $W_{\kappa ,\mu }\left(z\right)$ we may integrate along outward rays from the origin in the sectors , with initial values obtained from connection formulas in §13.2(vii), §13.14(vii). … ►

13.29.8 $$w(n)\sim \frac{\sqrt{\pi }{\mathrm{e}}^{\frac{1}{2}z}{z}^{\frac{1}{4}(4a-2b+1)}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a+1-b\right)}{n}^{\frac{1}{4}(4a-2b-3)}{\mathrm{e}}^{-2\sqrt{nz}},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.29.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §13.29(iv), §13.29(iv), §13.29 and Ch.13

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