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21: 2.4 Contour Integrals

§2.4 Contour Integrals

… ►Then by integration by parts the integral … ►By subtraction from (2.4.3) … ►In consequence, the asymptotic expansion obtained from (2.4.14) is no longer null. … ►For integral representations of the $b_{2s}$ and their asymptotic behavior as $s\to \mathrm{\infty }$ see Boyd (1995). …

22: 36.11 Leading-Order Asymptotics

§36.11 Leading-Order Asymptotics

… ►and far from the bifurcation set, the cuspoid canonical integrals are approximated by … ►

36.11.5 $${\mathrm{\Psi }}_3\left(0,y,0\right)=\overline{{\mathrm{\Psi }}_3(0,-y,0)}=\exp \left(\frac{1}{4}\mathrm{i}\pi \right)\sqrt{\pi /y}\left(1-(\mathrm{i}/\sqrt{3})\exp \left(\frac{3}{2}\mathrm{i}{(2y/5)}^{5/3}\right)+o\left(1\right)\right),$$ $y\to +\mathrm{\infty }$.

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Symbols:

${\mathrm{\Psi }}_K\left(𝐱\right)$: canonical integral function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $y$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

… ►

36.11.7 $${\mathrm{\Psi }}^{(\mathrm{E})}\left(0,0,z\right)=\frac{\pi }{z}\left(\mathrm{i}+\sqrt{3}\exp \left(\frac{4}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

►

36.11.8 $${\mathrm{\Psi }}^{(\mathrm{H})}\left(0,0,z\right)=\frac{2\pi }{z}\left(1-\frac{\mathrm{i}}{\sqrt{3}}\exp \left(\frac{1}{27}\mathrm{i}{z}^3\right)+o\left(1\right)\right),$$ $z\to \pm \mathrm{\infty }$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, ${\mathrm{\Psi }}^{(\mathrm{H})}\left(𝐱\right)$: hyperbolic umbilic canonical integral function, $\mathrm{i}$: imaginary unit, $o\left(x\right)$: order less than and $z$: real parameter

Permalink:

http://dlmf.nist.gov/36.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §36.11, §36.11 and Ch.36

23: Morris Newman

… ► 2007) received his early training in mathematics from New York University and Columbia University, obtaining his Ph. …from the University of Pennsylvania in 1952. From 1952–1977 he served as a research mathematician at the National Bureau of Standards. … ►Newman wrote the book Matrix Representations of Groups, published by the National Bureau of Standards in 1968, and the book Integral Matrices, published by Academic Press in 1972, which became a classic. … ►Newman was a member of the original editorial committee for the DLMF project, in existence from the mid-1990’s to the mid-2010’s. …

24: 8.21 Generalized Sine and Cosine Integrals

§8.21 Generalized Sine and Cosine Integrals

… ►From §§8.2(i) and 8.2(ii) it follows that each of the four functions $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ is a multivalued function of $z$ with branch point at $z=0$. … ►From here on it is assumed that unless indicated otherwise the functions $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$ have their principal values. … ►(obtained from (5.2.1) by rotation of the integration path) is also needed. ►

§8.21(iii) Integral Representations

…

25: 8.6 Integral Representations

§8.6 Integral Representations

►

§8.6(i) Integrals Along the Real Line

… ►

§8.6(ii) Contour Integrals

… ►

Mellin–Barnes Integrals

… ►

§8.6(iii) Compendia

…

26: Bibliography M

… ►

Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables. Ellis Horwood Ltd./John Wiley & Sons, Inc, Chichester/New York.

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Notes:

Edited by F. D. Gakhov, Translated from the Russian by L. W. Longdon

External Links:

ISBN 0-85312-538-7, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.32(iv), §10.43(vi), §10.9(iv), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iii), §13.23(i), §14.17(vi), §15.14, §18.17(ix), §6.14(iii), §7.14(iii), §8.14.

Encodings:

BibTeX

►

On the Representation of Meijer’s $G$-Function in the Vicinity of Singular Unity. In Complex Analysis and Applications ’81 (Varna, 1981), pp. 383–398.

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External Links:

MathReview (B. D. Agrawal), ZentralBlatt

Referenced by:

§16.21.

Encodings:

BibTeX

… ►

An integral representation for the Bessel form. J. Comput. Appl. Math. 57 (1-2), pp. 251–260.

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External Links:

ISSN 0377-0427, Document, MathReview (Hans-Jürgen Glaeske), ZentralBlatt

Referenced by:

§18.34(ii).

Encodings:

BibTeX

… ►

An integral representation for the Lambert $W$ function.

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External Links:

2012.02480

Referenced by:

(4.13.16), §4.13.

Encodings:

BibTeX

… ►

On the convergence of the Chebyshev series for functions possessing a singularity in the range of representation. SIAM J. Numer. Anal. 3 (3), pp. 390–409.

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External Links:

ISSN 1095-7170, Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

…

27: 25.2 Definition and Expansions

… ►

25.2.2 $$\zeta \left(s\right)=\frac{1}{1-2^{-s}}\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{(2n+1)}^s},$$ $\mathrm{\Re }s>1$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\mathrm{\Re }$: real part, $n$: nonnegative integer and $s$: complex variable

Keywords:

infinite series

Proof sketch:

Derivable from (25.2.1).

A&S Ref:

23.2.20 (is the special case with integer values of $s$)

Referenced by:

(25.11.11)

Permalink:

http://dlmf.nist.gov/25.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(ii), §25.2 and Ch.25

… ►

§25.2(iii) Representations by the Euler–Maclaurin Formula

►

25.2.8 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-s{\int }_N^{\mathrm{\infty }}\frac{x-\lfloor x\rfloor }{x^{s+1}}dx,$$ $\mathrm{\Re }s>0$, $N=1,2,3,\mathrm{\dots }$.

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\lfloor x\rfloor$: floor of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Source:

Apostol (1976, (25), p. 269)

Proof sketch:

Derivable from Apostol (1976, Theorem 12.21, p. 269) by setting $a=1$ and $N\mapsto N-1$.

A&S Ref:

23.2.9

Referenced by:

(25.2.9)

Permalink:

http://dlmf.nist.gov/25.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

►

25.2.9 $$\zeta \left(s\right)=\sum _{k=1}^N\frac{1}{k^s}+\frac{N^{1-s}}{s-1}-\frac{1}{2}{N}^{-s}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}{N}^{1-s-2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_N^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$; $n,N=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula, improper integral

Proof sketch:

Derivable from (25.2.8) by repeated integration by parts.

Referenced by:

§25.18(i), (25.2.10)

Permalink:

http://dlmf.nist.gov/25.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

►

25.2.10 $$\zeta \left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum _{k=1}^n\left(\genfrac{}{}{0.0pt}{}{s+2k-2}{2k-1}\right)\frac{B_{2k}}{2k}-\left(\genfrac{}{}{0.0pt}{}{s+2n}{2n+1}\right){\int }_1^{\mathrm{\infty }}\frac{{\tilde{B}}_{2n+1}\left(x\right)}{x^{s+2n+1}}dx,$$ $\mathrm{\Re }s>-2n$, $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\int$: integral, ${\tilde{B}}_n\left(x\right)$: periodic Bernoulli functions, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $s$: complex variable

Keywords:

Euler–Maclaurin formula

Proof sketch:

Derivable from (25.2.9) with $N=1$.

A&S Ref:

23.2.3

Permalink:

http://dlmf.nist.gov/25.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iii), §25.2 and Ch.25

…

28: 13.29 Methods of Computation

… ►For $M(a,b,z)$ and $M_{\kappa ,\mu }\left(z\right)$ this means that in the sector $|\mathrm{ph}z|\le \pi$ we may integrate along outward rays from the origin with initial values obtained from (13.2.2) and (13.14.2). ►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). In the sector the integration has to be towards the origin, with starting values computed from asymptotic expansions (§§13.7 and 13.19). … ►

§13.29(iii) Integral Representations

►The integral representations (13.4.1) and (13.4.4) can be used to compute the Kummer functions, and (13.16.1) and (13.16.5) for the Whittaker functions. …

29: 12.5 Integral Representations

§12.5 Integral Representations

►

§12.5(i) Integrals Along the Real Line

… ►

§12.5(ii) Contour Integrals

… ►

§12.5(iii) Mellin–Barnes Integrals

… ►

§12.5(iv) Compendia

…

30: 11.5 Integral Representations

§11.5 Integral Representations

►

§11.5(i) Integrals Along the Real Line

… ►

§11.5(ii) Contour Integrals

… ►

Mellin–Barnes Integrals

… ►

§11.5(iii) Compendia

…