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11: 8.6 Integral Representations

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§8.6(ii) Contour Integrals

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12: 36.15 Methods of Computation

… ►Close to the origin $𝐱=\mathrm{𝟎}$ of parameter space, the series in §36.8 can be used. … ►Close to the bifurcation set but far from $𝐱=\mathrm{𝟎}$, the uniform asymptotic approximations of §36.12 can be used. ►

§36.15(iii) Integration along Deformed Contour

►Direct numerical evaluation can be carried out along a contour that runs along the segment of the real $t$-axis containing all real critical points of $\mathrm{\Phi }$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\exp \left(\mathrm{i}\mathrm{\Phi }\right)$. … ►

§36.15(iv) Integration along Finite Contour

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13: 2.4 Contour Integrals

§2.4 Contour Integrals

… ►is seen to converge absolutely at each limit, and be independent of $\sigma \in [c,\mathrm{\infty })$. … ►The most successful results are obtained on moving the integration contour as far to the left as possible. … ►Let $𝒫$ denote the path for the contour integral … ►and assigning an appropriate value to $c$ to modify the contour, the approximating integral is reducible to an Airy function or a Scorer function (§§9.2, 9.12). …

14: 21.1 Special Notation

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$g,h$	positive integers.
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$𝜶,𝜷$	$g$-dimensional vectors, with all elements in $[0,1)$, unless stated otherwise.
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$a\circ b$	intersection index of $a$ and $b$, two cycles lying on a closed surface. $a\circ b=0$ if $a$ and $b$ do not intersect. Otherwise $a\circ b$ gets an additive contribution from every intersection point. This contribution is $1$ if the basis of the tangent vectors of the $a$ and $b$ cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is $-1$.
${\oint }_a\omega$	line integral of the differential $\omega$ over the cycle $a$.

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15: 21.7 Riemann Surfaces

… ►On this surface, we choose $2g$ cycles (that is, closed oriented curves, each with at most a finite number of singular points) $a_j$, $b_j$, $j=1,2,\mathrm{\dots },g$, such that their intersection indices satisfy … ►

21.7.5 $${\oint }_{a_k}{\omega }_j={\delta }_{j,k},$$ $j,k=1,2,\mathrm{\dots },g$.

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $g$: positive integer, $\omega$: differential and $a_j$: cycles

Permalink:

http://dlmf.nist.gov/21.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(i), §21.7 and Ch.21

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21.7.6 $${\mathrm{\Omega }}_{jk}={\oint }_{b_k}{\omega }_j,$$ $j,k=1,2,\mathrm{\dots },g$,

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Symbols:

$g$: positive integer, $\omega$: differential and $b_j$: cycles

Permalink:

http://dlmf.nist.gov/21.7.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §21.7(i), §21.7 and Ch.21

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16: 11.5 Integral Representations

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§11.5(ii) Contour Integrals

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17: Bibliography K

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Closed-form representations of the Lambert $W$ function. Fract. Calc. Appl. Anal. 7 (2), pp. 177–190.

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External Links:

ISSN 1311-0454, MathReview (Lewis H. Ryder), ZentralBlatt

Referenced by:

§4.13.

Encodings:

BibTeX

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An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives. Computer Physics Comm. 132 (1-2), pp. 142–165.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§36.15(iii).

Encodings:

BibTeX

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