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1: 5.21 Methods of Computation

… ►Another approach is to apply numerical quadrature (§3.5) to the integral (5.9.2), using paths of steepest descent for the contour. …

2: 26.14 Permutations: Order Notation

… ►Equivalently, this is the sum over of the number of integers less than $\sigma (j)$ that lie in positions to the right of the $j$th position: $inv(35247816)=2+3+1+1+2+2+0=11.$ ►A descent of a permutation is a pair of adjacent elements for which the first is larger than the second. The permutation $35247816$ has two descents: $52$ and $81$. The major index is the sum of all positions that mark the first element of a descent: … ►The Eulerian number, denoted , is the number of permutations in ${𝔖}_n$ with exactly $k$ descents. …

3: 9.17 Methods of Computation

… ►In the first method the integration path for the contour integral (9.5.4) is deformed to coincide with paths of steepest descent (§2.4(iv)). …

4: 2.4 Contour Integrals

… ►Paths on which $\mathrm{\Im }\left(zp(t)\right)$ is constant are also the ones on which $|\exp \left(-zp(t)\right)|$ decreases most rapidly. For this reason the name method of steepest descents is often used. However, for the purpose of simply deriving the asymptotic expansions the use of steepest descent paths is not essential. …

5: 3.5 Quadrature

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§3.5(ix) Other Contour Integrals

… ►For example, steepest descent paths can be used; see §2.4(iv). … ►The steepest descent path is given by $\mathrm{\Im }\left(t-2\sqrt{t}\right)=0$, or in polar coordinates $t=r{\mathrm{e}}^{\mathrm{i}\theta }$ we have $r={\sec }^2\left(\frac{1}{2}\theta \right)$. … ►A special case is the rule for Hilbert transforms (§1.14(v)): …

6: Bibliography P

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Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.

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External Links:

ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§10.20(ii).

Encodings:

BibTeX

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7: Bibliography T

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Steepest descent paths for integrals defining the modified Bessel functions of imaginary order. Methods Appl. Anal. 1 (1), pp. 14–24.

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External Links:

ISSN 1073-2772, FullText, MathReview (Anatoly Kilbas), ZentralBlatt

Referenced by:

§10.74(viii).

Encodings:

BibTeX

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8: Bibliography B

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Error bounds for the method of steepest descents. Proc. Roy. Soc. London Ser. A 440, pp. 493–518.

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External Links:

ISSN 0962-8444, FullText, MathReview (Pablo Martín), ZentralBlatt

Referenced by:

§2.3(iii), §2.4(iii), §9.10(ii).

Encodings:

BibTeX

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Gamma function asymptotics by an extension of the method of steepest descents. Proc. Roy. Soc. London Ser. A 447, pp. 609–630.

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External Links:

ISSN 0962-8444, FullText, MathReview (Jens Bolte), ZentralBlatt

Referenced by:

§5.11(i), §5.11(ii), §5.11(ii), (5.9.11_1), (5.9.11_2).

Encodings:

BibTeX

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Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents. Methods Appl. Anal. 2 (4), pp. 475–489.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Dan Polis̆evski), ZentralBlatt

Referenced by:

§2.4(iv).

Encodings:

BibTeX

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9: 36.5 Stokes Sets

… ►where $j$ denotes a real critical point (36.4.1) or (36.4.2), and $\mu$ denotes a critical point with complex $t$ or $s,t$, connected with $j$ by a steepest-descent path (that is, a path where $\mathrm{\Re }\mathrm{\Phi }=\mathrm{constant}$) in complex $t$ or $(s,t)$ space. …

10: 13.20 Uniform Asymptotic Approximations for Large $\mu$

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