steepest-descent paths

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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: 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)): …

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: 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).

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5: 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. …

6: 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. …However, for the purpose of simply deriving the asymptotic expansions the use of steepest descent paths is not essential. …