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21: 3.5 Quadrature

… ►with a function $f$ that is analytic in a strip containing $ℝ$. …

22: Bibliography

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Evaluation of Coulomb wave functions along the transition line. Physical Rev. (2) 96, pp. 77–79.

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External Links:

Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§33.12(i).

Encodings:

BibTeX

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Spectrum line profiles: The Voigt function. J. Quant. Spectrosc. Radiat. Transfer 7, pp. 61–88.

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External Links:

Document

Referenced by:

§7.19(iv), §7.21.

Encodings:

BibTeX

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Normal forms of functions near degenerate critical points, the Weyl groups $A_k,D_k,E_k$ and Lagrangian singularities. Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).

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Notes:

In Russian. English translation: Functional Anal. Appl., 6(1973), pp. 254–272.

External Links:

MathReview (G. R. Belickii), ZentralBlatt

Referenced by:

§36.2(i).

Encodings:

BibTeX

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Normal forms of functions in the neighborhood of degenerate critical points. Uspehi Mat. Nauk 29 (2(176)), pp. 11–49 (Russian).

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Notes:

Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901–1973), I. In Russian. English translation: Russian Math. Surveys, 29(1974), no. 2, pp. 10–50.

External Links:

Document, MathReview (G. R. Belickii), ZentralBlatt

Referenced by:

§36.2(i).

Encodings:

BibTeX

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Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 30(1975), no. 5, pp. 1–75.

External Links:

Document, MathReview (D. Siersma), ZentralBlatt

Referenced by:

§36.2(i), Ch.36.

Encodings:

BibTeX

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23: 31.15 Stieltjes Polynomials

… ►The zeros $z_k$, $k=1,2,\mathrm{\dots },n$, of the Stieltjes polynomial $S(z)$ are the critical points of the function $G$, that is, points at which $\partial G/\partial {\zeta }_k=0$, $k=1,2,\mathrm{\dots },n$, where …

24: 36.11 Leading-Order Asymptotics

… ►With real critical points (36.4.1) ordered so that … ►

Asymptotics along Symmetry Lines

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25: 36.4 Bifurcation Sets

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§36.4(i) Formulas

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Critical Points for Cuspoids

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Critical Points for Umbilics

… ►Swallowtail self-intersection line: … ►Swallowtail cusp lines (ribs): …

26: DLMF Project News

error generating summary

27: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in $𝐱$ space, across which ${\mathrm{\Psi }}_K(𝐱;k)$ or ${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)$ acquires an exponentially-small asymptotic contribution (in $k$), associated with a complex critical point of ${\mathrm{\Phi }}_K$ or . …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. … ►the intersection lines with the bifurcation set are generated by $|X|=X_2=0.45148$, $Y=Y_2=0.59693$. … ►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions. …