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1: 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

… ►This is the codimension-one surface in $𝐱$ space where critical points coalesce, satisfying (36.4.1) and … ►This is the codimension-one surface in $𝐱$ space where critical points coalesce, satisfying (36.4.2) and …

2: 9.16 Physical Applications

… ►The frequent appearances of the Airy functions in both classical and quantum physics is associated with wave equations with turning points, for which asymptotic (WKBJ) solutions are exponential on one side and oscillatory on the other. The Airy functions constitute uniform approximations whose region of validity includes the turning point and its neighborhood. … ►These examples of transitions to turbulence are presented in detail in Drazin and Reid (1981) with the problem of hydrodynamic stability. The investigation of the transition between subsonic and supersonic of a two-dimensional gas flow leads to the Euler–Tricomi equation (Landau and Lifshitz (1987)). … ►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point. …

3: 25.17 Physical Applications

… ►See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). ►The zeta function arises in the calculation of the partition function of ideal quantum gases (both Bose–Einstein and Fermi–Dirac cases), and it determines the critical gas temperature and density for the Bose–Einstein condensation phase transition in a dilute gas (Lifshitz and Pitaevskiĭ (1980)). …

4: Bibliography O

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

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Error bounds for asymptotic expansions in turning-point problems. J. Soc. Indust. Appl. Math. 12 (1), pp. 200–214.

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Document, MathReview (N. D. Kazarinoff), ZentralBlatt

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§2.8(iii).

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Second-order linear differential equations with two turning points. Philos. Trans. Roy. Soc. London Ser. A 278, pp. 137–174.

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FullText, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§2.8(vi).

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Improved error bounds for second-order differential equations with two turning points. J. Res. Nat. Bur. Standards Sect. B 80B (4), pp. 437–440.

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MathReview (Peter Deuflhard), ZentralBlatt

Referenced by:

§2.8(vi).

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Second-order differential equations with fractional transition points. Trans. Amer. Math. Soc. 226, pp. 227–241.

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FullText, MathReview (Anthony Leung), ZentralBlatt

Referenced by:

§2.8(v).

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5: 2.9 Difference Equations

… ►For discussions of turning points, transition points, and uniform asymptotic expansions for solutions of linear difference equations of the second order see Wang and Wong (2003, 2005). …

6: 2.8 Differential Equations with a Parameter

… ►The form of the asymptotic expansion depends on the nature of the transition points in $𝐃$, that is, points at which $f(z)$ has a zero or singularity. … ►In Case I there are no transition points in $𝐃$ and $g(z)$ is analytic. … ►

§2.8(ii) Case I: No Transition Points

… ►For connection formulas for Liouville–Green approximations across these transition points see Olver (1977b, a, 1978). ►

§2.8(vi) Coalescing Transition Points

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7: Bibliography W

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Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.

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ISSN 0029-599X, Document, MathReview (Sergey M. Zagorodnyuk), ZentralBlatt

Referenced by:

§2.9(iii).

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Linear difference equations with transition points. Math. Comp. 74 (250), pp. 629–653.

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ISSN 0025-5718, Document, MathReview (B. G. Pachpatte), ZentralBlatt

Referenced by:

§2.9(iii).

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The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt

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§29.19(ii).

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Linear Turning Point Theory. Applied Mathematical Sciences No. 54, Springer-Verlag, New York.

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ISBN 0-387-96046-5, MathReview (Anthony Leung), ZentralBlatt

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Ch.9.

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Asymptotic behaviour of the inflection points of Bessel functions. Proc. Roy. Soc. London Ser. A 431, pp. 509–518.

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ISSN 0962-8444, FullText, Document, MathReview (A. McD. Mercer), ZentralBlatt

Referenced by:

§10.21(vi), §10.21(vi).

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8: 33.12 Asymptotic Expansions for Large $\eta$

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§33.12(i) Transition Region

►When $\mathrm{\ell }=0$ and $\eta >0$, the outer turning point is given by ${\rho }_{\mathrm{tp}}(\eta ,0)=2\eta$; compare (33.2.2). …

9: Bibliography

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Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.

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Document, MathReview (Mark Adler)

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§32.11(ii), §32.13(i), §32.13(ii), §32.13(iii), §32.5.

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

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Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§33.12(i).

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On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

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§24.12(iii).

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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

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§10.43(iii).

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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.

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Document, MathReview (D. Siersma), ZentralBlatt

Referenced by:

§36.2(i), Ch.36.

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10: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

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The zeros of special functions from a fixed point method. SIAM J. Numer. Anal. 40 (1), pp. 114–133.

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ISSN 1095-7170, Document, MathReview, ZentralBlatt

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§3.8(v).

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§28.8(iv).

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Atomic Spectra and Radiative Transitions. 2nd edition, Springer-Verlag, Berlin.

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Notes:

A second printing with corrections was published in 1996.

Referenced by:

§34.12.

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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

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Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

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3rd item.

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