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1: 36.4 Bifurcation Sets

… ►

§36.4(i) Formulas

►

Critical Points for Cuspoids

… ►

Critical Points for Umbilics

… ►This is the codimension-one surface in

\mathbf{x}

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

\mathbf{x}

space where critical points coalesce, satisfying (36.4.2) and …

2: 36.12 Uniform Approximation of Integrals

§36.12 Uniform Approximation of Integrals

… ►Correspondence between the

u_{j}(\mathbf{y})

and the

t_{j}(\mathbf{x})

is established by the order of critical points along the real axis when

\mathbf{y}

and

\mathbf{x}

are such that these critical points are all real, and by continuation when some or all of the critical points are complex. …In (36.12.10), both second derivatives vanish when critical points coalesce, but their ratio remains finite. The square roots are real and positive when

\mathbf{y}

is such that all the critical points are real, and are defined by analytic continuation elsewhere. … ►For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).

3: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in

\mathbf{x}

space, across which

\Psi_{K}(\mathbf{x};k)

\Psi^{(\mathrm{U})}(\mathbf{x};k)

acquires an exponentially-small asymptotic contribution (in

k

), associated with a complex critical point of

\Phi_{K}

\Phi^{(\mathrm{U})}

. …where

j

denotes a real critical point (36.4.1) or (36.4.2), and

\mu

denotes a critical point with complex

t

s, t

, connected with

j

by a steepest-descent path (that is, a path where

\Re\Phi=\mathrm{constant}

) in complex

t

(s,t)

space. … ►

36.5.7

X=\dfrac{9}{20}+20u^{4}-\frac{Y^{2}}{20u^{2}}+6u^{2}\operatorname{sign}\left(z% \right),

ⓘ

Symbols:: $\operatorname{sign}\NVar{x}$ : sign of, $z$ : real parameter, $X$ : scaled coordinate and $Y$ : scaled coordinate
Permalink:: http://dlmf.nist.gov/36.5.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §36.5(ii), §36.5(ii), §36.5 and Ch.36

… ►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. …

4: 36.15 Methods of Computation

… ►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

\Phi

and is deformed outside this range so as to reach infinity along the asymptotic valleys of

\exp\left(i\Phi\right)

. … ►This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of

\Phi

, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. …

5: 3.8 Nonlinear Equations

… ► … ►The choice of

x_{0}

here is critical. … ►Consider

x=20

and

j=19

. … ►

§3.8(viii) Fixed-Point Iterations: Fractals

… ►

6: 19.35 Other Applications

… ►

§19.35(ii) Physical

►Elliptic integrals appear in lattice models of critical phenomena (Guttmann and Prellberg (1993)); theories of layered materials (Parkinson (1969)); fluid dynamics (Kida (1981)); string theory (Arutyunov and Staudacher (2004)); astrophysics (Dexter and Agol (2009)). …

7: Foreword

… ►²² 2 D. R. Lide (ed.), A Century of Excellence in Measurement, Standards, and Technology, CRC Press, 2001. The success of the original handbook, widely referred to as “Abramowitz and Stegun” (“A&S”), derived not only from the fact that it provided critically useful scientific data in a highly accessible format, but also because it served to standardize definitions and notations for special functions. … ►November 20, 2009 …

8: 15.18 Physical Applications

… ►The hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). ►More varied applications include photon scattering from atoms (Gavrila (1967)), energy distributions of particles in plasmas (Mace and Hellberg (1995)), conformal field theory of critical phenomena (Burkhardt and Xue (1991)), quantum chromo-dynamics (Atkinson and Johnson (1988)), and general parametrization of the effective potentials of interaction between atoms in diatomic molecules (Herrick and O’Connor (1998)).

9: 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. … ►Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. … ►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. …

10: 25.10 Zeros

… ►

§25.10(i) Distribution

… ►In the region

0<\Re s<1

, called the critical strip,

\zeta\left(s\right)

has infinitely many zeros, distributed symmetrically about the real axis and about the critical line

\Re s=\frac{1}{2}

. … ►Calculations relating to the zeros on the critical line make use of the real-valued function … ►Calculations based on the Riemann–Siegel formula reveal that the first ten billion zeros of

\zeta\left(s\right)

in the critical strip are on the critical line (van de Lune et al. (1986)). More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …