coalescing transition points

SearchAdvancedHelp

(0.001 seconds)

21—30 of 162 matching pages

21: Michael V. Berry

… ►

36 Integrals with Coalescing Saddles

…

22: 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)). …

23: Bibliography N

… ►

Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

ⓘ

Referenced by:

§2.8(vi).

Encodings:

BibTeX

…

24: 2.3 Integrals of a Real Variable

… ►However, cancellation does not take place near the endpoints, owing to lack of symmetry, nor in the neighborhoods of zeros of $p^{\prime }(t)$ because $p(t)$ changes relatively slowly at these stationary points. … ►For the more general integral (2.3.19) we assume, without loss of generality, that the stationary point (if any) is at the left endpoint. … ►

§2.3(v) Coalescing Peak and Endpoint: Bleistein’s Method

… ►Assume also that ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are continuous in $\alpha$ and $t$, and for each $\alpha$ the minimum value of $p(\alpha ,t)$ in $[0,k)$ is at $t=\alpha$, at which point $\partial p(\alpha ,t)/\partial t$ vanishes, but both ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are nonzero. … ►where $a$ and $b$ are functions of $\alpha$ chosen in such a way that $t=0$ corresponds to $w=0$, and the stationary points $t=\alpha$ and $w=a$ correspond. …

25: 33.12 Asymptotic Expansions for Large $\eta$

… ►

§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). …

26: Bibliography L

… ►

The two-point connection problem for differential equations of the Heun class. Teoret. Mat. Fiz. 101 (3), pp. 360–368 (Russian).

ⓘ

Notes:

In Russian. English translation: Theoret. and Math. Phys. 101(1994), no. 3, pp. 1413–1418 (1995)

External Links:

ISSN 0564-6162, Document, MathReview, ZentralBlatt

Referenced by:

§31.18.

Encodings:

BibTeX

… ►

The central two-point connection problem for the Heun class of ODEs. J. Phys. A 31 (18), pp. 4249–4261.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§31.12.

Encodings:

BibTeX

… ►

A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points. J. Phys. A 19 (3), pp. 329–335.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§2.4(vi).

Encodings:

BibTeX

… ►

A systematic “saddle point near a pole” asymptotic method with application to the Gauss hypergeometric function. Stud. Appl. Math. 127 (1), pp. 24–37.

ⓘ

External Links:

ISSN 0022-2526, Document, FullText, MathReview (M. S. Mahadeva Naika), ZentralBlatt

Referenced by:

§15.12(ii), §15.12(ii).

Encodings:

BibTeX

… ►

On the points of inflection of Bessel functions of positive order. I. Canad. J. Math. 42 (5), pp. 933–948.

ⓘ

Notes:

Corrigenda see same journal, 42(6), p. 1132.

External Links:

ISSN 0008-414X, MathReview (A. McD. Mercer), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

BibTeX

…

27: 16.8 Differential Equations

… ►

§16.8(i) Classification of Singularities

►An ordinary point of the differential equation …If $z_0$ is not an ordinary point but ${(z-z_0)}^{n-j}{f}_j(z)$, $j=0,1,\mathrm{\dots },n-1$, are analytic at $z=z_0$, then $z_0$ is a regular singularity. … ►Thus in the case $p=q$ the regular singularities of the function on the left-hand side at $\alpha$ and $\mathrm{\infty }$ coalesce into an irregular singularity at $\mathrm{\infty }$. …

28: Bibliography W

… ►

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

ⓘ

External Links:

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

Referenced by:

§2.9(iii).

Encodings:

BibTeX

►

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

ⓘ

External Links:

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

Referenced by:

§2.9(iii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

Ch.9.

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

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

Referenced by:

§10.21(vi), §10.21(vi).

Encodings:

BibTeX

►

On the points of inflection of Bessel functions of positive order. II. Canad. J. Math. 43 (3), pp. 628–651.

ⓘ

External Links:

ISSN 0008-414X, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§10.21(iv).

Encodings:

BibTeX

…

29: Bibliography T

… ►

The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.

ⓘ

Notes:

Algol procedures, maximum accuracy 14S.

External Links:

FullText, ZentralBlatt

Referenced by:

§12.21(ii).

Encodings:

BibTeX

… ►

Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.

ⓘ

External Links:

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

Referenced by:

§9.1, Notations E, Notations E.

Encodings:

BibTeX

…

30: Bibliography B

… ►

Coefficient functions for an inhomogeneous turning-point problem. Mathematika 38 (2), pp. 217–238.

ⓘ

External Links:

ISSN 0025-5793, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

… ►

Unfolding the high orders of asymptotic expansions with coalescing saddles: Singularity theory, crossover and duality. Proc. Roy. Soc. London Ser. A 443, pp. 107–126.

ⓘ

External Links:

ISSN 0962-8444, FullText, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§36.12(i), §36.12(ii), §36.12(iii).

Encodings:

BibTeX

… ►

Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math. 19, pp. 353–370.

ⓘ

External Links:

Document, MathReview (J. G. van der Corput), ZentralBlatt

Referenced by:

§2.3(v).

Encodings:

BibTeX

►

Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities. J. Math. Mech. 17, pp. 533–559.

ⓘ

External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§36.12(iii).

Encodings:

BibTeX

… ►

Asymptotic expansions for the coefficient functions that arise in turning-point problems. Proc. Roy. Soc. London Ser. A 410, pp. 35–60.

ⓘ

External Links:

ISSN 0080-4630, FullText, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

…