connection formulas across transition points

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11: 10.4 Connection Formulas

§10.4 Connection Formulas

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12: 9.16 Physical Applications

… ►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)). … ►These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). … ►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. …

13: 25.13 Periodic Zeta Function

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25.13.2 $$F(x,s)=\frac{\mathrm{\Gamma }\left(1-s\right)}{{(2\pi )}^{1-s}}\left({\mathrm{e}}^{\pi \mathrm{i}(1-s)/2}\zeta (1-s,x)+{\mathrm{e}}^{\pi \mathrm{i}(s-1)/2}\zeta (1-s,1-x)\right),$$ , $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connectionformula

Source:

Apostol (1976, Exercise 2, p. 273)

Permalink:

http://dlmf.nist.gov/25.13.E2

Encodings:

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See also:

Annotations for §25.13 and Ch.25

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25.13.3 $$\zeta (1-s,x)=\frac{\mathrm{\Gamma }\left(s\right)}{{(2\pi )}^s}\left({\mathrm{e}}^{-\pi \mathrm{i}s/2}F(x,s)+{\mathrm{e}}^{\pi \mathrm{i}s/2}F(-x,s)\right),$$ $\mathrm{\Re }s>0$ if ; $\mathrm{\Re }s>1$ if $x=1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $F(x,s)$: periodic zeta function, $\mathrm{\Re }$: real part, $x$: real variable and $s$: complex variable

Keywords:

connectionformula

Source:

Apostol (1976, (10), p. 257)

Notes:

This formula is equivalent to (25.11.9).

Referenced by:

Erratum (V1.1.4) for Equation (25.13.3)

Permalink:

http://dlmf.nist.gov/25.13.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §25.13 and Ch.25

14: Bibliography I

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IEEE Standard for Floating-Point Arithmetic. The Institute of Electrical and Electronics Engineers, Inc..

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Notes:

IEEE Std 754-2008, Revision of IEEE Std 754-1985

External Links:

ISBN 978-0-7381-5753-5, Document

Referenced by:

§3.1(i).

Encodings:

BibTeX

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IEEE International Standard for Information Technology—Microprocessor Systems—Floating-Point arithmetic: IEEE Std 754-2019. The Institute of Electrical and Electronics Engineers, Inc..

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Notes:

IEEE Std ISO/IEC/IEEE 60559, Revision of IEEE Std 754-1985

External Links:

ISBN 978-2-88912-566-1, Document

Referenced by:

Figure 3.1.1, Figure 3.1.1, §3.1(i), §3.1(i), Erratum (V1.0.25) for Section 3.1.

Encodings:

BibTeX

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The method of isomonodromic deformations and relation formulas for the second Painlevé transcendent. Izv. Akad. Nauk SSSR Ser. Mat. 51 (4), pp. 878–892, 912 (Russian).

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Notes:

In Russian. English translation: Math. USSR-Izv. 31(1988), no. 1, pp. 193–207

External Links:

ISSN 0373-2436, Document, MathReview (Alexander A. Pankov), ZentralBlatt

Referenced by:

§32.11(iii).

Encodings:

BibTeX

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Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution. J. Phys. A 31 (17), pp. 4073–4113.

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External Links:

ISSN 0305-4470, Document, MathReview (Yoshitsugu Takei), ZentralBlatt

Referenced by:

§32.11(v).

Encodings:

BibTeX

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15: 8.23 Statistical Applications

§8.23 Statistical Applications

… ►The function ${\mathrm{B}}_x(a,b)$ and its normalization $I_x(a,b)$ play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). …

16: 13.27 Mathematical Applications

… ►Confluent hypergeometric functions are connected with representations of the group of third-order triangular matrices. … ►For applications of Whittaker functions to the uniform asymptotic theory of differential equations with a coalescing turning point and simple pole see §§2.8(vi) and 18.15(i). …

17: Mark J. Ablowitz

… ►ODEs which do not have moveable branch point singularities. …Widespread interest in Painlevé equations re-emerged in the 1970s and thereafter partially due to the connection with IST and integrable systems. …

18: 3.3 Interpolation

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Three-Point Formula

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Four-Point Formula

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Five-Point Formula

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Six-Point Formula

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Seven-Point Formula

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19: Bibliography J

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Tables of Functions with Formulae and Curves. 4th edition, Dover Publications, New York.

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Notes:

Contains corrections and enlargements of earlier editions. Originally published by B. G. Teubner, Leipzig, in 1933. In German and English.

External Links:

MathReview, ZentralBlatt

Referenced by:

2nd item, §20.15, §5.1, Notations P, E. Jahnke, F. Emde, and F. Lösch (1966).

Encodings:

BibTeX

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Asymptotic formulas for the zeros of the Meixner polynomials. J. Approx. Theory 96 (2), pp. 281–300.

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External Links:

ISSN 0021-9045, Document, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§18.24.

Encodings:

BibTeX

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Efficient implementation of the Hardy-Ramanujan-Rademacher formula. LMS J. Comput. Math. 15, pp. 341–359.

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External Links:

ISSN 1461-1570, Document, Link, MathReview (Tim Huber), ZentralBlatt

Referenced by:

§27.20.

Encodings:

BibTeX

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Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations. V. Angular oblate spheroidal wavefunctions ${\overline{ps}}_n^r(\eta ,h)$ and ${\overline{qs}}_n^r(\eta ,h)$ for large $h$ . Proc. Roy. Soc. London Ser. A 321, pp. 545–555.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§30.9(ii).

Encodings:

BibTeX

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The Painlevé connection problem: An asymptotic approach. I. Stud. Appl. Math. 86 (4), pp. 315–376.

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External Links:

ISSN 0022-2526, MathReview (I. P. Martynov), ZentralBlatt

Referenced by:

§32.11(i), §32.12(ii).

Encodings:

BibTeX

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20: 27.5 Inversion Formulas

§27.5 Inversion Formulas

… ►Generating functions yield many relations connecting number-theoretic functions. …which, in turn, is the basis for the Möbius inversion formula relating sums over divisors: … ►Special cases of Möbius inversion pairs are: … ►Other types of Möbius inversion formulas include: …