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41: 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. … ►Other applications appear in the study of instability of Couette flow of an inviscid fluid. … ►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). … ►Solutions of the Schrödinger equation involving the Airy functions are given for other potentials in Vallée and Soares (2010). …

42: 6.2 Definitions and Interrelations

… ►

6.2.2

E_{1}\left(z\right)=e^{-z}\int_{0}^{\infty}\frac{e^{-t}}{t+z}\,\mathrm{d}t,

|\operatorname{ph}z|<\pi

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\int$ : integral, $\operatorname{ph}$ : phase and $z$ : complex variable
A&S Ref:: 5.1.1 (in modified form)
Referenced by:: §2.5(iii), §6.18(i), §6.7(iii)
Permalink:: http://dlmf.nist.gov/6.2.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §6.2(i), §6.2 and Ch.6

… ►

6.2.7

\operatorname{Ei}\left(\pm x\right)=-\operatorname{Ein}\left(\mp x\right)+\ln x% +\gamma.

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\ln\NVar{z}$ : principal branch of logarithm function and $x$ : real variable
A&S Ref:: 5.1.40 (in modified form)
Permalink:: http://dlmf.nist.gov/6.2.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §6.2(i), §6.2 and Ch.6

… ►The logarithmic integral is defined by … ►

6.2.10

\operatorname{si}\left(z\right)=-\int_{z}^{\infty}\frac{\sin t}{t}\,\mathrm{d}% t=\operatorname{Si}\left(z\right)-\tfrac{1}{2}\pi.

ⓘ

Defines:: $\operatorname{si}\left(\NVar{z}\right)$ : sine integral
Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $\operatorname{Si}\left(\NVar{z}\right)$ : sine integral and $z$ : complex variable
A&S Ref:: 5.2.5 (in modified form) 5.2.26 (in modified form)
Referenced by:: §6.5
Permalink:: http://dlmf.nist.gov/6.2.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §6.2(ii), §6.2 and Ch.6

… ►

6.2.13

\operatorname{Ci}\left(z\right)=-\operatorname{Cin}\left(z\right)+\ln z+\gamma.

ⓘ

Symbols:: $\gamma$ : Euler’s constant, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\operatorname{Cin}\left(\NVar{z}\right)$ : cosine integral, $\ln\NVar{z}$ : principal branch of logarithm function and $z$ : complex variable
A&S Ref:: 5.2.2 (in modified form)
Referenced by:: §6.4
Permalink:: http://dlmf.nist.gov/6.2.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §6.2(ii), §6.2 and Ch.6

…

43: 2.9 Difference Equations

… ►Provided that

\alpha_{2}-\alpha_{1}

is not zero or an integer, (2.9.1) has independent solutions

w_{j}(n)

j=1,2

, of the form … ►For analogous results for difference equations of the form … ►

§2.9(iii) Other Approximations

…

44: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►These are based on the Liouville normal form of (1.13.29). … ► … ► … ►compare (1.18.30) and (1.18.45), and the eigenfunction expansions are of the form … ►Consider formally self-adjoint operators of the form …

45: 9.13 Generalized Airy Functions

… ►Equations of the form … ►Swanson and Headley (1967) define independent solutions

A_{n}\left(z\right)

and

B_{n}\left(z\right)

of (9.13.1) by … ►

-A_{n}'\left(0\right)=p^{1/2}B_{n}'\left(0\right)=\frac{p^{p}}{\Gamma\left(p% \right)}.

…

46: 11.13 Methods of Computation

… ►Other integrals that appear in §11.5(i) have highly oscillatory integrands unless

z

is small. … ►Then from the limiting forms for small argument (§§11.2(i), 10.7(i), 10.30(i)), limiting forms for large argument (§§11.6(i), 10.7(ii), 10.30(ii)), and the connection formulas (11.2.5) and (11.2.6), it is seen that

\mathbf{H}_{\nu}\left(x\right)

and

\mathbf{L}_{\nu}\left(x\right)

can be computed in a stable manner by integrating forwards, that is, from the origin toward infinity. …

47: 27.16 Cryptography

… ►To code a piece

x

, raise

x

to the power

r

and reduce

x^{r}

modulo

n

to obtain an integer

y

(the coded form of

x

) between

1

and

n

. … ►In other words, to recover

x

from

y

we simply raise

y

to the power

s

and reduce modulo

n

. … ►For further information see Apostol and Niven (1994, p. 24), and for other applications to cryptography see Menezes et al. (1997) and Schroeder (2006).

48: About the Project

… ►The results have been published in book form as the NIST Handbook of Mathematical Functions, by Cambridge University Press, and disseminated in the free electronic Digital Library of Mathematical Functions. … ► Olver, Editor-in-Chief and Mathematics Editor of the DLMF, the other Editors initiated an effort aimed at updating the organizational structure of the DLMF project. … ►They will be called upon to help deal with reports of suspected errors and suggestions for additions or other modifications to the chapters. … ►The complete list of Editors, Senior Associate Editors, Associate Editors, and other currently active contributors to the DLMF Project are listed on the Staff page. …

49: 15.14 Integrals

… ►Integrals of the form

\int x^{\alpha}(x+t)^{\beta}F\left(a,b;c;x\right)\,\mathrm{d}x

and more complicated forms are given in Apelblat (1983, pp. 370–387), Prudnikov et al. (1990, §§1.15 and 2.21), Gradshteyn and Ryzhik (2000, §7.5) and Koornwinder (2015). … ►For other integral transforms see Erdélyi et al. (1954b), Prudnikov et al. (1992b, §4.3.43), and also §15.9(ii). …

50: 27.2 Functions

… ►An equivalent form states that the

n

th prime

p_{n}

(when the primes are listed in increasing order) is asymptotic to

n\ln n

n\to\infty

: …Other examples of number-theoretic functions treated in this chapter are as follows. …