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1: 33.24 Tables

§33.24 Tables

►

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_{0}\left(\eta,\rho\right)$ , $G_{0}\left(\eta,\rho\right)$ , $F_{0}'\left(\eta,\rho\right)$ , and $G_{0}'\left(\eta,\rho\right)$ for $\eta=0.5(.5)20$ and $\rho=1(1)20$ , 5S; $C_{0}\left(\eta\right)$ for $\eta=0(.05)3$ , 6S.

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•

Curtis (1964a) tabulates $P_{\ell}(\epsilon,r)$ , $Q_{\ell}(\epsilon,r)$ (§33.1), and related functions for $\ell=0,1,2$ and $\epsilon=-2(.2)2$ , with $x=0(.1)4$ for $\epsilon<0$ and $x=0(.1)10$ for $\epsilon\geq 0$ ; 6D.

…

2: 8.12 Uniform Asymptotic Expansions for Large Parameter

… ►Define …where the branch of the square root is continuous and satisfies

\eta(\lambda)\sim\lambda-1

\lambda\to 1

. … ►With

\mu=\lambda-1

, the coefficients

c_{k}(\eta)

are given by …The right-hand sides of equations (8.12.9), (8.12.10) have removable singularities at

\eta=0

, and the Maclaurin series expansion of

c_{k}(\eta)

is given by … ►For the asymptotic behavior of

c_{k}(\eta)

k\to\infty

see Dunster et al. (1998) and Olde Daalhuis (1998c). …

3: 33.8 Continued Fractions

§33.8 Continued Fractions

►With arguments

\eta,\rho

suppressed, … ►

a=1+\ell\pm\mathrm{i}\eta,

►

b=-\ell\pm\mathrm{i}\eta,

… ►If we denote

u=\ifrac{F_{\ell}'}{F_{\ell}}

and

p+\mathrm{i}q=\ifrac{{H^{+}_{\ell}}'}{{H^{+}_{\ell}}}

, then …

4: 13.28 Physical Applications

… ►The reduced wave equation

\nabla^{2}w=k^{2}w

in paraboloidal coordinates,

x=2\sqrt{\xi\eta}\cos\phi

y=2\sqrt{\xi\eta}\sin\phi

z=\xi-\eta

, can be solved via separation of variables

w=f_{1}(\xi)f_{2}(\eta)e^{\mathrm{i}p\phi}

, where ►

f_{1}(\xi)=\xi^{-\frac{1}{2}}V_{\kappa,\frac{1}{2}p}^{(1)}(2\mathrm{i}k\xi)

►

f_{2}(\eta)=\eta^{-\frac{1}{2}}V_{\kappa,\frac{1}{2}p}^{(2)}(-2\mathrm{i}k\eta)

…

5: 23.21 Physical Applications

… ►The Weierstrass function

\wp

plays a similar role for cubic potentials in canonical form

g_{3}+g_{2}x-4x^{3}

. … ►Airault et al. (1977) applies the function

\wp

to an integrable classical many-body problem, and relates the solutions to nonlinear partial differential equations. For applications to soliton solutions of the Korteweg–de Vries (KdV) equation see McKean and Moll (1999, p. 91), Deconinck and Segur (2000), and Walker (1996, §8.1). … ►Ellipsoidal coordinates

(\xi,\eta,\zeta)

may be defined as the three roots

\rho

of the equation … ►

•

Statistical mechanics. See Baxter (1982, p. 434) and Itzykson and Drouffe (1989, §9.3).

…

6: 21.7 Riemann Surfaces

… ►by setting

\lambda=\tilde{\lambda}/\tilde{\eta}

\mu=\tilde{\mu}/\tilde{\eta}

, and then clearing fractions. … ►Here

\sqrt{\zeta(P)}

is such that

\sqrt{\zeta(P)}^{2}=\zeta(P)

P\in\Gamma

. … ►Next, define an isomorphism

\boldsymbol{{\eta}}

which maps every subset

T

B

with an even number of elements to a

2g

-dimensional vector

\boldsymbol{{\eta}}(T)

with elements either

0

\tfrac{1}{2}

. …Also,

T^{c}=B\setminus T

\boldsymbol{{\eta}}^{1}(T)=(\eta_{1}(T),\eta_{2}(T),\dots,\eta_{g}(T))

, and

\boldsymbol{{\eta}}^{2}(T)=(\eta_{g+1}(T),\eta_{g+2}(T),\dots,\eta_{2g}(T))

. … ►Furthermore, let

\boldsymbol{{\eta}}(P_{\infty})=\boldsymbol{{0}}

and

\boldsymbol{{\eta}}(P_{j})=\boldsymbol{{\eta}}(\{P_{j},P_{\infty}\})

. …

7: 7.8 Inequalities

… ►Let

\mathsf{M}\left(x\right)

denote Mills’ ratio: ►

7.8.1

\mathsf{M}\left(x\right)=\frac{\int_{x}^{\infty}e^{-t^{2}}\,\mathrm{d}t}{e^{-x% ^{2}}}=e^{x^{2}}\int_{x}^{\infty}e^{-t^{2}}\,\mathrm{d}t.

ⓘ

Defines:: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio
Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $x$ : real variable
Permalink:: http://dlmf.nist.gov/7.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §7.8 and Ch.7

… ►

7.8.2

\frac{1}{x+\sqrt{x^{2}+2}}<\mathsf{M}\left(x\right)\leq\frac{1}{x+\sqrt{x^{2}+% (4/\pi)}},

x\geq 0

ⓘ

Symbols:: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio, $\pi$ : the ratio of the circumference of a circle to its diameter and $x$ : real variable
A&S Ref:: 7.1.13
Referenced by:: §7.8
Permalink:: http://dlmf.nist.gov/7.8.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §7.8 and Ch.7

►

7.8.3

\frac{\sqrt{\pi}}{2\sqrt{\pi}x+2}\leq\mathsf{M}\left(x\right)<\frac{1}{x+1},

x\geq 0

ⓘ

Symbols:: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio, $\pi$ : the ratio of the circumference of a circle to its diameter and $x$ : real variable
Referenced by:: §7.8
Permalink:: http://dlmf.nist.gov/7.8.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §7.8 and Ch.7

►

7.8.4

\mathsf{M}\left(x\right)<\frac{2}{3x+\sqrt{x^{2}+4}},

x>-\tfrac{1}{2}\sqrt{2}

ⓘ

Symbols:: $\mathsf{M}\left(\NVar{x}\right)$ : Mills’ ratio and $x$ : real variable
Referenced by:: §7.8
Permalink:: http://dlmf.nist.gov/7.8.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §7.8 and Ch.7

…

8: 32.6 Hamiltonian Structure

… ►

\mbox{P}_{\mbox{\scriptsize I}}

–

\mbox{P}_{\mbox{\scriptsize VI}}

can be written as a Hamiltonian system … ►The Hamiltonian for

\mbox{P}_{\mbox{\scriptsize I}}

is … ►The Hamiltonian for

\mbox{P}_{\mbox{\scriptsize II}}

is …Then

q=w

satisfies

\mbox{P}_{\mbox{\scriptsize II}}

and

p

satisfies … ►The Hamiltonian for

\mbox{P}^{\prime}_{\mbox{\scriptsize III}}

(§32.2(iii)) is …

9: 28.31 Equations of Whittaker–Hill and Ince

… ►When

k^{2}<0

, we substitute … ►When

p

is a nonnegative integer, the parameter

\eta

can be chosen so that solutions of (28.31.3) are trigonometric polynomials, called Ince polynomials. … ►The values of

\eta

corresponding to

C_{p}^{m}(z,\xi)

S_{p}^{m}(z,\xi)

are denoted by

a_{p}^{m}(\xi)

b_{p}^{m}(\xi)

, respectively. … ►with

\eta=a_{p}^{m}(\xi)

\eta=b_{p}^{m}(\xi)

, respectively. … ►and also for all

p_{1},p_{2},m_{1},m_{2}

, given by …

10: 33.1 Special Notation

… ► ►►►

$k,\ell$	nonnegative integers.
…
$\epsilon,\eta$	real parameters.
…

►The main functions treated in this chapter are first the Coulomb radial functions

F_{\ell}\left(\eta,\rho\right)

G_{\ell}\left(\eta,\rho\right)

{H^{\pm}_{\ell}}\left(\eta,\rho\right)

(Sommerfeld (1928)), which are used in the case of repulsive Coulomb interactions, and secondly the functions

f\left(\epsilon,\ell;r\right)

h\left(\epsilon,\ell;r\right)

s\left(\epsilon,\ell;r\right)

c\left(\epsilon,\ell;r\right)

(Seaton (1982, 2002a)), which are used in the case of attractive Coulomb interactions. … ►

Curtis (1964a):

$P_{\ell}(\epsilon,r)=(2\ell+1)!f\left(\epsilon,\ell;r\right)/2^{\ell+1}$ , $Q_{\ell}(\epsilon,r)=-(2\ell+1)!h\left(\epsilon,\ell;r\right)/(2^{\ell+1}A(% \epsilon,\ell))$ .

►

Greene et al. (1979):

$f^{(0)}(\epsilon,\ell;r)=f\left(\epsilon,\ell;r\right)$ , $f(\epsilon,\ell;r)=s\left(\epsilon,\ell;r\right)$ , $g(\epsilon,\ell;r)=c\left(\epsilon,\ell;r\right)$ .