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11: 33.19 Power-Series Expansions in $r$

§33.19 Power-Series Expansions in $r$

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33.19.1 $$f(ϵ,\mathrm{\ell };r)=r^{\mathrm{\ell }+1}\sum _{k=0}^{\mathrm{\infty }}{\alpha }_k{r}^k,$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and ${\alpha }_k$: term

Referenced by:

§33.18, §33.19

Permalink:

http://dlmf.nist.gov/33.19.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

… ►Here $\kappa$ is defined by (33.14.6), $A(ϵ,\mathrm{\ell })$ is defined by (33.14.11) or (33.14.12), ${\gamma }_0=1$, ${\gamma }_1=1$, and … ►

33.19.6 $$k(k+2\mathrm{\ell }+1){\delta }_k+2{\delta }_{k-1}+ϵ{\delta }_{k-2}+2(2k+2\mathrm{\ell }+1)A(ϵ,\mathrm{\ell }){\alpha }_k=0,$$ $k=2,3,\mathrm{\dots }$,

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Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter, $A(ϵ,\mathrm{\ell })$: function, ${\alpha }_k$: term and ${\delta }_k$: term

Permalink:

http://dlmf.nist.gov/33.19.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

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33.19.7 $${\beta }_k-{\beta }_{k-1}+\frac{1}{4}(k-1)(k-2\mathrm{\ell }-2)ϵ{\beta }_{k-2}+\frac{1}{2}(k-1)ϵ{\gamma }_{k-2}=0,$$ $k=2,3,\mathrm{\dots }$.

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Symbols:

$k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $ϵ$: real parameter, ${\gamma }_k$: term and ${\beta }_k$: term

Permalink:

http://dlmf.nist.gov/33.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.19 and Ch.33

…

12: 27.16 Cryptography

… ►If $p$ and $q$ are known, $s$ and $y^s$ can be determined (mod $n$) by straightforward calculations that require only a few minutes of machine time. …

13: 33.21 Asymptotic Approximations for Large $|r|$

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§33.21(i) Limiting Forms

►We indicate here how to obtain the limiting forms of $f(ϵ,\mathrm{\ell };r)$, $h(ϵ,\mathrm{\ell };r)$, $s(ϵ,\mathrm{\ell };r)$, and $c(ϵ,\mathrm{\ell };r)$ as $r\to \pm \mathrm{\infty }$, with $ϵ$ and $\mathrm{\ell }$ fixed, in the following cases: ►

When $r\to \pm \mathrm{\infty }$ with $ϵ>0$, Equations (33.16.4)–(33.16.7) are combined with (33.10.1).

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When $r\to \pm \mathrm{\infty }$ with $ϵ=0$, combine (33.20.1), (33.20.2) with §§10.7(ii), 10.30(ii).

… ►For asymptotic expansions of $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$ as $r\to \pm \mathrm{\infty }$ with $ϵ$ and $\mathrm{\ell }$ fixed, see Curtis (1964a, §6).

14: 33.16 Connection Formulas

… ►When $ϵ>0$ denote …and again define $A(ϵ,\mathrm{\ell })$ by (33.14.11) or (33.14.12). … ►When denote …and again define $A(ϵ,\mathrm{\ell })$ by (33.14.11) or (33.14.12). … ►When $ϵ>0$, again denote $\tau$ by (33.16.3). …

15: Bibliography C

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Über die Fälle, wenn die Reihe von der Form $y=1+\frac{\alpha }{1}\cdot \frac{\beta }{\gamma }x+\frac{\alpha \cdot \alpha +1}{1\cdot 2}\cdot \frac{\beta \cdot \beta +1}{\gamma \cdot \gamma +1}{x}^2+$ etc. ein Quadrat von der Form $z=1+\frac{{\alpha }^{\prime }}{1}\cdot \frac{{\beta }^{\prime }}{{\gamma }^{\prime }}\cdot \frac{{\delta }^{\prime }}{{ϵ}^{\prime }}x+\frac{{\alpha }^{\prime }\cdot {\alpha }^{\prime }+1}{1\cdot 2}\cdot \frac{{\beta }^{\prime }\cdot {\beta }^{\prime }+1}{{\gamma }^{\prime }\cdot {\gamma }^{\prime }+1}\cdot \frac{{\delta }^{\prime }\cdot {\delta }^{\prime }+1}{{ϵ}^{\prime }\cdot {ϵ}^{\prime }+1}{x}^2+$ etc. hat. J. Reine Angew. Math. 3, pp. 89–91.

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External Links:

Link, MathReview Entry, ZentralBlatt

Referenced by:

§16.12.

Encodings:

BibTeX

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An algorithm for the machine calculation of complex Fourier series. Math. Comp. 19 (90), pp. 297–301.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§3.11(v).

Encodings:

BibTeX

…

16: 32.10 Special Function Solutions

… ►For example, if $\alpha =\frac{1}{2}\epsilon$, with $\epsilon =\pm 1$, then the Riccati equation is … ►with $n\in \mathbb{Z}$, and ${\epsilon }_1=\pm 1$, ${\epsilon }_2=\pm 1$, independently. In the case ${\epsilon }_1\alpha +{\epsilon }_2\beta =2$, the Riccati equation is …with $\zeta =\sqrt{{\epsilon }_1{\epsilon }_2}z$, $\nu =\frac{1}{2}\alpha {\epsilon }_1$, and $C_1$, $C_2$ arbitrary constants. … ►where $n\in \mathbb{Z}$, $a={\epsilon }_1\sqrt{2\alpha }$, $b={\epsilon }_2\sqrt{-2\beta }$, $c={\epsilon }_3\sqrt{2\gamma }$, and $d={\epsilon }_4\sqrt{1-2\delta }$, with ${\epsilon }_j=\pm 1$, $j=1,2,3,4$, independently. …

17: 31.2 Differential Equations

… ►This equation has regular singularities at $0,1,a,\mathrm{\infty }$, with corresponding exponents $\{0,1-\gamma \}$, $\{0,1-\delta \}$, $\{0,1-ϵ\}$, $\{\alpha ,\beta \}$, respectively (§2.7(i)). … ►The parameters play different roles: $a$ is the singularity parameter; $\alpha ,\beta ,\gamma ,\delta ,ϵ$ are exponent parameters; $q$ is the accessory parameter. … ► $w(z)=z^{1-\gamma }{w}_1(z)$ satisfies (31.2.1) if $w_1$ is a solution of (31.2.1) with transformed parameters $q_1=q+(a\delta +ϵ)(1-\gamma )$; ${\alpha }_1=\alpha +1-\gamma$, ${\beta }_1=\beta +1-\gamma$, ${\gamma }_1=2-\gamma$. …Lastly, $w(z)={(z-a)}^{1-ϵ}{w}_3(z)$ satisfies (31.2.1) if $w_3$ is a solution of (31.2.1) with transformed parameters $q_3=q+\gamma (1-ϵ)$; ${\alpha }_3=\alpha +1-ϵ$, ${\beta }_3=\beta +1-ϵ$, ${ϵ}_3=2-ϵ$. … ►For example, if $\tilde{z}=z/a$, then the parameters are $\tilde{a}=1/a$, $\tilde{q}=q/a$; $\tilde{\delta }=ϵ$, $\widetilde{ϵ}=\delta$. …

18: 22.21 Tables

§22.21 Tables

►Spenceley and Spenceley (1947) tabulates $\mathrm{sn}(Kx,k)$, $\mathrm{cn}(Kx,k)$, $\mathrm{dn}(Kx,k)$, $\mathrm{am}(Kx,k)$, $ℰ(Kx,k)$ for $\arcsin k=1^{\circ }(1^{\circ }){89}^{\circ }$ and $x=0\left(\frac{1}{90}\right)1$ to 12D, or 12 decimals of a radian in the case of $\mathrm{am}(Kx,k)$. … ►Lawden (1989, pp. 280–284 and 293–297) tabulates $\mathrm{sn}(x,k)$, $\mathrm{cn}(x,k)$, $\mathrm{dn}(x,k)$, $ℰ(x,k)$, $\mathrm{Z}\left(x|k\right)$ to 5D for $k=0.1(.1)0.9$, $x=0(.1)X$, where $X$ ranges from 1. …

19: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►

$𝗄$ Scaling

… ►In these applications, the $Z$-scaled variables $r$ and $ϵ$ are more convenient. ►

$Z$ Scaling

►The $Z$-scaled variables $r$ and $ϵ$ of §33.14 are given by … ►

$\mathrm{i}𝗄$ Scaling

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20: 32.2 Differential Equations

… ►thus in the limit as $ϵ\to 0$, $W(\zeta )$ satisfies ${\text{P}}_{\text{I}}$ with $z=\zeta$. … ►then as $ϵ\to 0$, $W(\zeta ;a)$ satisfies ${\text{P}}_{\text{II}}$ with $z=\zeta$, $\alpha =a$. … ►then as $ϵ\to 0$, $W(\zeta ;a)$ satisfies ${\text{P}}_{\text{II}}$ with $z=\zeta$, $\alpha =a$. … ►then as $ϵ\to 0$, $W(\zeta ;a,b,c,d)$ satisfies ${\text{P}}_{\text{III}}$ with $z=\zeta$, $\alpha =a$, $\beta =b$, $\gamma =c$, $\delta =d$. … ►then as $ϵ\to 0$, $W(\zeta ;a,b)$ satisfies ${\text{P}}_{\text{IV}}$ with $z=\zeta$, $\alpha =a$, $\beta =b$. …