Coulomb%20phase%20shift

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21: 3.8 Nonlinear Equations

… ►For the computation of zeros of Bessel functions, Coulomb functions, and conical functions as eigenvalues of finite parts of infinite tridiagonal matrices, see Grad and Zakrajšek (1973), Ikebe (1975), Ikebe et al. (1991), Ball (2000), and Gil et al. (2007a, pp. 205–213). … ►Initial approximations to the zeros can often be found from asymptotic or other approximations to $f(z)$, or by application of the phase principle or Rouché’s theorem; see §1.10(iv). … ►

3.8.15 $$p(x)=(x-1)(x-2)\mathrm{\cdots }(x-20)$$

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Permalink:

http://dlmf.nist.gov/3.8.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

… ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

22: 33.22 Particle Scattering and Atomic and Molecular Spectra

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§33.22(i) Schrödinger Equation

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§33.22(iv) Klein–Gordon and Dirac Equations

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§33.22(vi) Solutions Inside the Turning Point

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23: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

BibTeX

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Coulomb Functions for Large Values of the Parameter $\eta$ . Technical report Atomic Energy of Canada Limited, Chalk River, Ontario.

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Notes:

Rep. CRT-526

External Links:

MathReview (A. Erdélyi)

Referenced by:

§33.7.

Encodings:

BibTeX

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

BibTeX

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Evaluation of negative energy Coulomb (Whittaker) functions. Comput. Phys. Comm. 159 (1), pp. 55–62.

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Notes:

Double-precision Fortran-90 code.

External Links:

Document, Code

Referenced by:

§33.23(vi), 8th item.

Encodings:

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24: 33.8 Continued Fractions

§33.8 Continued Fractions

… ►If we denote $u=F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $p+\mathrm{i}q=H_{\mathrm{\ell }}^{+}{}_{}{}^{\prime }/H_{\mathrm{\ell }}^{+}$, then … ►

$F_{\mathrm{\ell }}^{\prime }=u{F}_{\mathrm{\ell }},$

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$G_{\mathrm{\ell }}=q^{-1}(u-p)F_{\mathrm{\ell }},$

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$G_{\mathrm{\ell }}^{\prime }=q^{-1}(up-p^2-q^2)F_{\mathrm{\ell }}.$

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25: 33.16 Connection Formulas

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§33.16(i) $F_{\mathrm{\ell }}$ and $G_{\mathrm{\ell }}$ in Terms of $f$ and $h$

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§33.16(ii) $f$ and $h$ in Terms of $F_{\mathrm{\ell }}$ and $G_{\mathrm{\ell }}$ when $ϵ>0$

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§33.16(iii) $f$ and $h$ in Terms of $W_{\kappa ,\mu }\left(z\right)$ when

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§33.16(iv) $s$ and $c$ in Terms of $F_{\mathrm{\ell }}$ and $G_{\mathrm{\ell }}$ when $ϵ>0$

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§33.16(v) $s$ and $c$ in Terms of $W_{\kappa ,\mu }\left(z\right)$ when

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26: 33 Coulomb Functions

Chapter 33 Coulomb Functions

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27: 5.20 Physical Applications

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Rutherford Scattering

►In nonrelativistic quantum mechanics, collisions between two charged particles are described with the aid of the Coulomb phase shift $\mathrm{ph}\mathrm{\Gamma }\left(\mathrm{\ell }+1+\mathrm{i}\eta \right)$; see (33.2.10) and Clark (1979). …

28: 8 Incomplete Gamma and Related
Functions

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29: 28 Mathieu Functions and Hill’s Equation

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30: 33.20 Expansions for Small $|ϵ|$

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§33.20(i) Case $ϵ=0$

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§33.20(ii) Power-Series in $ϵ$ for the Regular Solution

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§33.20(iii) Asymptotic Expansion for the Irregular Solution

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§33.20(iv) Uniform Asymptotic Expansions

►For a comprehensive collection of asymptotic expansions that cover $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$ as $ϵ\to 0\pm$ and are uniform in $r$, including unbounded values, see Curtis (1964a, §7). …