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1: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ 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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2: 33.3 Graphics

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§33.3(i) Line Graphs of the Coulomb Radial Functions $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$

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Figure 33.3.1: $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$ with $\mathrm{\ell }=0$, $\eta =-2$. Magnify

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Figure 33.3.2: $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$ with $\mathrm{\ell }=0$, $\eta =0$. Magnify

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Figure 33.3.3: $F_{\mathrm{\ell }}(\eta ,\rho )$, $G_{\mathrm{\ell }}(\eta ,\rho )$ with $\mathrm{\ell }=0$, $\eta =2$. … Magnify

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§33.3(ii) Surfaces of the Coulomb Radial Functions $F_0(\eta ,\rho )$ and $G_0(\eta ,\rho )$

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3: 33.2 Definitions and Basic Properties

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§33.2(i) Coulomb Wave Equation

… ►This differential equation has a regular singularity at $\rho =0$ with indices $\mathrm{\ell }+1$ and $-\mathrm{\ell }$, and an irregular singularity of rank 1 at $\rho =\mathrm{\infty }$ (§§2.7(i), 2.7(ii)). … ►

§33.2(ii) Regular Solution $F_{\mathrm{\ell }}(\eta ,\rho )$

►The function $F_{\mathrm{\ell }}(\eta ,\rho )$ is recessive (§2.7(iii)) at $\rho =0$, and is defined by … ► $F_{\mathrm{\ell }}(\eta ,\rho )$ is a real and analytic function of $\rho$ on the open interval , and also an analytic function of $\eta$ when . …

4: 31.12 Confluent Forms of Heun’s Equation

… ►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. … ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. … ►This has irregular singularities at $z=0$ and $\mathrm{\infty }$, each of rank $1$. … ►This has a regular singularity at $z=0$, and an irregular singularity at $\mathrm{\infty }$ of rank $2$. … ►This has one singularity, an irregular singularity of rank $3$ at $z=\mathrm{\infty }$. …

5: 16.21 Differential Equation

… ►With the classification of §16.8(i), when the only singularities of (16.21.1) are a regular singularity at $z=0$ and an irregular singularity at $z=\mathrm{\infty }$. When $p=q$ the only singularities of (16.21.1) are regular singularities at $z=0$, ${(-1)}^{p-m-n}$, and $\mathrm{\infty }$. …

6: 31.14 General Fuchsian Equation

… ►The general second-order Fuchsian equation with $N+1$ regular singularities at $z=a_j$, $j=1,2,\mathrm{\dots },N$, and at $\mathrm{\infty }$, is given by ►

31.14.1 $$\frac{d^{2}w}{{dz}^2}+\left(\sum _{j=1}^N\frac{{\gamma }_j}{z-a_j}\right)\frac{dw}{dz}+\left(\sum _{j=1}^N\frac{q_j}{z-a_j}\right)w=0,$$ ${\sum }_{j=1}^N{q}_j=0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(i)

Permalink:

http://dlmf.nist.gov/31.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14 and Ch.31

►The exponents at the finite singularities $a_j$ are $\{0,1-{\gamma }_j\}$ and those at $\mathrm{\infty }$ are $\{\alpha ,\beta \}$, where …The three sets of parameters comprise the singularity parameters $a_j$, the exponent parameters $\alpha ,\beta ,{\gamma }_j$, and the $N-2$ free accessory parameters $q_j$. … ►

31.14.3 $$w(z)=\left(\prod _{j=1}^N{(z-a_j)}^{-{\gamma }_j/2}\right)W(z),$$

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $W(z)$: solution

Permalink:

http://dlmf.nist.gov/31.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

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7: 33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

… ►Again, there is a regular singularity at $r=0$ with indices $\mathrm{\ell }+1$ and $-\mathrm{\ell }$, and an irregular singularity of rank 1 at $r=\mathrm{\infty }$. … ►

§33.14(ii) Regular Solution $f(ϵ,\mathrm{\ell };r)$

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33.14.4 $$f(ϵ,\mathrm{\ell };r)={\kappa }^{\mathrm{\ell }+1}{M}_{\kappa ,\mathrm{\ell }+\frac{1}{2}}\left(2r/\kappa \right)/(2\mathrm{\ell }+1)!,$$

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Defines:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and $\kappa$: quantity

Referenced by:

§13.18(vi), (33.14.14)

Permalink:

http://dlmf.nist.gov/33.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.14(ii), §33.14 and Ch.33

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33.14.5 $$f(ϵ,\mathrm{\ell };r)={(2r)}^{\mathrm{\ell }+1}{\mathrm{e}}^{-r/\kappa }M(\mathrm{\ell }+1-\kappa ,2\mathrm{\ell }+2,2r/\kappa )/(2\mathrm{\ell }+1)!,$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and $\kappa$: quantity

Referenced by:

§13.6(vii)

Permalink:

http://dlmf.nist.gov/33.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §33.14(ii), §33.14 and Ch.33

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8: Bibliography

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Computation of the regular confluent hypergeometric function. The Mathematica Journal 5 (4), pp. 74–76.

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External Links:

ISSN 1047-5974, FullText

Referenced by:

§13.32(iii).

Encodings:

BibTeX

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Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.

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External Links:

MathReview (E. Weber), ZentralBlatt

Referenced by:

§33.9(ii).

Encodings:

BibTeX

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Unsteady lifting-line theory as a singular-perturbation problem. J. Fluid Mech 153, pp. 59–81.

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External Links:

Document, ZentralBlatt

Referenced by:

§11.12.

Encodings:

BibTeX

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Scattering by singular potentials with a perturbation – Theoretical introduction to Mathieu functions. J. Mathematical Phys. 16, pp. 961–970.

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Notes:

Erratum: same journal 17 (1975), no. 7, p. 1361

External Links:

Document, MathReview (K. Veselic), ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

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Singular Continuous Spectrum for a Class of Almost Periodic Jacobi Matrices. Bulletin of the American Mathematical Society 6 (1), pp. 81–85.

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External Links:

MathReview Entry

Referenced by:

§1.18(vii).

Encodings:

BibTeX

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9: 16.8 Differential Equations

… ►If $z_0$ is not an ordinary point but ${(z-z_0)}^{n-j}{f}_j(z)$, $j=0,1,\mathrm{\dots },n-1$, are analytic at $z=z_0$, then $z_0$ is a regular singularity. All other singularities are irregular. … … ►Equation (16.8.4) has a regular singularity at $z=0$, and an irregular singularity at $z=\mathrm{\infty }$, whereas (16.8.5) has regular singularities at $z=0$, $1$, and $\mathrm{\infty }$. … ►Thus in the case $p=q$ the regular singularities of the function on the left-hand side at $\alpha$ and $\mathrm{\infty }$ coalesce into an irregular singularity at $\mathrm{\infty }$. …

10: Bibliography B

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Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.

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External Links:

Document

Referenced by:

§33.22(iv).

Encodings:

BibTeX

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An algorithm for regular and irregular Coulomb and Bessel functions of real order to machine accuracy. Comput. Phys. Comm. 21 (3), pp. 297–314.

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External Links:

Document

Referenced by:

§3.10(iii), §33.22(iv).

Encodings:

BibTeX

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Singularities in Waves and Rays. In Les Houches Lecture Series Session XXXV, R. Balian, M. Kléman, and J.-P. Poirier (Eds.), Vol. 35, pp. 453–543.

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Referenced by:

§36.14(i).

Encodings:

BibTeX

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Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math. 19, pp. 353–370.

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External Links:

Document, MathReview (J. G. van der Corput), ZentralBlatt

Referenced by:

§2.3(v).

Encodings:

BibTeX

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Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions. SIAM J. Math. Anal. 17 (2), pp. 422–450.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.15(iv), §14.15(iv), §14.26, §2.8(vi).

Encodings:

BibTeX

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