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11: 13.2 Definitions and Basic Properties

… ►This equation has a regular singularity at the origin with indices $0$ and $1-b$, and an irregular singularity at infinity of rank one. …In effect, the regular singularities of the hypergeometric differential equation at $b$ and $\mathrm{\infty }$ coalesce into an irregular singularity at $\mathrm{\infty }$. …

12: Bibliography D

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Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one. Methods Appl. Anal. 3 (1), pp. 109–134.

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External Links:

ISSN 1073-2772, Pdf, MathReview (Roderick Wong), ZentralBlatt

Referenced by:

§2.11(v).

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13: 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)). …

14: 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 $r=\mathrm{\infty }$. …

15: 10.47 Definitions and Basic Properties

… ►Equations (10.47.1) and (10.47.2) each have a regular singularity at $z=0$ with indices $n$, $-n-1$, and an irregular singularity at $z=\mathrm{\infty }$ of rank $1$; compare §§2.7(i)–2.7(ii). …

16: 13.14 Definitions and Basic Properties

… ►It has a regular singularity at the origin with indices $\frac{1}{2}\pm \mu$, and an irregular singularity at infinity of rank one. …

17: 28.2 Definitions and Basic Properties

… ►This equation has regular singularities at 0 and 1, both with exponents 0 and $\frac{1}{2}$, and an irregular singular point at $\mathrm{\infty }$. …

18: Bibliography E

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The Fuchsian equation of second order with four singularities. Duke Math. J. 9 (1), pp. 48–58.

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External Links:

Document, MathReview (R. E. Langer), ZentralBlatt

Referenced by:

§31.11(i).

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Generalized Bernoulli numbers, generalized irregular primes, and class number. Ann. Univ. Turku. Ser. A I 178, pp. 1–72.

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External Links:

ISSN 0082-7002, MathReview (Wells Johnson), ZentralBlatt

Referenced by:

§24.10(ii).

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19: Bibliography

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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).

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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.

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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.

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Singularities of Differentiable Maps. Vol. II. Birkhäuser, Boston-Berlin.

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Notes:

Monodromy and asymptotics of integrals, Translated from the Russian by Hugh Porteous, Translation revised by the authors and James Montaldi

External Links:

ISBN 0-8176-3185-2, MathReview, ZentralBlatt

Referenced by:

§36.12(iii).

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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).

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20: 18.39 Applications in the Physical Sciences

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Figure 18.39.2: Coulomb–Pollaczek weight functions, $x\in [-1,1]$, (18.39.50) for $s=10$, $l=0$, and $Z=\pm 1$. For $Z=+1$ the weight function, red curve, has an essential singularity at $x=-1$, as all derivatives vanish as $x\to -1+$; the green curve is ${\int }_{-1}^x{w}^{\mathrm{CP}}(y)dy$, to be compared with its histogram approximation in §18.40(ii). For $Z=-1$ the weight function, blue curve, is non-zero at $x=-1$, but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as $k\to \mathrm{\infty }$, $x_k\to -1-$. Magnify

… ►The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as $x\to -1-$. … ►See Yamani and Fishman (1975) for $L^2$ for expansions of both the regular and irregular spherical Bessel functions, which are the Pollaczeks with $a=Z=0$, and Coulomb functions for fixed $l$, Broad and Reinhardt (1976) for a many particle example, and the overview of Alhaidari et al. (2008). …