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1: 33.23 Methods of Computation

… ►The power-series expansions of §§33.6 and 33.19 converge for all finite values of the radii $\rho$ and $r$, respectively, and may be used to compute the regular and irregular solutions. … ►Thus the regular solutions can be computed from the power-series expansions (§§33.6, 33.19) for small values of the radii and then integrated in the direction of increasing values of the radii. … ►This implies decreasing $\mathrm{\ell }$ for the regular solutions and increasing $\mathrm{\ell }$ for the irregular solutions of §§33.2(iii) and 33.14(iii). …

2: 33.2 Definitions and Basic Properties

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§33.2(i) Coulomb Wave Equation

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

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§33.2(iii) Irregular Solutions $G_{\mathrm{\ell }}(\eta ,\rho ),H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$

… ►As in the case of $F_{\mathrm{\ell }}(\eta ,\rho )$, the solutions $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ are analytic functions of $\rho$ when . …

3: 33.14 Definitions and Basic Properties

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§33.14(i) Coulomb Wave Equation

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

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4: 33.20 Expansions for Small $|ϵ|$

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

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5: 16.21 Differential Equation

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16.21.1 $$\left({(-1)}^{p-m-n}z(\mathit{\vartheta }-a_1+1)\mathrm{\cdots }(\mathit{\vartheta }-a_p+1)-(\mathit{\vartheta }-b_1)\mathrm{\cdots }(\mathit{\vartheta }-b_q)\right)w=0,$$

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Symbols:

$p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $w$: solution and $\mathit{\vartheta }$: differential operator

Referenced by:

§16.21, §16.21

Permalink:

http://dlmf.nist.gov/16.21.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.21 and Ch.16

… ►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 $\mathrm{\infty }$. ►A fundamental set of solutions of (16.21.1) is given by …

6: 2.6 Distributional Methods

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§2.6(iv) Regularization

… ►However, in the theory of generalized functions (distributions), there is a method, known as “regularization”, by which these integrals can be interpreted in a meaningful manner. …For rigorous derivations of these results and also order estimates for ${\delta }_n(x)$, see Wong (1979) and Wong (1989, Chapter 6).

7: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►For scattering problems, the interior solution is then matched to a linear combination of a pair of Coulomb functions, $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$, or $f(ϵ,\mathrm{\ell };r)$ and $h(ϵ,\mathrm{\ell };r)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). …

8: 14.2 Differential Equations

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§14.2(i) Legendre’s Equation

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§14.2(ii) Associated Legendre Equation

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§14.2(iii) Numerically Satisfactory Solutions

►Equation (14.2.2) has regular singularities at $x=1$, $-1$, and $\mathrm{\infty }$, with exponent pairs $\{-\frac{1}{2}\mu ,\frac{1}{2}\mu \}$, $\{-\frac{1}{2}\mu ,\frac{1}{2}\mu \}$, and $\{\nu +1,-\nu \}$, respectively; compare §2.7(i). … ►Hence they comprise a numerically satisfactory pair of solutions of (14.2.2) in the interval . …

9: Bibliography B

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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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10: 15.11 Riemann’s Differential Equation

… ►The importance of (15.10.1) is that any homogeneous linear differential equation of the second order with at most three distinct singularities, all regular, in the extended plane can be transformed into (15.10.1). The most general form is given by … ►denotes the set of solutions of (15.10.1). ►

§15.11(ii) Transformation Formulas

… ►The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …