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31: 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 …

32: 1.15 Summability Methods

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§1.15(ii) Regularity

►Methods of summation are regular if they are consistent with conventional summation. All of the methods described in §1.15(i) are regular. …

33: 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. …

34: 1.13 Differential Equations

… ►on a finite interval $[a,b]\subset ℝ$, this is then a regular Sturm-Liouville system. … ►A regular Sturm-Liouville system will only have solutions for certain (real) values of $\lambda$, these are eigenvalues. … ►For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda$; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called nodes, for $t\in (0,c)$ as for $x\in (a,b)$; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …

35: 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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36: 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)). All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $ℂ\cup \{\mathrm{\infty }\}$, can be transformed into (31.2.1). …

37: 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). … ►The penetrability of repulsive Coulomb potential barriers is normally expressed in terms of the quantity $\rho /(F_{\mathrm{\ell }}^2(\eta ,\rho )+G_{\mathrm{\ell }}^2(\eta ,\rho ))$ (Mott and Massey (1956, pp. 63–65)). …

38: 1.16 Distributions

… ►A distribution $\mathrm{\Lambda }$ is called regular if there is a locally integrable function $f$ on $I$ (i. …We denote a regular distribution by ${\mathrm{\Lambda }}_f$, or simply $f$, where $f$ is the function giving rise to the distribution. (If a distribution is not regular, it is called singular.) More generally, for $\alpha :[a,b]\to [-\mathrm{\infty },\mathrm{\infty }]$ a nondecreasing function the corresponding Lebesgue–Stieltjes measure ${\mu }_{\alpha }$ (see §1.4(v)) can be considered as a distribution: …

39: 10.2 Definitions

… ►This differential equation has a regular singularity at $z=0$ with indices $\pm \nu$, and an irregular singularity at $z=\mathrm{\infty }$ of rank $1$; compare §§2.7(i) and 2.7(ii). …

40: 27.2 Functions

… ►They tend to thin out among the large integers, but this thinning out is not completely regular. …