# regular solutions

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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 $\ell$ for the regular solutions and increasing $\ell$ for the irregular solutions of §§33.2(iii) and 33.14(iii). …
##### 2: 33.2 Definitions and Basic Properties
###### §33.2(iii) Irregular Solutions$G_{\ell}\left(\eta,\rho\right),{H^{\pm}_{\ell}}\left(\eta,\rho\right)$
As in the case of $F_{\ell}\left(\eta,\rho\right)$, the solutions ${H^{\pm}_{\ell}}\left(\eta,\rho\right)$ and $G_{\ell}\left(\eta,\rho\right)$ are analytic functions of $\rho$ when $0<\rho<\infty$. …
##### 5: 1.13 Differential Equations
A regular Sturm-Liouville system will only have solutions for certain (real) values of $\lambda$, these are eigenvalues. …
##### 6: 16.21 Differential Equation
16.21.1 $\left((-1)^{p-m-n}z(\vartheta-a_{1}+1)\cdots(\vartheta-a_{p}+1)-(\vartheta-b_{% 1})\cdots(\vartheta-b_{q})\right)w=0,$
With the classification of §16.8(i), when $p the only singularities of (16.21.1) are a regular singularity at $z=0$ and an irregular singularity at $z=\infty$. When $p=q$ the only singularities of (16.21.1) are regular singularities at $z=0$, $(-1)^{p-m-n}$, and $\infty$. A fundamental set of solutions of (16.21.1) is given by …
##### 7: 2.6 Distributional Methods
###### §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).
##### 8: 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, $F_{\ell}\left(\eta,\rho\right)$ and $G_{\ell}\left(\eta,\rho\right)$, or $f\left(\epsilon,\ell;r\right)$ and $h\left(\epsilon,\ell;r\right)$, to determine the scattering $S$-matrix and also the correct normalization of the interior wave solutions; see Bloch et al. (1951). …
##### 9: 14.2 Differential Equations
###### §14.2(iii) Numerically Satisfactory Solutions
Equation (14.2.2) has regular singularities at $x=1$, $-1$, and $\infty$, with exponent pairs $\left\{-\frac{1}{2}\mu,\frac{1}{2}\mu\right\}$, $\left\{-\frac{1}{2}\mu,\frac{1}{2}\mu\right\}$, and $\left\{\nu+1,-\nu\right\}$, respectively; compare §2.7(i). … Hence they comprise a numerically satisfactory pair of solutions2.7(iv)) of (14.2.2) in the interval $-1. …
##### 10: Bibliography B
• W. G. C. Boyd and T. M. Dunster (1986) 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.