regular%20solutions

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1: 33.24 Tables

•

Abramowitz and Stegun (1964, Chapter 14) tabulates $F_{0}\left(\eta,\rho\right)$ , $G_{0}\left(\eta,\rho\right)$ , $F_{0}'\left(\eta,\rho\right)$ , and $G_{0}'\left(\eta,\rho\right)$ for $\eta=0.5(.5)20$ and $\rho=1(1)20$ , 5S; $C_{0}\left(\eta\right)$ for $\eta=0(.05)3$ , 6S.

…

2: 33.23 Methods of Computation

… â–ºThe 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. … â–ºThus 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. … â–ºThis implies decreasing

\ell

for the regular solutions and increasing

\ell

for the irregular solutions of §§33.2(iii) and 33.14(iii). … â–º§33.8 supplies continued fractions for

F_{\ell}'/F_{\ell}

and

{H^{\pm}_{\ell}}'/{H^{\pm}_{\ell}}

. Combined with the Wronskians (33.2.12), the values of

F_{\ell}

G_{\ell}

, and their derivatives can be extracted. …

3: 33.3 Graphics

… â–º

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

â–º

See accompanying text — Figure 33.3.1: $F_{\ell}\left(\eta,\rho\right)$ , $G_{\ell}\left(\eta,\rho\right)$ with $\ell=0$ , $\eta=-2$ . Magnify

â–º

… â–º

§33.3(ii) Surfaces of the Coulomb Radial Functions $F_{0}\left(\eta,\rho\right)$ and $G_{0}\left(\eta,\rho\right)$

…

4: 33.2 Definitions and Basic Properties

… â–º

§33.2(i) Coulomb Wave Equation

… â–ºThis differential equation has a regular singularity at

\rho=0

with indices

\ell+1

and

-\ell

, and an irregular singularity of rank 1 at

\rho=\infty

(§§2.7(i), 2.7(ii)). … â–º

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

… â–º

§33.2(iii) Irregular Solutions $G_{\ell}\left(\eta,\rho\right),{H^{\pm}_{\ell}}\left(\eta,\rho\right)$

… â–ºAs 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: 33.14 Definitions and Basic Properties

… â–º

§33.14(i) Coulomb Wave Equation

… â–ºAgain, there is a regular singularity at

r=0

with indices

\ell+1

and

-\ell

, and an irregular singularity of rank 1 at

r=\infty

. … â–º

§33.14(ii) Regular Solution $f\left(\epsilon,\ell;r\right)$

… â–º

§33.14(iii) Irregular Solution $h\left(\epsilon,\ell;r\right)$

… â–º

§33.14(iv) Solutions $s\left(\epsilon,\ell;r\right)$ and $c\left(\epsilon,\ell;r\right)$

…

6: 31.12 Confluent Forms of Heun’s Equation

… â–ºConfluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. … â–ºThis has regular singularities at

z=0

and

1

, and an irregular singularity of rank 1 at

z=\infty

. â–ºMathieu functions (Chapter 28), spheroidal wave functions (Chapter 30), and Coulomb spheroidal functions (§30.12) are special cases of solutions of the confluent Heun equation. … â–ºThis has a regular singularity at

z=0

, and an irregular singularity at

\infty

of rank

2

. … â–ºThis has one singularity, an irregular singularity of rank

3

z=\infty

. …

7: 16.21 Differential Equation

… â–º

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,

â“˜

Symbols:: $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $w$ : solution and $\vartheta$ : 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

… â–ºWith the classification of §16.8(i), when

p<q

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

. â–ºA fundamental set of solutions of (16.21.1) is given by …

8: Bibliography I

… â–º

Y. Ikebe (1975) The zeros of regular Coulomb wave functions and of their derivatives. Math. Comp. 29, pp. 878–887.

â“˜

External Links:: ISSN 0025-5718, Document, MathReview, ZentralBlatt
Referenced by:: §3.8(iii).
Encodings:: BibTeX

… â–º

K. Inkeri (1959) The real roots of Bernoulli polynomials. Ann. Univ. Turku. Ser. A I 37, pp. 1–20.

â“˜

External Links:: MathReview (D. H. Lehmer), ZentralBlatt
Referenced by:: §24.12(i).
Encodings:: BibTeX

… â–º

A. R. Its and A. A. Kapaev (1998) Connection formulae for the fourth Painlevé transcendent; Clarkson-McLeod solution. J. Phys. A 31 (17), pp. 4073–4113.

â“˜

External Links:: ISSN 0305-4470, Document, MathReview (Yoshitsugu Takei), ZentralBlatt
Referenced by:: §32.11(v).
Encodings:: BibTeX

…

9: 15.10 Hypergeometric Differential Equation

… â–º

§15.10(i) Fundamental Solutions

… â–ºIt has regular singularities at

z=0,1,\infty

, with corresponding exponent pairs

\{0,1-c\}

\{0,c-a-b\}

\{a,b\}

, respectively. … â–º â–º

§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

… â–ºThe

\genfrac{(}{)}{0.0pt}{}{6}{3}=20

connection formulas for the principal branches of Kummer’s solutions are: …

10: 33.20 Expansions for Small $|\epsilon|$

… â–º

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

â–º

33.20.3

f\left(\epsilon,\ell;r\right)=\sum_{k=0}^{\infty}\epsilon^{k}{\sf F}_{k}(\ell;% r),

â“˜

Symbols:: $f\left(\NVar{\epsilon},\NVar{\ell};\NVar{r}\right)$ : regular Coulomb function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter and $\mathsf{F}_{k}(\ell;r)$ : coefficient
Referenced by:: §33.20(ii)
Permalink:: http://dlmf.nist.gov/33.20.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §33.20(ii), §33.20 and Ch.33

… â–º

§33.20(iii) Asymptotic Expansion for the Irregular Solution

…

regular%20solutions

1—10 of 313 matching pages

1: 33.24 Tables

2: 33.23 Methods of Computation

3: 33.3 Graphics

§33.3(i) Line Graphs of the Coulomb Radial Functions F â„“ â¡ ( Î· , Ï ) and G â„“ â¡ ( Î· , Ï )

§33.3(ii) Surfaces of the Coulomb Radial Functions F 0 â¡ ( Î· , Ï ) and G 0 â¡ ( Î· , Ï )

4: 33.2 Definitions and Basic Properties

§33.2(i) Coulomb Wave Equation

§33.2(ii) Regular Solution F â„“ â¡ ( Î· , Ï )

§33.2(iii) Irregular Solutions G â„“ â¡ ( Î· , Ï ) , H â„“ ± â¡ ( Î· , Ï )

5: 33.14 Definitions and Basic Properties

§33.14(i) Coulomb Wave Equation

§33.14(ii) Regular Solution f â¡ ( Ïµ , â„“ ; r )

§33.14(iii) Irregular Solution h â¡ ( Ïµ , â„“ ; r )

§33.14(iv) Solutions s â¡ ( Ïµ , â„“ ; r ) and c â¡ ( Ïµ , â„“ ; r )

6: 31.12 Confluent Forms of Heun’s Equation

7: 16.21 Differential Equation

8: Bibliography I

9: 15.10 Hypergeometric Differential Equation

§15.10(i) Fundamental Solutions

§15.10(ii) Kummer’s 24 Solutions and Connection Formulas

10: 33.20 Expansions for Small | Ïµ |

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

§33.20(iii) Asymptotic Expansion for the Irregular Solution

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

§33.3(ii) Surfaces of the Coulomb Radial Functions $F_{0}\left(\eta,\rho\right)$ and $G_{0}\left(\eta,\rho\right)$

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

§33.2(iii) Irregular Solutions $G_{\ell}\left(\eta,\rho\right),{H^{\pm}_{\ell}}\left(\eta,\rho\right)$

§33.14(ii) Regular Solution $f\left(\epsilon,\ell;r\right)$

§33.14(iii) Irregular Solution $h\left(\epsilon,\ell;r\right)$

§33.14(iv) Solutions $s\left(\epsilon,\ell;r\right)$ and $c\left(\epsilon,\ell;r\right)$

10: 33.20 Expansions for Small $|\epsilon|$

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