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11: 33.12 Asymptotic Expansions for Large $\eta$

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33.12.2 $$\genfrac{}{}{0pt}{}{F_0(\eta ,\rho )}{G_0(\eta ,\rho )}\sim {\pi }^{1/2}{(2\eta )}^{1/6}\left\{\genfrac{}{}{0pt}{}{\mathrm{Ai}\left(x\right)}{\mathrm{Bi}\left(x\right)}\left(1+\frac{B_1}{\mu }+\frac{B_2}{{\mu }^2}+\mathrm{\cdots }\right)+\genfrac{}{}{0pt}{}{{\mathrm{Ai}}^{\prime }\left(x\right)}{{\mathrm{Bi}}^{\prime }\left(x\right)}\left(\frac{A_1}{\mu }+\frac{A_2}{{\mu }^2}+\mathrm{\cdots }\right)\right\},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable, $\eta$: real parameter, $x$: variable, $\mu$: variable, $A_j$: coefficients and $B_j$: coefficients

A&S Ref:

14.4.9

Permalink:

http://dlmf.nist.gov/33.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

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33.12.3 $$\genfrac{}{}{0pt}{}{F_0^{\prime }(\eta ,\rho )}{G_0^{\prime }(\eta ,\rho )}\sim -{\pi }^{1/2}{(2\eta )}^{-1/6}\left\{\genfrac{}{}{0pt}{}{\mathrm{Ai}\left(x\right)}{\mathrm{Bi}\left(x\right)}\left(\frac{B_1^{\prime }+x{A}_1}{\mu }+\frac{B_2^{\prime }+x{A}_2}{{\mu }^2}+\mathrm{\cdots }\right)+\genfrac{}{}{0pt}{}{{\mathrm{Ai}}^{\prime }\left(x\right)}{{\mathrm{Bi}}^{\prime }\left(x\right)}\left(\frac{B_1+A_1^{\prime }}{\mu }+\frac{B_2+A_2^{\prime }}{{\mu }^2}+\mathrm{\cdots }\right)\right\},$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\rho$: nonnegative real variable, $\eta$: real parameter, $x$: variable, $\mu$: variable, $A_j$: coefficients and $B_j$: coefficients

A&S Ref:

14.4.10

Permalink:

http://dlmf.nist.gov/33.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

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33.12.6 $$\genfrac{}{}{0pt}{}{F_0(\eta ,2\eta )}{3^{-1/2}{G}_0(\eta ,2\eta )}\sim \frac{\mathrm{\Gamma }\left(\frac{1}{3}\right){\omega }^{1/2}}{2\sqrt{\pi }}\left(1\mp \frac{2}{35}\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\frac{1}{{\omega }^4}-\frac{8}{2025}\frac{1}{{\omega }^6}\mp \frac{5792}{\mathrm{46\hspace{0.17em}06875}}\frac{\mathrm{\Gamma }\left(\frac{2}{3}\right)}{\mathrm{\Gamma }\left(\frac{1}{3}\right)}\frac{1}{{\omega }^{10}}-\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $G_{\mathrm{\ell }}(\eta ,\rho )$: irregular Coulomb radial function, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\eta$: real parameter and $\omega$: factor

Permalink:

http://dlmf.nist.gov/33.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §33.12(i), §33.12 and Ch.33

… ►For asymptotic expansions of $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ when $\eta \to \pm \mathrm{\infty }$ see Temme (2015, Chapter 31). … ►Then, by application of the results given in §§2.8(iii) and 2.8(iv), two sets of asymptotic expansions can be constructed for $F_{\mathrm{\ell }}(\eta ,\rho )$ and $G_{\mathrm{\ell }}(\eta ,\rho )$ when $\eta \to \mathrm{\infty }$. …

12: 33.6 Power-Series Expansions in $\rho$

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33.6.1 $$F_{\mathrm{\ell }}(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}{A}_k^{\mathrm{\ell }}(\eta ){\rho }^k,$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

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33.6.2 $$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )=C_{\mathrm{\ell }}\left(\eta \right)\sum _{k=\mathrm{\ell }+1}^{\mathrm{\infty }}k{A}_k^{\mathrm{\ell }}(\eta ){\rho }^{k-1},$$

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Symbols:

$C_{\mathrm{\ell }}\left(\eta \right)$: normalizing constant for Coulomb radial functions, $F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter and $A_{\mathrm{\ell }+1}^{\mathrm{\ell }}$: coefficient

A&S Ref:

14.1.4, 14.1.5

Referenced by:

§33.6

Permalink:

http://dlmf.nist.gov/33.6.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.6 and Ch.33

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13: 33.15 Graphics

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14: 33.17 Recurrence Relations and Derivatives

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33.17.1 $$(\mathrm{\ell }+1)rf(ϵ,\mathrm{\ell }-1;r)-(2\mathrm{\ell }+1)\left(\mathrm{\ell }(\mathrm{\ell }+1)-r\right)f(ϵ,\mathrm{\ell };r)+\mathrm{\ell }\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r)=0,$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

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33.17.3 $$(\mathrm{\ell }+1)r{f}^{\prime }(ϵ,\mathrm{\ell };r)=\left({(\mathrm{\ell }+1)}^2-r\right)f(ϵ,\mathrm{\ell };r)-\left(1+{(\mathrm{\ell }+1)}^2ϵ\right)rf(ϵ,\mathrm{\ell }+1;r),$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Permalink:

http://dlmf.nist.gov/33.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §33.17 and Ch.33

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15: 33.18 Limiting Forms for Large $\mathrm{\ell }$

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16: 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 $r=\mathrm{\infty }$. … ►

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

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33.14.4 $$f(ϵ,\mathrm{\ell };r)={\kappa }^{\mathrm{\ell }+1}{M}_{\kappa ,\mathrm{\ell }+\frac{1}{2}}\left(2r/\kappa \right)/(2\mathrm{\ell }+1)!,$$

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Defines:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function

Symbols:

$M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and $\kappa$: quantity

Referenced by:

§13.18(vi), (33.14.14)

Permalink:

http://dlmf.nist.gov/33.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §33.14(ii), §33.14 and Ch.33

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33.14.5 $$f(ϵ,\mathrm{\ell };r)={(2r)}^{\mathrm{\ell }+1}{\mathrm{e}}^{-r/\kappa }M(\mathrm{\ell }+1-\kappa ,2\mathrm{\ell }+2,2r/\kappa )/(2\mathrm{\ell }+1)!,$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and $\kappa$: quantity

Referenced by:

§13.6(vii)

Permalink:

http://dlmf.nist.gov/33.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §33.14(ii), §33.14 and Ch.33

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17: 33.11 Asymptotic Expansions for Large $\rho$

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$F_{\mathrm{\ell }}(\eta ,\rho )=g(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}+f(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$

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$F_{\mathrm{\ell }}^{\prime }(\eta ,\rho )=\widehat{g}(\eta ,\rho )\cos {\theta }_{\mathrm{\ell }}+\widehat{f}(\eta ,\rho )\sin {\theta }_{\mathrm{\ell }},$

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18: 31.12 Confluent Forms of Heun’s Equation

… ►Confluent forms of Heun’s differential equation (31.2.1) arise when two or more of the regular singularities merge to form an irregular singularity. … ►This has regular singularities at $z=0$ and $1$, and an irregular singularity of rank 1 at $z=\mathrm{\infty }$. … ►This has a regular singularity at $z=0$, and an irregular singularity at $\mathrm{\infty }$ of rank $2$. …

19: 16.21 Differential Equation

… ►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 $\mathrm{\infty }$. …

20: 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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33.20.3 $$f(ϵ,\mathrm{\ell };r)=\sum _{k=0}^{\mathrm{\infty }}{ϵ}^k{𝖥}_k(\mathrm{\ell };r),$$

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Symbols:

$f(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $k$: nonnegative integer, $\mathrm{\ell }$: nonnegative integer, $r$: real variable, $ϵ$: real parameter and ${𝖥}_k(\mathrm{\ell };r)$: 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

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