§33.20 Expansions for Small $|\epsilon|$

Permalink:: http://dlmf.nist.gov/33.20

§33.20(i) Case $\epsilon=0$
§33.20(ii) Power-Series in $\epsilon$ for the Regular Solution
§33.20(iii) Asymptotic Expansion for the Irregular Solution
§33.20(iv) Uniform Asymptotic Expansions

§33.20(i) Case $\epsilon=0$

Notes:: See Seaton (2002a, Eqs. 96, 98, 100, 102(corrected)).
Keywords:: Coulomb functions: variables $r,\epsilon$
Permalink:: http://dlmf.nist.gov/33.20.i

33.20.1

$\mathop{f\/}\nolimits\!\left(0,\ell;r\right)=(2r)^{{1/2}}\mathop{J_{{2\ell+1}}\/}\nolimits\!\left(\sqrt{8r}\right),$

$\mathop{h\/}\nolimits\!\left(0,\ell;r\right)=-(2r)^{{1/2}}\mathop{Y_{{2\ell+1}}\/}\nolimits\!\left(\sqrt{8r}\right)$ , $r>0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $\ell$ : nonnegative integer and $r$ : real variable
Referenced by:: §33.21(i)
Permalink:: http://dlmf.nist.gov/33.20.E1
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33.20.2

$\mathop{f\/}\nolimits\!\left(0,\ell;r\right)=(-1)^{{\ell+1}}(2|r|)^{{1/2}}\mathop{I_{{2\ell+1}}\/}\nolimits\!\left(\sqrt{8|r|}\right),$

$\mathop{h\/}\nolimits\!\left(0,\ell;r\right)=(-1)^{\ell}(2/\pi)(2|r|)^{{1/2}}\mathop{K_{{2\ell+1}}\/}\nolimits\!\left(\sqrt{8|r|}\right)$ , $r<0$ .

Symbols:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : regular Coulomb function, $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\ell$ : nonnegative integer and $r$ : real variable
Referenced by:: §33.21(i)
Permalink:: http://dlmf.nist.gov/33.20.E2
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For the functions $\mathop{J\/}\nolimits$ , $\mathop{Y\/}\nolimits$ , $\mathop{I\/}\nolimits$ , and $\mathop{K\/}\nolimits$ see §§10.2(ii), 10.25(ii).

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

Notes:: See Seaton (2002a, Eqs. 58, 64, 67, 68).
Keywords:: Coulomb functions: variables $r,\epsilon$
Referenced by:: §33.14(ii)
Permalink:: http://dlmf.nist.gov/33.20.ii

33.20.3 $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)=\sum _{{k=0}}^{\infty}\epsilon^{k}{\sf F}_{k}(\ell;r),$

Symbols:: $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : 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
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where

33.20.4 ${\sf F}_{k}(\ell;r)=\sum _{{p=2k}}^{{3k}}(2r)^{{(p+1)/2}}C_{{k,p}}\mathop{J_{{2\ell+1+p}}\/}\nolimits\!\left(\sqrt{8r}\right),$ $r>0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\mathsf{F}_{k}(\ell;r)$ : coefficient and $C_{{k,p}}$ : coefficients
Permalink:: http://dlmf.nist.gov/33.20.E4
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33.20.5 ${\sf F}_{k}(\ell;r)=\sum _{{p=2k}}^{{3k}}(-1)^{{\ell+1+p}}(2|r|)^{{(p+1)/2}}C_{{k,p}}\mathop{I_{{2\ell+1+p}}\/}\nolimits\!\left(\sqrt{8|r|}\right),$ $r<0$ .

Symbols:: $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\mathsf{F}_{k}(\ell;r)$ : coefficient and $C_{{k,p}}$ : coefficients
Permalink:: http://dlmf.nist.gov/33.20.E5
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The functions $\mathop{J\/}\nolimits$ and $\mathop{I\/}\nolimits$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{{k,p}}$ are given by $C_{{0,0}}=1$ , $C_{{1,0}}=0$ , and

33.20.6

$C_{{k,p}}=0,$ $p<2k$ or $p>3k$ ,

$C_{{k,p}}=\left(-(2\ell+p)C_{{k-1,p-2}}+C_{{k-1,p-3}}\right)/(4p),$ $k>0$ , $2k\leq p\leq 3k$ .

Symbols:: $k$ : nonnegative integer, $\ell$ : nonnegative integer and $C_{{k,p}}$ : coefficients
Referenced by:: §33.20(iii)
Permalink:: http://dlmf.nist.gov/33.20.E6
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The series (33.20.3) converges for all $r$ and $\epsilon$ .

§33.20(iii) Asymptotic Expansion for the Irregular Solution

Notes:: See Seaton (2002a, Eqs. 59, 69, 70).
Keywords:: Coulomb functions: variables $r,\epsilon$
Permalink:: http://dlmf.nist.gov/33.20.iii

As $\epsilon\to 0$ with $\ell$ and $r$ fixed,

33.20.7 $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)\sim-A(\epsilon,\ell)\sum _{{k=0}}^{\infty}\epsilon^{k}{\sf H}_{k}(\ell;r),$

Symbols:: $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ : irregular Coulomb function, $\sim$ : asymptotic equality, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $\epsilon$ : real parameter, $A(\epsilon,\ell)$ : function and $\mathsf{H}_{k}(\ell;r)$ : coefficient
Permalink:: http://dlmf.nist.gov/33.20.E7
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where $A(\epsilon,\ell)$ is given by (33.14.11), (33.14.12), and

33.20.8 ${\sf H}_{k}(\ell;r)=\sum _{{p=2k}}^{{3k}}(2r)^{{(p+1)/2}}C_{{k,p}}\mathop{Y_{{2\ell+1+p}}\/}\nolimits\!\left(\sqrt{8r}\right),$ $r>0$ ,

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $C_{{k,p}}$ : coefficients and $\mathsf{H}_{k}(\ell;r)$ : coefficient
Permalink:: http://dlmf.nist.gov/33.20.E8
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33.20.9 ${\sf H}_{k}(\ell;r)=(-1)^{{\ell+1}}\frac{2}{\pi}\sum _{{p=2k}}^{{3k}}(2|r|)^{{(p+1)/2}}C_{{k,p}}\mathop{K_{{2\ell+1+p}}\/}\nolimits\!\left(\sqrt{8|r|}\right),$ $r<0$ .

Symbols:: $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $r$ : real variable, $C_{{k,p}}$ : coefficients and $\mathsf{H}_{k}(\ell;r)$ : coefficient
Permalink:: http://dlmf.nist.gov/33.20.E9
Encodings:: TeX, pMML, png

The functions $\mathop{Y\/}\nolimits$ and $\mathop{K\/}\nolimits$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{{k,p}}$ are given by (33.20.6).

§33.20(iv) Uniform Asymptotic Expansions

Keywords:: Coulomb functions: variables $r,\epsilon$
Referenced by:: §2.8(iii), §2.8(iv), §33.12(ii)
Permalink:: http://dlmf.nist.gov/33.20.iv

For a comprehensive collection of asymptotic expansions that cover $\mathop{f\/}\nolimits\!\left(\epsilon,\ell;r\right)$ and $\mathop{h\/}\nolimits\!\left(\epsilon,\ell;r\right)$ as $\epsilon\to 0\pm$ and are uniform in $r$ , including unbounded values, see Curtis (1964a, §7). These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\ell+1$ and $2\ell+2$ .

§33.20 Expansions for Small

Contents

§33.20(i) Case

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

§33.20(iii) Asymptotic Expansion for the Irregular Solution

§33.20(iv) Uniform Asymptotic Expansions

§33.20 Expansions for Small $|\epsilon|$

§33.20(i) Case $\epsilon=0$

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