§33.9 Expansions in Series of Bessel Functions

Permalink:: http://dlmf.nist.gov/33.9

§33.9(i) Spherical Bessel Functions
§33.9(ii) Bessel Functions and Modified Bessel Functions

§33.9(i) Spherical Bessel Functions

Notes:: The convergence of (33.9.1) follows from the asymptotic forms, for large $k$ , of $a_{k}$ (obtained by application of §2.9(i)) and $\mathop{\mathsf{j}_{{\ell+k}}\/}\nolimits\!\left(\rho\right)$ (obtained from (10.19.1) and (10.47.3)).
Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.9.i

33.9.1 $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=\rho\sum _{{k=0}}^{\infty}a_{k}\mathop{\mathsf{j}_{{\ell+k}}\/}\nolimits\!\left(\rho\right),$

Symbols:: $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter and $a_{k}$ : coefficients
A&S Ref:: 14.4.5
Referenced by:: §33.9(i), §33.9(i)
Permalink:: http://dlmf.nist.gov/33.9.E1
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where the function $\mathop{\mathsf{j}\/}\nolimits$ is as in §10.47(ii), $a_{{-1}}=0$ , $a_{0}=(2\ell+1)!!\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ , and

33.9.2 $\frac{k(k+2\ell+1)}{2k+2\ell+1}a_{k}-2\eta a_{{k-1}}+\frac{(k-2)(k+2\ell-1)}{2k+2\ell-3}a_{{k-2}}=0,$ $k=1,2,\dots$ .

Symbols:: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $a_{k}$ : coefficients
A&S Ref:: 14.4.6
Permalink:: http://dlmf.nist.gov/33.9.E2
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The series (33.9.1) converges for all finite values of $\eta$ and $\rho$ .

§33.9(ii) Bessel Functions and Modified Bessel Functions

Notes:: For (33.9.3) see Yost et al. (1936), Abramowitz (1954), and Humblet (1985). For (33.9.4) see Curtis (1964a, §5.1). For (33.9.6) see Yost et al. (1936) and Abramowitz (1954).
Keywords:: Coulomb functions: variables $\rho,\eta$
Permalink:: http://dlmf.nist.gov/33.9.ii

In this subsection the functions $\mathop{J\/}\nolimits$ , $\mathop{I\/}\nolimits$ , and $\mathop{K\/}\nolimits$ are as in §§10.2(ii) and 10.25(ii).

With $t=2\left|\eta\right|\rho$ ,

33.9.3 $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)\frac{(2\ell+1)!}{(2\eta)^{{2\ell+1}}}\rho^{{-\ell}}\*\sum _{{k=2\ell+1}}^{\infty}b_{k}t^{{k/2}}\mathop{I_{{k}}\/}\nolimits\!\left(\textstyle 2\sqrt{t}\right),$ $\eta>0$ ,

Symbols:: $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $!$ : $n!$ : factorial, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $t$ : parameter and $b_{k}$ : coefficient
A&S Ref:: 14.4.1
Referenced by:: §33.9(ii), §33.9(ii)
Permalink:: http://dlmf.nist.gov/33.9.E3
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33.9.4 $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)=\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)\frac{(2\ell+1)!}{(2\left|\eta\right|)^{{2\ell+1}}}\rho^{{-\ell}}\*\sum _{{k=2\ell+1}}^{\infty}\!\! b_{k}t^{{k/2}}\mathop{J_{{k}}\/}\nolimits\!\left(\textstyle 2\sqrt{t}\right),$ $\eta<0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)$ : normalizing constant for Coulomb radial functions, $!$ : $n!$ : factorial, $\mathop{F_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)$ : regular Coulomb radial function, $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\rho$ : nonnegative real variable, $\eta$ : real parameter, $t$ : parameter and $b_{k}$ : coefficient
Referenced by:: §33.9(ii), §33.9(ii)
Permalink:: http://dlmf.nist.gov/33.9.E4
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Here $b_{{2\ell}}=b_{{2\ell+2}}=0$ , $b_{{2\ell+1}}=1$ , and

33.9.5 ${4\eta^{2}(k-2\ell)b_{{k+1}}+kb_{{k-1}}+b_{{k-2}}=0},$ $k=2\ell+2,2\ell+3,\dots$ .

Symbols:: $k$ : nonnegative integer, $\ell$ : nonnegative integer, $\eta$ : real parameter and $b_{k}$ : coefficient
A&S Ref:: 14.4.3 ((Early printings contained an error. All editions miss $b_{{2L}}=0$ .))
Permalink:: http://dlmf.nist.gov/33.9.E5
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The series (33.9.3) and (33.9.4) converge for all finite positive values of $\left|\eta\right|$ and $\rho$ .

Next, as $\eta\to+\infty$ with $\rho$ ( $>0$ ) fixed,

33.9.6 $\mathop{G_{{\ell}}\/}\nolimits\!\left(\eta,\rho\right)\sim\frac{\rho^{{-\ell}}}{(\ell+\frac{1}{2})\lambda _{\ell}(\eta)\mathop{C_{{\ell}}\/}\nolimits\!\left(\eta\right)}\*\sum _{{k=2\ell+1}}^{\infty}(-1)^{k}b_{k}t^{{k/2}}\mathop{K_{{k}}\/}\nolimits\!\left(\textstyle 2\sqrt{t}\right),$