10 Bessel Functions Spherical Bessel Functions 10.52 Limiting Forms 10.54 Integral Representations

§10.53 Power Series

Notes:: Combine (10.2.2) and (10.25.2) with (10.47.3), (10.47.4), and (10.47.7).
Keywords:: power series, spherical Bessel functions
Referenced by:: §10.52(i)
Permalink:: http://dlmf.nist.gov/10.53

10.53.1 $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)=z^{n}\sum_{{k=0}}^{\infty% }\frac{(-\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1)!!},$

Symbols:: : : double factorial, : : factorial, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, : integer, : nonnegative integer and : complex variable
A&S Ref:: 10.1.2
Referenced by:: §10.53
Permalink:: http://dlmf.nist.gov/10.53.E1
Encodings:: TeX, pMML, png

10.53.2 $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)=-\frac{1}{z^{{n+1}}}\sum_% {{k=0}}^{n}\frac{(2n-2k-1)!!(\frac{1}{2}z^{2})^{k}}{k!}+\frac{(-1)^{{n+1}}}{z^% {{n+1}}}\sum_{{k=n+1}}^{\infty}\frac{(-\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)!!}.$

Symbols:: : : double factorial, : : factorial, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, : integer, : nonnegative integer and : complex variable
A&S Ref:: 10.1.3
Referenced by:: §10.53
Permalink:: http://dlmf.nist.gov/10.53.E2
Encodings:: TeX, pMML, png

10.53.3 $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)=z^{n}\sum_{{k=0% }}^{\infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2n+2k+1)!!},$

Symbols:: : : double factorial, : : factorial, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : integer, : nonnegative integer and : complex variable
A&S Ref:: 10.2.5
Referenced by:: §10.53
Permalink:: http://dlmf.nist.gov/10.53.E3
Encodings:: TeX, pMML, png

10.53.4 $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)=\frac{(-1)^{n}}% {z^{{n+1}}}\sum_{{k=0}}^{n}\frac{(2n-2k-1)!!(-\frac{1}{2}z^{2})^{k}}{k!}+\frac% {1}{z^{{n+1}}}\sum_{{k=n+1}}^{\infty}\frac{(\frac{1}{2}z^{2})^{k}}{k!(2k-2n-1)% !!}.$

Symbols:: : : double factorial, : : factorial, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : integer, : nonnegative integer and : complex variable
A&S Ref:: 10.2.6
Referenced by:: §10.53
Permalink:: http://dlmf.nist.gov/10.53.E4
Encodings:: TeX, pMML, png

For $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ and $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ combine (10.47.10), (10.53.1), and (10.53.2). For $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ combine (10.47.11), (10.53.3), and (10.53.4).

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