10 Bessel Functions Spherical Bessel Functions 10.51 Recurrence Relations and Derivatives 10.53 Power Series

§10.52 Limiting Forms

Keywords:: spherical Bessel functions
Referenced by:: §10.47(iii)
Permalink:: http://dlmf.nist.gov/10.52

§10.52(i) $z\to 0$
§10.52(ii) $z\to\infty$

§10.52(i) $z\to 0$

Notes:: Use §10.53.
Referenced by:: §10.50
Permalink:: http://dlmf.nist.gov/10.52.i

10.52.1 $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right),\mathop{{\mathsf{i}^{{(1)% }}_{{n}}}\/}\nolimits\!\left(z\right)\sim z^{n}/(2n+1)!!,$

Symbols:: : : double factorial, $\sim$ : asymptotic equality, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, : integer and : complex variable
A&S Ref:: 10.1.4
Permalink:: http://dlmf.nist.gov/10.52.E1
Encodings:: TeX, pMML, png

10.52.2 $-\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right),i\mathop{{\mathsf{h}^{{(% 1)}}_{{n}}}\/}\nolimits\!\left(z\right),-i\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/% }\nolimits\!\left(z\right),(-1)^{n}\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}% \nolimits\!\left(z\right),(2/\pi)\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z% \right)\sim(2n-1)!!/z^{{n+1}}.$

Symbols:: : : double factorial, $\sim$ : asymptotic equality, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, : integer and : complex variable
A&S Ref:: 10.1.5
Permalink:: http://dlmf.nist.gov/10.52.E2
Encodings:: TeX, pMML, png

§10.52(ii) $z\to\infty$

Notes:: Use (10.49.2), (10.49.4), (10.49.6)–(10.49.8), (10.49.10), and (10.49.12).
Referenced by:: §10.50
Permalink:: http://dlmf.nist.gov/10.52.ii

10.52.3

$\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)=z^{{-1}}\mathop{\sin\/}% \nolimits(z-\tfrac{1}{2}n\pi)+e^{{|\imagpart{z}|}}\mathop{O\/}\nolimits(z^{{-2% }}),$

$\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)=-z^{{-1}}\mathop{\cos\/}% \nolimits(z-\tfrac{1}{2}n\pi)+e^{{|\imagpart{z}|}}\mathop{O\/}\nolimits(z^{{-2% }}),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\mathop{\cos\/}\nolimits z$ : cosine function, : base of exponential function, $\imagpart{}$ : imaginary part, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, $\mathop{\mathsf{y}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the second kind, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/10.52.E3
Encodings:: TeX, TeX, pMML, pMML, png, png

10.52.4

$\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)\sim i^{{-n-1}}z% ^{{-1}}e^{{iz}},$

$\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)\sim i^{{n+1}}z^% {{-1}}e^{{-iz}},$

Symbols:: : base of exponential function, $\sim$ : asymptotic equality, $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, $\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the third kind, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/10.52.E4
Encodings:: TeX, TeX, pMML, pMML, png, png

10.52.5 $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)\sim\mathop{{% \mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)\sim\tfrac{1}{2}z^{{-1}}e% ^{z},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{2}\pi-\delta(<\tfrac{1}{2}\pi),$

Symbols:: : base of exponential function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $\mathop{{\mathsf{i}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, $\mathop{{\mathsf{i}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : integer, : complex variable and $\delta$ : small positive constant
Permalink:: http://dlmf.nist.gov/10.52.E5
Encodings:: TeX, pMML, png

10.52.6 $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)\sim\tfrac{1}{2}\pi z^{{-1% }}e^{{-z}}.$

Symbols:: : base of exponential function, $\sim$ : asymptotic equality, $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/10.52.E6
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 10.51 Recurrence Relations and Derivatives 10.53 Power Series

§10.52 Limiting Forms

Contents

§10.52(i)

§10.52(ii)

§10.52(i) $z\to 0$

§10.52(ii) $z\to\infty$