§10.54 Integral Representations

Notes:: See Watson (1944, pp. 50 and 174–175). For (10.54.1) use (10.9.4).
Keywords:: spherical Bessel functions
Permalink:: http://dlmf.nist.gov/10.54

10.54.1 $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)=\frac{z^{n}}{2^{{n+1}}n!}% \int_{0}^{\pi}\mathop{\cos\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta% \right)(\mathop{\sin\/}\nolimits\theta)^{{2n+1}}d\theta.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , : : factorial, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, : integer and : complex variable
A&S Ref:: 10.1.13
Referenced by:: §10.54
Permalink:: http://dlmf.nist.gov/10.54.E1
Encodings:: TeX, pMML, png

10.54.2 $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)=\frac{(-i)^{n}}{2}\int_{0% }^{\pi}e^{{iz\mathop{\cos\/}\nolimits\theta}}\mathop{P_{{n}}\/}\nolimits\!% \left(\mathop{\cos\/}\nolimits\theta\right)\mathop{\sin\/}\nolimits\theta d\theta.$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , : base of exponential function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, : integer and : complex variable
A&S Ref:: 10.1.14
Permalink:: http://dlmf.nist.gov/10.54.E2
Encodings:: TeX, pMML, png

10.54.3 $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)=\frac{\pi}{2}\int_{1}^{% \infty}e^{{-zt}}\mathop{P_{{n}}\/}\nolimits\!\left(t\right)dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi.$

Symbols:: $\mathop{P^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the first kind, : differential of , : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathsf{k}_{{n}}\/}\nolimits\!\left(z\right)$ : modified spherical Bessel function, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/10.54.E3
Encodings:: TeX, pMML, png

10.54.4 $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)=\frac{(-i)^{{n+1}}}{2\pi}% \int_{{i\infty}}^{{(-1+,1+)}}e^{{izt}}\mathop{Q_{{n}}\/}\nolimits\!\left(t% \right)dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi.$

Symbols:: $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, : differential of , : base of exponential function, $\int$ : integral, : open interval, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathsf{j}_{{n}}\/}\nolimits\!\left(z\right)$ : spherical Bessel function of the first kind, : integer and : complex variable
Permalink:: http://dlmf.nist.gov/10.54.E4
Encodings:: TeX, pMML, png

10.54.5

$\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)=\frac{(-i)^{{n+% 1}}}{\pi}\int_{{i\infty}}^{{(1+)}}e^{{izt}}\mathop{Q_{{n}}\/}\nolimits\!\left(% t\right)dt,$

$\mathop{{\mathsf{h}^{{(2)}}_{{n}}}\/}\nolimits\!\left(z\right)=\frac{(-i)^{{n+% 1}}}{\pi}\int_{{i\infty}}^{{(-1+)}}e^{{izt}}\mathop{Q_{{n}}\/}\nolimits\!\left% (t\right)dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi.$

Symbols:: $\mathop{Q^{{\mu}}_{{\nu}}\/}\nolimits\!\left(z\right)$ : associated Legendre function of the second kind, : differential of , : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{{\mathsf{h}^{{(1)}}_{{n}}}\/}\nolimits\!\left(z\right)$ : 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/10.54.E5
Encodings:: TeX, TeX, pMML, pMML, png, png

For the Legendre polynomial $\mathop{P_{{n}}\/}\nolimits$ and the associated Legendre function $\mathop{Q_{{n}}\/}\nolimits$ see §§18.3 and 14.21(i), with $\mu=0$ and $\nu=n$ .

Additional integral representations can be obtained by combining the definitions (10.47.3)–(10.47.9) with the results given in §10.9 and §10.32.