10 Bessel Functions Spherical Bessel Functions 10.53 Power Series 10.55 Continued Fractions

§10.54 Integral Representations

ⓘ

Keywords:: integral representations, spherical Bessel functions
Notes:: See Watson (1944, pp. 50 and 174–175). For (10.54.1) use (10.9.4).
Referenced by:: Erratum (V1.1.6) for Chapters 10 Bessel Functions, 18 Orthogonal Polynomials, 34 3j, 6j, 9j Symbols
Permalink:: http://dlmf.nist.gov/10.54
See also:: Annotations for Ch.10

10.54.1		$\mathsf{j}_{n}\left(z\right)=\frac{z^{n}}{2^{n+1}n!}\int_{0}^{\pi}\cos\left(z% \cos\theta\right)(\sin\theta)^{2n+1}\,\mathrm{d}\theta.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $!$ : factorial (as in $n!$ ), $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable A&S Ref: 10.1.13 Referenced by: §10.54 Permalink: http://dlmf.nist.gov/10.54.E1 Encodings: TeX, pMML, png See also: Annotations for §10.54 and Ch.10

10.54.2	$\displaystyle\mathsf{j}_{n}\left(z\right)$	$\displaystyle=\frac{(-i)^{n}}{2}\int_{0}^{\pi}e^{iz\cos\theta}P_{n}\left(\cos% \theta\right)\sin\theta\,\mathrm{d}\theta.$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable A&S Ref: 10.1.14 Permalink: http://dlmf.nist.gov/10.54.E2 Encodings: TeX, pMML, png See also: Annotations for §10.54 and Ch.10
10.54.3	$\displaystyle\mathsf{k}_{n}\left(z\right)$	$\displaystyle=\frac{\pi}{2}\int_{1}^{\infty}e^{-zt}P_{n}\left(t\right)\,% \mathrm{d}t,$
$\|\operatorname{ph}z\|<\tfrac{1}{2}\pi.$
ⓘ Symbols: $P_{\NVar{n}}\left(\NVar{x}\right)$ : Legendre polynomial, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $\mathsf{k}_{\NVar{n}}\left(\NVar{z}\right)$ : modified spherical Bessel function, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.54.E3 Encodings: TeX, pMML, png See also: Annotations for §10.54 and Ch.10
10.54.4	$\displaystyle\mathsf{j}_{n}\left(z\right)$	$\displaystyle=\frac{(-i)^{n+1}}{2\pi}\int_{i\infty}^{(-1+,1+)}e^{izt}Q_{n}% \left(t\right)\,\mathrm{d}t,$
$\|\operatorname{ph}z\|<\tfrac{1}{2}\pi.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $\operatorname{ph}$ : phase, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, $\mathsf{j}_{\NVar{n}}\left(\NVar{z}\right)$ : spherical Bessel function of the first kind, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.54.E4 Encodings: TeX, pMML, png See also: Annotations for §10.54 and Ch.10

10.54.5		$\displaystyle{\mathsf{h}^{(1)}_{n}}\left(z\right)$	$\displaystyle=\frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(1+)}e^{izt}Q_{n}\left(t% \right)\,\mathrm{d}t,$
		$\displaystyle{\mathsf{h}^{(2)}_{n}}\left(z\right)$	$\displaystyle=\frac{(-i)^{n+1}}{\pi}\int_{i\infty}^{(-1+)}e^{izt}Q_{n}\left(t% \right)\,\mathrm{d}t,$
	$\|\operatorname{ph}z\|<\tfrac{1}{2}\pi.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $Q_{\NVar{\nu}}\left(\NVar{z}\right)=Q^{0}_{\nu}\left(z\right)$ : Legendre function of the second kind, ${\mathsf{h}^{(1)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, ${\mathsf{h}^{(2)}_{\NVar{n}}}\left(\NVar{z}\right)$ : spherical Bessel function of the third kind, $n$ : integer and $z$ : complex variable Permalink: http://dlmf.nist.gov/10.54.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.54 and Ch.10

For the Legendre polynomial $P_{n}$ and the associated Legendre function $Q_{n}$ 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.

10.53 Power Series 10.55 Continued Fractions

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