10 Bessel Functions Bessel and Hankel Functions 10.8 Power Series 10.10 Continued Fractions

§10.9 Integral Representations

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Referenced by:: §10.54
Permalink:: http://dlmf.nist.gov/10.9
See also:: Annotations for Ch.10

§10.9(i) Integrals along the Real Line

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Keywords:: Bessel functions, along the real line, integral representations
Notes:: See Watson (1944, pp. 19–21, 47–48, 68–71, 169–170, 175–180). For (10.9.3) see Olver (1997b, p. 244) (with “Exercises 2.2 and 9.5” corrected to “Exercises 2.3 and 9.5”). For (10.9.5), (10.9.10), (10.9.11), (10.9.13), (10.9.14) see Erdélyi et al. (1953b, pp. 18, 21, 82). (The condition $\Re z\pm\zeta>0$ in (10.9.14) is weaker than the corresponding condition in Erdélyi et al. (1953b, p. 82, Eq. (18)).) (10.9.15), (10.9.16) follow from (10.9.10), (10.9.11) by change of variables $z=\zeta\cosh\phi$ , $t\to t-\ln\tanh\left(\tfrac{1}{2}\phi\right)$ , $\phi>0$ .
Permalink:: http://dlmf.nist.gov/10.9.i
See also:: Annotations for §10.9 and Ch.10

Bessel’s Integral

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Keywords:: Bessel functions, Bessel’s integral
See also:: Annotations for §10.9(i), §10.9 and Ch.10

10.9.1		$J_{0}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta\right)\,% \mathrm{d}\theta=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\cos\theta\right)\,% \mathrm{d}\theta,$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 9.1.18 Permalink: http://dlmf.nist.gov/10.9.E1 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.2		$J_{n}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta-n\theta% \right)\,\mathrm{d}\theta=\frac{i^{-n}}{\pi}\int_{0}^{\pi}e^{iz\cos\theta}\cos% \left(n\theta\right)\,\mathrm{d}\theta,$
$n\in\mathbb{Z}$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\mathbb{Z}$ : set of all integers, $\int$ : integral, $\sin\NVar{z}$ : sine function, $n$ : integer and $z$ : complex variable A&S Ref: 9.1.21 Referenced by: §11.10(vii) Permalink: http://dlmf.nist.gov/10.9.E2 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

Neumann’s Integral

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Keywords:: Bessel functions, Neumann’s integral
See also:: Annotations for §10.9(i), §10.9 and Ch.10

10.9.3		$Y_{0}\left(z\right)=\frac{4}{\pi^{2}}\int_{0}^{\frac{1}{2}\pi}\cos\left(z\cos% \theta\right)\left(\gamma+\ln\left(2z{\sin}^{2}\theta\right)\right)\,\mathrm{d% }\theta,$
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\gamma$ : Euler’s constant, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 9.19 Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E3 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

where $\gamma$ is Euler’s constant (§5.2(ii)).

Poisson’s and Related Integrals

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Keywords:: Bessel functions, Poisson’s integral
See also:: Annotations for §10.9(i), §10.9 and Ch.10

10.9.4		$J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma% \left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi}\cos\left(z\cos\theta\right)(\sin% \theta)^{2\nu}\,\mathrm{d}\theta=\frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2% }}\Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}% \cos\left(zt\right)\,\mathrm{d}t,$
$\Re\nu>-\tfrac{1}{2}$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.20 Referenced by: §10.54 Permalink: http://dlmf.nist.gov/10.9.E4 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.5		$Y_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma% \left(\nu+\tfrac{1}{2}\right)}\left(\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}% \sin\left(zt\right)\,\mathrm{d}t-\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{% 1}{2}}\,\mathrm{d}t\right),$
$\Re\nu>-\tfrac{1}{2},\|\operatorname{ph}z\|<\tfrac{1}{2}\pi$ .
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : 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, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E5 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

Schläfli’s and Related Integrals

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Keywords:: Bessel functions, Schläfli’s integrals
See also:: Annotations for §10.9(i), §10.9 and Ch.10

10.9.6		$J_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta-\nu% \theta\right)\,\mathrm{d}\theta-\frac{\sin\left(\nu\pi\right)}{\pi}\int_{0}^{% \infty}e^{-z\sinh t-\nu t}\,\mathrm{d}t,$
$\|\operatorname{ph}z\|<\tfrac{1}{2}\pi$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : 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, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.22 Permalink: http://dlmf.nist.gov/10.9.E6 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.7		$Y_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\sin\left(z\sin\theta-\nu% \theta\right)\,\mathrm{d}\theta-\frac{1}{\pi}\int_{0}^{\infty}\left(e^{\nu t}+% e^{-\nu t}\cos\left(\nu\pi\right)\right)e^{-z\sinh t}\,\mathrm{d}t,$
$\|\operatorname{ph}z\|<\tfrac{1}{2}\pi$ .
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : 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, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\operatorname{ph}$ : phase, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.22 Permalink: http://dlmf.nist.gov/10.9.E7 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

Mehler–Sonine and Related Integrals

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Keywords:: Bessel and Hankel functions, Bessel functions, Hankel functions, Mehler–Sonine integrals, along real line, along the real line, integral representations
See also:: Annotations for §10.9(i), §10.9 and Ch.10

10.9.8		$\displaystyle J_{\nu}\left(x\right)$	$\displaystyle=\frac{2}{\pi}\int_{0}^{\infty}\sin\left(x\cosh t-\tfrac{1}{2}\nu% \pi\right)\cosh\left(\nu t\right)\,\mathrm{d}t,$
		$\displaystyle Y_{\nu}\left(x\right)$	$\displaystyle=-\frac{2}{\pi}\int_{0}^{\infty}\cos\left(x\cosh t-\tfrac{1}{2}% \nu\pi\right)\cosh\left(\nu t\right)\,\mathrm{d}t,$
	$\|\Re\nu\|<1,x>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $x$ : real variable and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.9.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

In particular,

10.9.9		$\displaystyle J_{0}\left(x\right)$	$\displaystyle=\frac{2}{\pi}\int_{0}^{\infty}\sin\left(x\cosh t\right)\,\mathrm% {d}t,$
	$x>0$ ,
		$\displaystyle Y_{0}\left(x\right)$	$\displaystyle=-\frac{2}{\pi}\int_{0}^{\infty}\cos\left(x\cosh t\right)\,% \mathrm{d}t,$
	$x>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\sin\NVar{z}$ : sine function and $x$ : real variable A&S Ref: 9.1.23 Permalink: http://dlmf.nist.gov/10.9.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.10		${H^{(1)}_{\nu}}\left(z\right)=\frac{e^{-\frac{1}{2}\nu\pi i}}{\pi i}\int_{-% \infty}^{\infty}e^{iz\cosh t-\nu t}\,\mathrm{d}t,$
$0<\operatorname{ph}z<\pi$ ,
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E10 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.11		${H^{(2)}_{\nu}}\left(z\right)=-\frac{e^{\frac{1}{2}\nu\pi i}}{\pi i}\int_{-% \infty}^{\infty}e^{-iz\cosh t-\nu t}\,\mathrm{d}t,$
$-\pi<\operatorname{ph}z<0$ .
ⓘ Symbols: ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E11 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.12		$\displaystyle J_{\nu}\left(x\right)$	$\displaystyle=\frac{2(\tfrac{1}{2}x)^{-\nu}}{\pi^{\frac{1}{2}}\Gamma\left(% \tfrac{1}{2}-\nu\right)}\int_{1}^{\infty}\frac{\sin\left(xt\right)\,\mathrm{d}% t}{(t^{2}-1)^{\nu+\frac{1}{2}}},$
		$\displaystyle Y_{\nu}\left(x\right)$	$\displaystyle=-\frac{2(\tfrac{1}{2}x)^{-\nu}}{\pi^{\frac{1}{2}}\Gamma\left(% \tfrac{1}{2}-\nu\right)}\int_{1}^{\infty}\frac{\cos\left(xt\right)\,\mathrm{d}% t}{(t^{2}-1)^{\nu+\frac{1}{2}}},$
	$\|\Re\nu\|<\tfrac{1}{2}$ , $x>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $x$ : real variable and $\nu$ : complex parameter A&S Ref: 9.1.24 Permalink: http://dlmf.nist.gov/10.9.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.13		$\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}J_{\nu}\left((z^{2}-\zeta% ^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{\zeta\cos\theta}\cos% \left(z\sin\theta-\nu\theta\right)\,\mathrm{d}\theta-\frac{\sin\left(\nu\pi% \right)}{\pi}\int_{0}^{\infty}e^{-\zeta\cosh t-z\sinh t-\nu t}\,\mathrm{d}t,$
$\Re\left(z+\zeta\right)>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : 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, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E13 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.14		$\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}Y_{\nu}\left((z^{2}-\zeta% ^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{\zeta\cos\theta}\sin% \left(z\sin\theta-\nu\theta\right)\,\mathrm{d}\theta-\frac{1}{\pi}\int_{0}^{% \infty}\left(e^{\nu t+\zeta\cosh t}+e^{-\nu t-\zeta\cosh t}\cos\left(\nu\pi% \right)\right)\*e^{-z\sinh t}\,\mathrm{d}t,$
$\Re\left(z\pm\zeta\right)>0$ .
ⓘ Symbols: $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : 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, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E14 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.15		$\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}{H^{(1)}_{\nu}}\left((z^{% 2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi i}e^{-\frac{1}{2}\nu\pi i}\int% _{-\infty}^{\infty}e^{iz\cosh t+i\zeta\sinh t-\nu t}\,\mathrm{d}t,$
$\Im\left(z\pm\zeta\right)>0$ ,
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E15 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

10.9.16		$\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}{H^{(2)}_{\nu}}\left((z^{% 2}-\zeta^{2})^{\frac{1}{2}}\right)=-\frac{1}{\pi i}e^{\frac{1}{2}\nu\pi i}\int% _{-\infty}^{\infty}e^{-iz\cosh t-i\zeta\sinh t-\nu t}\,\mathrm{d}t,$
$\Im\left(z\pm\zeta\right)<0$ .
ⓘ Symbols: ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.9(i) Permalink: http://dlmf.nist.gov/10.9.E16 Encodings: TeX, pMML, png See also: Annotations for §10.9(i), §10.9(i), §10.9 and Ch.10

§10.9(ii) Contour Integrals

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Keywords:: Bessel functions, contour integrals, integral representations
Notes:: See Watson (1944, pp. 160–167 and 174–178).
Permalink:: http://dlmf.nist.gov/10.9.ii
Clarification (effective with 1.2.1):: In regard to the contour of integration, just below (10.9.19), we inserted “(see Figure 5.9.1)”.
See also:: Annotations for §10.9 and Ch.10

Schläfli–Sommerfeld Integrals

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Keywords:: Bessel and Hankel functions, Hankel functions, Schläfli–Sommerfeld integrals, contour integrals, integral representations
See also:: Annotations for §10.9(ii), §10.9 and Ch.10

When $|\operatorname{ph}z|<\frac{1}{2}\pi$ ,

10.9.17		$J_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{\infty-\pi i}^{\infty+\pi i}e^{z% \sinh t-\nu t}\,\mathrm{d}t,$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.9.E17 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

and

10.9.18	$\displaystyle{H^{(1)}_{\nu}}\left(z\right)$	$\displaystyle=\frac{1}{\pi i}\int_{-\infty}^{\infty+\pi i}e^{z\sinh t-\nu t}\,% \mathrm{d}t,$
10.9.18	$\displaystyle{H^{(2)}_{\nu}}\left(z\right)$	$\displaystyle=-\frac{1}{\pi i}\int_{-\infty}^{\infty-\pi i}e^{z\sinh t-\nu t}% \,\mathrm{d}t.$
ⓘ Symbols: ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable and $\nu$ : complex parameter A&S Ref: 9.1.25 Referenced by: §10.74(iii) Permalink: http://dlmf.nist.gov/10.9.E18 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

Schläfli’s Integral

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Keywords:: Bessel functions, Schläfli’s integrals
See also:: Annotations for §10.9(ii), §10.9 and Ch.10

10.9.19		$J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{2\pi i}\int_{-\infty}^{(0+)% }\exp\left(t-\frac{z^{2}}{4t}\right)\frac{\,\mathrm{d}t}{t^{\nu+1}},$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.9(ii), Erratum (V1.2.1) for Figure 5.9.1 Permalink: http://dlmf.nist.gov/10.9.E19 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

where the integration path is a simple loop contour (see Figure 5.9.1), and $t^{\nu+1}$ is continuous on the path and takes its principal value at the intersection with the positive real axis.

Hankel’s Integrals

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See also:: Annotations for §10.9(ii), §10.9 and Ch.10

In (10.9.20) and (10.9.21) the integration paths are simple loop contours not enclosing $t=-1$ . Also, $(t^{2}-1)^{\nu-\frac{1}{2}}$ is continuous on the path, and takes its principal value at the intersection with the interval $(1,\infty)$ .

10.9.20		$J_{\nu}\left(z\right)=\frac{\Gamma\left(\frac{1}{2}-\nu\right)(\frac{1}{2}z)^{% \nu}}{\pi^{\frac{3}{2}}i}\int_{0}^{(1+)}\cos\left(zt\right)(t^{2}-1)^{\nu-% \frac{1}{2}}\,\mathrm{d}t,$
$\nu\neq\tfrac{1}{2},\tfrac{3}{2},\dotsc$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.9(ii) Permalink: http://dlmf.nist.gov/10.9.E20 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

10.9.21		$\displaystyle{H^{(1)}_{\nu}}\left(z\right)$	$\displaystyle=\frac{\Gamma\left(\tfrac{1}{2}-\nu\right)(\tfrac{1}{2}z)^{\nu}}{% \pi^{\frac{3}{2}}i}\int_{1+i\infty}^{(1+)}e^{izt}(t^{2}-1)^{\nu-\frac{1}{2}}\,% \mathrm{d}t,$
		$\displaystyle{H^{(2)}_{\nu}}\left(z\right)$	$\displaystyle=\frac{\Gamma\left(\tfrac{1}{2}-\nu\right)(\tfrac{1}{2}z)^{\nu}}{% \pi^{\frac{3}{2}}i}\int_{1-i\infty}^{(1+)}e^{-izt}(t^{2}-1)^{\nu-\frac{1}{2}}% \,\mathrm{d}t,$
	$\nu\neq\tfrac{1}{2},\tfrac{3}{2},\dotsc,\|\operatorname{ph}z\|<\tfrac{1}{2}\pi$ .
	ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : 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, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.9(ii) Permalink: http://dlmf.nist.gov/10.9.E21 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

Mellin–Barnes Type Integrals

ⓘ

Keywords:: Bessel functions, Mellin–Barnes type, contour integrals, integral representations
See also:: Annotations for §10.9(ii), §10.9 and Ch.10

10.9.22		$J_{\nu}\left(x\right)=\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma% \left(-t\right)(\tfrac{1}{2}x)^{\nu+2t}}{\Gamma\left(\nu+t+1\right)}\,\mathrm{% d}t,$
$\Re\nu>0$ , $x>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $x$ : real variable and $\nu$ : complex parameter Keywords: Mellin transform A&S Ref: 9.1.26 Referenced by: §10.9(ii) Permalink: http://dlmf.nist.gov/10.9.E22 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

where the integration path passes to the left of $t=0,1,2,\dotsc$ .

10.9.23		$J_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{-\infty-ic}^{-\infty+ic}\frac{% \Gamma\left(t\right)}{\Gamma\left(\nu-t+1\right)}(\tfrac{1}{2}z)^{\nu-2t}\,% \mathrm{d}t,$
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable, $\nu$ : complex parameter and $c$ : constant Keywords: Mellin transform Permalink: http://dlmf.nist.gov/10.9.E23 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

where $c$ is a positive constant and the integration path encloses the points $t=0,-1,-2,\dotsc$ .

In (10.9.24) and (10.9.25) $c$ is any constant exceeding $\max(\Re\nu,0)$ .

10.9.24	$\displaystyle{H^{(1)}_{\nu}}\left(z\right)$	$\displaystyle=-\frac{e^{-\frac{1}{2}\nu\pi i}}{2\pi^{2}}\*\int_{c-i\infty}^{c+% i\infty}\Gamma\left(t\right)\Gamma\left(t-\nu\right)(-\tfrac{1}{2}iz)^{\nu-2t}% \,\mathrm{d}t,$
$0<\operatorname{ph}z<\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : 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, $z$ : complex variable, $\nu$ : complex parameter and $c$ : constant Keywords: Mellin transform Referenced by: §10.9(ii) Permalink: http://dlmf.nist.gov/10.9.E24 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10
10.9.25	$\displaystyle{H^{(2)}_{\nu}}\left(z\right)$	$\displaystyle=\frac{e^{\frac{1}{2}\nu\pi i}}{2\pi^{2}}\int_{c-i\infty}^{c+i% \infty}\Gamma\left(t\right)\Gamma\left(t-\nu\right)(\tfrac{1}{2}iz)^{\nu-2t}\,% \mathrm{d}t,$
$-\pi<\operatorname{ph}z<0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)$ : Bessel function of the third kind (or Hankel function), $\pi$ : 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, $z$ : complex variable, $\nu$ : complex parameter and $c$ : constant Keywords: Mellin transform Referenced by: §10.9(ii), §10.9(ii) Permalink: http://dlmf.nist.gov/10.9.E25 Encodings: TeX, pMML, png See also: Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

For (10.9.22)–(10.9.25) and further integrals of this type see Paris and Kaminski (2001, pp. 114–116).

§10.9(iii) Products

ⓘ

Keywords:: Bessel functions, integral representations, products
Notes:: See Watson (1944, pp. 150, 436, 438, 441–444). For (10.9.27) see Erdélyi et al. (1953b, p. 47). See also Olver (1997b, pp. 340–341).
Permalink:: http://dlmf.nist.gov/10.9.iii
See also:: Annotations for §10.9 and Ch.10

10.9.26		$J_{\mu}\left(z\right)J_{\nu}\left(z\right)=\frac{2}{\pi}\int_{0}^{\pi/2}J_{\mu% +\nu}\left(2z\cos\theta\right)\cos\left((\mu-\nu)\theta\right)\,\mathrm{d}\theta,$
$\Re\left(\mu+\nu\right)>-1$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $\nu$ : complex parameter Source: Watson (1944, (5.43.1), p. 150, originally $\cos(\mu-\nu)\theta$ . Clarified to $\cos\left((\mu-\nu)\theta\right)$ in 1.0.18.) Referenced by: Erratum (V1.0.18) for (10.9.26) Permalink: http://dlmf.nist.gov/10.9.E26 Encodings: TeX, pMML, png Clarification (effective with 1.0.18): The factor on the right-hand side of the integral, originally $\cos(\mu-\nu)\theta$ , was replaced with $\cos\left((\mu-\nu)\theta\right)$ to clarify the meaning. Note that this ambiguity originates with Watson (1944, p. 150). See also: Annotations for §10.9(iii), §10.9 and Ch.10

10.9.27		$J_{\nu}\left(z\right)J_{\nu}\left(\zeta\right)=\frac{2}{\pi}\int_{0}^{\pi/2}J_% {2\nu}\left(2(z\zeta)^{\frac{1}{2}}\sin\theta\right)\cos\left((z-\zeta)\cos% \theta\right)\,\mathrm{d}\theta,$
$\Re\nu>-\tfrac{1}{2}$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : sine function, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.9(iii) Permalink: http://dlmf.nist.gov/10.9.E27 Encodings: TeX, pMML, png See also: Annotations for §10.9(iii), §10.9 and Ch.10

where the square root has its principal value.

10.9.28		$J_{\nu}\left(z\right)J_{\nu}\left(\zeta\right)=\frac{1}{2\pi i}\int_{c-i\infty% }^{c+i\infty}\*\exp\left(\frac{1}{2}t-\frac{z^{2}+\zeta^{2}}{2t}\right)I_{\nu}% \left(\frac{z\zeta}{t}\right)\frac{\,\mathrm{d}t}{t},$
$\Re\nu>-1$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $\Re$ : real part, $z$ : complex variable, $\nu$ : complex parameter and $c$ : constant Permalink: http://dlmf.nist.gov/10.9.E28 Encodings: TeX, pMML, png See also: Annotations for §10.9(iii), §10.9 and Ch.10

where $c$ is a positive constant. For the function $I_{\nu}$ see §10.25(ii).

Mellin–Barnes Type

ⓘ

See also:: Annotations for §10.9(iii), §10.9 and Ch.10

10.9.29		$J_{\mu}\left(x\right)J_{\nu}\left(x\right)=\frac{1}{2\pi i}\int_{-i\infty}^{i% \infty}\frac{\Gamma\left(-t\right)\Gamma\left(2t+\mu+\nu+1\right)(\tfrac{1}{2}% x)^{\mu+\nu+2t}}{\Gamma\left(t+\mu+1\right)\Gamma\left(t+\nu+1\right)\Gamma% \left(t+\mu+\nu+1\right)}\,\mathrm{d}t,$
$x>0$ ,
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $x$ : real variable and $\nu$ : complex parameter Keywords: Mellin transform Permalink: http://dlmf.nist.gov/10.9.E29 Encodings: TeX, pMML, png See also: Annotations for §10.9(iii), §10.9(iii), §10.9 and Ch.10

where the path of integration separates the poles of $\Gamma\left(-t\right)$ from those of $\Gamma\left(2t+\mu+\nu+1\right)$ . See Paris and Kaminski (2001, p. 116) for related results.

Nicholson’s Integral

ⓘ

Keywords:: Bessel functions, Nicholson’s integral, integral representations, products
See also:: Annotations for §10.9(iii), §10.9 and Ch.10

10.9.30		${J_{\nu}}^{2}\left(z\right)+{Y_{\nu}}^{2}\left(z\right)=\frac{8}{\pi^{2}}\int_% {0}^{\infty}\cosh\left(2\nu t\right)K_{0}\left(2z\sinh t\right)\,\mathrm{d}t,$
$\|\operatorname{ph}z\|<\tfrac{1}{2}\pi$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $Y_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : hyperbolic cosine function, $\sinh\NVar{z}$ : hyperbolic sine function, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, $z$ : complex variable and $\nu$ : complex parameter Permalink: http://dlmf.nist.gov/10.9.E30 Encodings: TeX, pMML, png See also: Annotations for §10.9(iii), §10.9(iii), §10.9 and Ch.10

For the function $K_{0}$ see §10.25(ii).

§10.9(iv) Compendia

ⓘ

Keywords:: Bessel functions, Bessel functions and Hankel functions, Hankel functions, Hankel’s integrals, compendia, integral representations
Permalink:: http://dlmf.nist.gov/10.9.iv
See also:: Annotations for §10.9 and Ch.10

For collections of integral representations of Bessel and Hankel functions see Erdélyi et al. (1953b, §§7.3 and 7.12), Erdélyi et al. (1954a, pp. 43–48, 51–60, 99–105, 108–115, 123–124, 272–276, and 356–357), Gröbner and Hofreiter (1950, pp. 189–192), Marichev (1983, pp. 191–192 and 196–210), Magnus et al. (1966, §3.6), and Watson (1944, Chapter 6).

10.8 Power Series 10.10 Continued Fractions

NIST

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