§10.9 Integral Representations

Referenced by:: §10.54
Permalink:: http://dlmf.nist.gov/10.9

§10.9(i) Integrals along the Real Line
§10.9(ii) Contour Integrals
§10.9(iii) Products
§10.9(iv) Compendia

§10.9(i) Integrals along the Real Line

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 $\realpart{(z\pm\zeta)}>0$ 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 $z=\zeta\mathop{\cosh\/}\nolimits\phi$ , $t\to t-\mathop{\ln\/}\nolimits\mathop{\tanh\/}\nolimits(\tfrac{1}{2}\phi)$ , $\phi>0$ .
Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/10.9.i

¶ Bessel’s Integral

Keywords:: Bessel functions, Bessel’s integral

10.9.1 $\mathop{J_{{0}}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int _{0}^{\pi}\mathop{\cos\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta\right)d\theta=\frac{1}{\pi}\int _{0}^{\pi}\mathop{\cos\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta\right)d\theta,$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 9.1.18
Permalink:: http://dlmf.nist.gov/10.9.E1
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10.9.2 $\mathop{J_{{n}}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int _{0}^{\pi}\mathop{\cos\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta-n\theta\right)d\theta=\frac{i^{{-n}}}{\pi}\int _{0}^{\pi}e^{{iz\mathop{\cos\/}\nolimits\theta}}\mathop{\cos\/}\nolimits\!\left(n\theta\right)d\theta,$ $n\in\Integer$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\in$ : element of, $e$ : base of exponential function, $\Integer$ : set of all integers, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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
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¶ Neumann’s Integral

Keywords:: Bessel functions, Neumann’s integral

10.9.3 $\mathop{Y_{{0}}\/}\nolimits\!\left(z\right)=\frac{4}{\pi^{2}}\int _{0}^{{\frac{1}{2}\pi}}\mathop{\cos\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta\right)\left(\EulerConstant+\mathop{\ln\/}\nolimits\!\left(2z{\mathop{\sin\/}\nolimits^{{2}}}\theta\right)\right)d\theta,$

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\EulerConstant$ : Euler’s constant, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 9.19
Referenced by:: §10.9(i)
Permalink:: http://dlmf.nist.gov/10.9.E3
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where $\EulerConstant$ is Euler’s constant (§5.2(ii)).

¶ Poisson’s and Related Integrals

Keywords:: Bessel functions, Poisson’s integral

10.9.4 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{{\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int _{0}^{\pi}\mathop{\cos\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta\right)(\mathop{\sin\/}\nolimits\theta)^{{2\nu}}d\theta=\frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{{\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int _{0}^{1}(1-t^{2})^{{\nu-\frac{1}{2}}}\mathop{\cos\/}\nolimits\!\left(zt\right)dt,$ $\realpart{\nu}>-\tfrac{1}{2}$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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
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10.9.5 $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{{\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\left(\int _{0}^{1}(1-t^{2})^{{\nu-\frac{1}{2}}}\mathop{\sin\/}\nolimits(zt)dt-\int _{0}^{\infty}e^{{-zt}}(1+t^{2})^{{\nu-\frac{1}{2}}}dt\right),$ $\realpart{\nu}>-\tfrac{1}{2},|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.9(i)
Permalink:: http://dlmf.nist.gov/10.9.E5
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¶ Schläfli’s and Related Integrals

Keywords:: Bessel functions, Schläfli’s integrals

10.9.6 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int _{0}^{\pi}\mathop{\cos\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta-\nu\theta\right)d\theta-\frac{\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)}{\pi}\int _{0}^{\infty}e^{{-z\mathop{\sinh\/}\nolimits t-\nu t}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.22
Permalink:: http://dlmf.nist.gov/10.9.E6
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10.9.7 $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int _{0}^{\pi}\mathop{\sin\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits\theta-\nu\theta\right)d\theta-\frac{1}{\pi}\int _{0}^{\infty}\left(e^{{\nu t}}+e^{{-\nu t}}\mathop{\cos\/}\nolimits\!\left(\nu\pi\right)\right)e^{{-z\mathop{\sinh\/}\nolimits t}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : complex parameter
A&S Ref:: 9.1.22
Permalink:: http://dlmf.nist.gov/10.9.E7
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¶ Mehler–Sonine and Related Integrals

Keywords:: Bessel functions, Hankel functions, Mehler–Sonine integrals

10.9.8

$\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{2}{\pi}\int _{0}^{\infty}\mathop{\sin\/}\nolimits(x\mathop{\cosh\/}\nolimits t-\tfrac{1}{2}\nu\pi)\mathop{\cosh\/}\nolimits(\nu t)dt,$

$\mathop{Y_{{\nu}}\/}\nolimits\!\left(x\right)=-\frac{2}{\pi}\int _{0}^{\infty}\mathop{\cos\/}\nolimits(x\mathop{\cosh\/}\nolimits t-\tfrac{1}{2}\nu\pi)\mathop{\cosh\/}\nolimits(\nu t)dt,$ $|\realpart{\nu}|<1,x>0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.9.E8
Encodings:: TeX, TeX, pMML, pMML, png, png

In particular,

10.9.9

$\mathop{J_{{0}}\/}\nolimits\!\left(x\right)=\frac{2}{\pi}\int _{0}^{\infty}\mathop{\sin\/}\nolimits\!\left(x\mathop{\cosh\/}\nolimits t\right)dt,$ $x>0$ ,

$\mathop{Y_{{0}}\/}\nolimits\!\left(x\right)=-\frac{2}{\pi}\int _{0}^{\infty}\mathop{\cos\/}\nolimits\!\left(x\mathop{\cosh\/}\nolimits t\right)dt,$ $x>0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $x$ : real variable
A&S Ref:: 9.1.23
Permalink:: http://dlmf.nist.gov/10.9.E9
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10.9.10 $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)=\frac{e^{{-\frac{1}{2}\nu\pi i}}}{\pi i}\int _{{-\infty}}^{\infty}e^{{iz\mathop{\cosh\/}\nolimits t-\nu t}}dt,$ $0<\mathop{\mathrm{ph}\/}\nolimits z<\pi$ ,

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.9(i)
Permalink:: http://dlmf.nist.gov/10.9.E10
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10.9.11 $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)=-\frac{e^{{\frac{1}{2}\nu\pi i}}}{\pi i}\int _{{-\infty}}^{\infty}e^{{-iz\mathop{\cosh\/}\nolimits t-\nu t}}dt,$ $-\pi<\mathop{\mathrm{ph}\/}\nolimits z<0$ .

Symbols:: $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.9(i)
Permalink:: http://dlmf.nist.gov/10.9.E11
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10.9.12

$\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{2(\tfrac{1}{2}x)^{{-\nu}}}{\pi^{{\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-\nu\right)}\int _{1}^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(xt\right)dt}{(t^{2}-1)^{{\nu+\frac{1}{2}}}},$

$\mathop{Y_{{\nu}}\/}\nolimits\!\left(x\right)=-\frac{2(\tfrac{1}{2}x)^{{-\nu}}}{\pi^{{\frac{1}{2}}}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-\nu\right)}\int _{1}^{\infty}\frac{\mathop{\cos\/}\nolimits\!\left(xt\right)dt}{(t^{2}-1)^{{\nu+\frac{1}{2}}}},$ $|\realpart{\nu}|<\tfrac{1}{2}$ , $x>0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.1.24
Permalink:: http://dlmf.nist.gov/10.9.E12
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10.9.13 $\left(\frac{z+\zeta}{z-\zeta}\right)^{{\frac{1}{2}\nu}}\mathop{J_{{\nu}}\/}\nolimits\!\left((z^{2}-\zeta^{2})^{{\frac{1}{2}}}\right)=\frac{1}{\pi}\int _{0}^{\pi}e^{{\zeta\mathop{\cos\/}\nolimits\theta}}\mathop{\cos\/}\nolimits(z\mathop{\sin\/}\nolimits\theta-\nu\theta)d\theta-\frac{\mathop{\sin\/}\nolimits(\nu\pi)}{\pi}\int _{0}^{\infty}e^{{-\zeta\mathop{\cosh\/}\nolimits t-z\mathop{\sinh\/}\nolimits t-\nu t}}dt,$ $\realpart{(z+\zeta)}>0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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

10.9.14 $\left(\frac{z+\zeta}{z-\zeta}\right)^{{\frac{1}{2}\nu}}\mathop{Y_{{\nu}}\/}\nolimits\!\left((z^{2}-\zeta^{2})^{{\frac{1}{2}}}\right)=\frac{1}{\pi}\int _{0}^{\pi}e^{{\zeta\mathop{\cos\/}\nolimits\theta}}\mathop{\sin\/}\nolimits(z\mathop{\sin\/}\nolimits\theta-\nu\theta)d\theta-\frac{1}{\pi}\int _{0}^{\infty}\left(e^{{\nu t+\zeta\mathop{\cosh\/}\nolimits t}}+e^{{-\nu t-\zeta\mathop{\cosh\/}\nolimits t}}\mathop{\cos\/}\nolimits(\nu\pi)\right)\* e^{{-z\mathop{\sinh\/}\nolimits t}}dt,$ $\realpart{(z\pm\zeta)}>0$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : 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

10.9.15 $\left(\frac{z+\zeta}{z-\zeta}\right)^{{\frac{1}{2}\nu}}\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left((z^{2}-\zeta^{2})^{{\frac{1}{2}}}\right)=\frac{1}{\pi i}e^{{-\frac{1}{2}\nu\pi i}}\int _{{-\infty}}^{\infty}e^{{iz\mathop{\cosh\/}\nolimits t+i\zeta\mathop{\sinh\/}\nolimits t-\nu t}}dt,$ $\imagpart{(z\pm\zeta)}>0$ ,

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\imagpart{}$ : imaginary part, $\int$ : 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

10.9.16 $\left(\frac{z+\zeta}{z-\zeta}\right)^{{\frac{1}{2}\nu}}\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left((z^{2}-\zeta^{2})^{{\frac{1}{2}}}\right)=-\frac{1}{\pi i}e^{{\frac{1}{2}\nu\pi i}}\int _{{-\infty}}^{\infty}e^{{-iz\mathop{\cosh\/}\nolimits t-i\zeta\mathop{\sinh\/}\nolimits t-\nu t}}dt,$ $\imagpart{(z\pm\zeta)}<0$ .

Symbols:: $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\imagpart{}$ : imaginary part, $\int$ : integral, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.9(i)
Permalink:: http://dlmf.nist.gov/10.9.E16
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§10.9(ii) Contour Integrals

Notes:: See Watson (1944, pp. 160–167 and 174–178).
Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/10.9.ii

¶ Schläfli–Sommerfeld Integrals

Keywords:: Hankel functions, Schläfli–Sommerfeld integrals

When $|\mathop{\mathrm{ph}\/}\nolimits z|<\frac{1}{2}\pi$ ,

10.9.17 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{1}{2\pi i}\int _{{\infty-\pi i}}^{{\infty+\pi i}}e^{{z\mathop{\sinh\/}\nolimits t-\nu t}}dt,$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.9.E17
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and

10.9.18

$\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)=\frac{1}{\pi i}\int _{{-\infty}}^{{\infty+\pi i}}e^{{z\mathop{\sinh\/}\nolimits t-\nu t}}dt,$

$\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)=-\frac{1}{\pi i}\int _{{-\infty}}^{{\infty-\pi i}}e^{{z\mathop{\sinh\/}\nolimits t-\nu t}}dt.$

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : 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
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¶ Schläfli’s Integral

Keywords:: Bessel functions, Schläfli’s integrals

10.9.19 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{2\pi i}\int _{{-\infty}}^{{(0+)}}\mathop{\exp\/}\nolimits\left(t-\frac{z^{2}}{4t}\right)\frac{dt}{t^{{\nu+1}}},$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.9.E19
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where the integration path is a simple loop contour, 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

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 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-\nu\right)(\frac{1}{2}z)^{\nu}}{\pi^{{\frac{3}{2}}}i}\int _{0}^{{(1+)}}\mathop{\cos\/}\nolimits(zt)(t^{2}-1)^{{\nu-\frac{1}{2}}}dt,$ $\nu\neq\tfrac{1}{2},\tfrac{3}{2},\ldots$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : 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

10.9.21

$\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-\nu\right)(\tfrac{1}{2}z)^{\nu}}{\pi^{{\frac{3}{2}}}i}\int _{{1+i\infty}}^{{(1+)}}e^{{izt}}(t^{2}-1)^{{\nu-\frac{1}{2}}}dt,$

$\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-\nu\right)(\tfrac{1}{2}z)^{\nu}}{\pi^{{\frac{3}{2}}}i}\int _{{1-i\infty}}^{{(1+)}}e^{{-izt}}(t^{2}-1)^{{\nu-\frac{1}{2}}}dt,$ $\nu\neq\tfrac{1}{2},\tfrac{3}{2},\ldots,|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: ¶ ‣ §10.9(ii)
Permalink:: http://dlmf.nist.gov/10.9.E21
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¶ Mellin–Barnes Type Integrals

Keywords:: Bessel functions

10.9.22 $\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{1}{2\pi i}\int _{{-i\infty}}^{{i\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(-t\right)(\tfrac{1}{2}x)^{{\nu+2t}}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+t+1\right)}dt,$ $\realpart{\nu}>0$ , $x>0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $x$ : real variable and $\nu$ : complex parameter
A&S Ref:: 9.1.26
Referenced by:: ¶ ‣ §10.9(ii)
Permalink:: http://dlmf.nist.gov/10.9.E22
Encodings:: TeX, pMML, png

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

10.9.23 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{1}{2\pi i}\int _{{-\infty-ic}}^{{-\infty+ic}}\frac{\mathop{\Gamma\/}\nolimits\!\left(t\right)}{\mathop{\Gamma\/}\nolimits\!\left(\nu-t+1\right)}(\tfrac{1}{2}z)^{{\nu-2t}}dt,$

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $\nu$ : complex parameter and $c$ : constant
Permalink:: http://dlmf.nist.gov/10.9.E23
Encodings:: TeX, pMML, png

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

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

10.9.24 $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)=-\frac{e^{{-\frac{1}{2}\nu\pi i}}}{2\pi^{2}}\*\int _{{c-i\infty}}^{{c+i\infty}}\mathop{\Gamma\/}\nolimits\!\left(t\right)\mathop{\Gamma\/}\nolimits\!\left(t-\nu\right)(-\tfrac{1}{2}iz)^{{\nu-2t}}dt,$ $0<\mathop{\mathrm{ph}\/}\nolimits z<\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable, $\nu$ : complex parameter and $c$ : constant
Referenced by:: ¶ ‣ §10.9(ii)
Permalink:: http://dlmf.nist.gov/10.9.E24
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10.9.25 $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)=\frac{e^{{\frac{1}{2}\nu\pi i}}}{2\pi^{2}}\int _{{c-i\infty}}^{{c+i\infty}}\mathop{\Gamma\/}\nolimits\!\left(t\right)\mathop{\Gamma\/}\nolimits\!\left(t-\nu\right)(\tfrac{1}{2}iz)^{{\nu-2t}}dt,$ $-\pi<\mathop{\mathrm{ph}\/}\nolimits z<0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{H^{{(2)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable, $\nu$ : complex parameter and $c$ : constant
Referenced by:: ¶ ‣ §10.9(ii), ¶ ‣ §10.9(ii)
Permalink:: http://dlmf.nist.gov/10.9.E25
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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

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).
Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/10.9.iii

10.9.26 $\mathop{J_{{\mu}}\/}\nolimits\!\left(z\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)=\frac{2}{\pi}\int _{0}^{{\pi/2}}\mathop{J_{{\mu+\nu}}\/}\nolimits\!\left(2z\mathop{\cos\/}\nolimits\theta\right)\mathop{\cos\/}\nolimits(\mu-\nu)\theta d\theta,$ $\realpart{(\mu+\nu)}>-1$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.9.E26
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10.9.27 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(\zeta\right)=\frac{2}{\pi}\int _{0}^{{\pi/2}}\mathop{J_{{2\nu}}\/}\nolimits\!\left(2(z\zeta)^{{\frac{1}{2}}}\mathop{\sin\/}\nolimits\theta\right)\mathop{\cos\/}\nolimits\left((z-\zeta)\mathop{\cos\/}\nolimits\theta\right)d\theta,$ $\realpart{\nu}>-\tfrac{1}{2}$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, $z$ : complex variable and $\nu$ : complex parameter
Referenced by:: §10.9(iii)
Permalink:: http://dlmf.nist.gov/10.9.E27
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where the square root has its principal value.

10.9.28 $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(\zeta\right)=\frac{1}{2\pi i}\int _{{c-i\infty}}^{{c+i\infty}}\*\mathop{\exp\/}\nolimits\left(\frac{1}{2}t-\frac{z^{2}+\zeta^{2}}{2t}\right)\mathop{I_{{\nu}}\/}\nolimits\!\left(\frac{z\zeta}{t}\right)\frac{dt}{t},$ $\realpart{\nu}>-1$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $dx$ : differential of $x$ , $\mathop{\exp\/}\nolimits z$ : exponential function, $\int$ : integral, $\mathop{I_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\realpart{}$ : real part, $z$ : complex variable, $\nu$ : complex parameter and $c$ : constant
Permalink:: http://dlmf.nist.gov/10.9.E28
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where $c$ is a positive constant. For the function $\mathop{I_{{\nu}}\/}\nolimits$ see §10.25(ii).

¶ Mellin–Barnes Type

10.9.29 $\mathop{J_{{\mu}}\/}\nolimits\!\left(x\right)\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right)=\frac{1}{2\pi i}\int _{{-i\infty}}^{{i\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(-t\right)\mathop{\Gamma\/}\nolimits\!\left(2t+\mu+\nu+1\right)(\tfrac{1}{2}x)^{{\mu+\nu+2t}}}{\mathop{\Gamma\/}\nolimits\!\left(t+\mu+1\right)\mathop{\Gamma\/}\nolimits\!\left(t+\nu+1\right)\mathop{\Gamma\/}\nolimits\!\left(t+\mu+\nu+1\right)}dt,$ $x>0$ ,

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.9.E29
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where the path of integration separates the poles of $\mathop{\Gamma\/}\nolimits\!\left(-t\right)$ from those of $\mathop{\Gamma\/}\nolimits\!\left(2t+\mu+\nu+1\right)$ . See Paris and Kaminski (2001, p. 116) for related results.

¶ Nicholson’s Integral

Keywords:: Bessel functions, Nicholson’s integral

10.9.30 ${\mathop{J_{{\nu}}\/}\nolimits^{{2}}}\!\left(z\right)+{\mathop{Y_{{\nu}}\/}\nolimits^{{2}}}\!\left(z\right)=\frac{8}{\pi^{2}}\int _{0}^{\infty}\mathop{\cosh\/}\nolimits(2\nu t)\mathop{K_{{0}}\/}\nolimits\!\left(2z\mathop{\sinh\/}\nolimits t\right)dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.9.E30
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For the function $\mathop{K_{{0}}\/}\nolimits$ see §10.25(ii).

§10.9(iv) Compendia

Keywords:: Bessel functions, Hankel functions, Hankel’s integrals
Permalink:: http://dlmf.nist.gov/10.9.iv

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).