§10.9 Integral Representations

§10.9(i) Integrals along the Real Line

Bessel’s Integral

 10.9.1 $J_{0}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta\right)% \mathrm{d}\theta=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\cos\theta\right)% \mathrm{d}\theta,$
 10.9.2 $J_{n}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta-n\theta% \right)\mathrm{d}\theta=\frac{i^{-n}}{\pi}\int_{0}^{\pi}e^{iz\cos\theta}\cos% \left(n\theta\right)\mathrm{d}\theta,$ $n\in\mathbb{Z}$.

Neumann’s Integral

 10.9.3 $Y_{0}\left(z\right)=\frac{4}{\pi^{2}}\int_{0}^{\frac{1}{2}\pi}\cos\left(z\cos% \theta\right)\left(\gamma+\ln\left(2z{\sin}^{2}\theta\right)\right)\mathrm{d}\theta,$

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

Poisson’s and Related Integrals

 10.9.4 $J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma% \left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi}\cos\left(z\cos\theta\right)(\sin% \theta)^{2\nu}\mathrm{d}\theta=\frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}% \Gamma\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}% \cos\left(zt\right)\mathrm{d}t,$ $\Re\nu>-\tfrac{1}{2}$.
 10.9.5 $Y_{\nu}\left(z\right)=\frac{2(\tfrac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\Gamma% \left(\nu+\tfrac{1}{2}\right)}\left(\int_{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}% \sin\left(zt\right)\mathrm{d}t-\int_{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}% {2}}\mathrm{d}t\right),$ $\Re\nu>-\tfrac{1}{2},|\operatorname{ph}z|<\tfrac{1}{2}\pi$.

Schläfli’s and Related Integrals

 10.9.6 $J_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\cos\left(z\sin\theta-\nu% \theta\right)\mathrm{d}\theta-\frac{\sin\left(\nu\pi\right)}{\pi}\int_{0}^{% \infty}e^{-z\sinh t-\nu t}\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$,
 10.9.7 $Y_{\nu}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}\sin\left(z\sin\theta-\nu% \theta\right)\mathrm{d}\theta-\frac{1}{\pi}\int_{0}^{\infty}\left(e^{\nu t}+e^% {-\nu t}\cos\left(\nu\pi\right)\right)e^{-z\sinh t}\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$.

Mehler–Sonine and Related Integrals

 10.9.8 $\displaystyle J_{\nu}\left(x\right)$ $\displaystyle=\frac{2}{\pi}\int_{0}^{\infty}\sin\left(x\cosh t-\tfrac{1}{2}\nu% \pi\right)\cosh\left(\nu t\right)\mathrm{d}t,$ $\displaystyle Y_{\nu}\left(x\right)$ $\displaystyle=-\frac{2}{\pi}\int_{0}^{\infty}\cos\left(x\cosh t-\tfrac{1}{2}% \nu\pi\right)\cosh\left(\nu t\right)\mathrm{d}t,$ $|\Re\nu|<1,x>0$.

In particular,

 10.9.9 $\displaystyle J_{0}\left(x\right)$ $\displaystyle=\frac{2}{\pi}\int_{0}^{\infty}\sin\left(x\cosh t\right)\mathrm{d% }t,$ $x>0$, $\displaystyle Y_{0}\left(x\right)$ $\displaystyle=-\frac{2}{\pi}\int_{0}^{\infty}\cos\left(x\cosh t\right)\mathrm{% d}t,$ $x>0$.
 10.9.10 ${H^{(1)}_{\nu}}\left(z\right)=\frac{e^{-\frac{1}{2}\nu\pi i}}{\pi i}\int_{-% \infty}^{\infty}e^{iz\cosh t-\nu t}\mathrm{d}t,$ $0<\operatorname{ph}z<\pi$,
 10.9.11 ${H^{(2)}_{\nu}}\left(z\right)=-\frac{e^{\frac{1}{2}\nu\pi i}}{\pi i}\int_{-% \infty}^{\infty}e^{-iz\cosh t-\nu t}\mathrm{d}t,$ $-\pi<\operatorname{ph}z<0$.
 10.9.12 $\displaystyle J_{\nu}\left(x\right)$ $\displaystyle=\frac{2(\tfrac{1}{2}x)^{-\nu}}{\pi^{\frac{1}{2}}\Gamma\left(% \tfrac{1}{2}-\nu\right)}\int_{1}^{\infty}\frac{\sin\left(xt\right)\mathrm{d}t}% {(t^{2}-1)^{\nu+\frac{1}{2}}},$ $\displaystyle Y_{\nu}\left(x\right)$ $\displaystyle=-\frac{2(\tfrac{1}{2}x)^{-\nu}}{\pi^{\frac{1}{2}}\Gamma\left(% \tfrac{1}{2}-\nu\right)}\int_{1}^{\infty}\frac{\cos\left(xt\right)\mathrm{d}t}% {(t^{2}-1)^{\nu+\frac{1}{2}}},$ $|\Re\nu|<\tfrac{1}{2}$, $x>0$.
 10.9.13 $\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}J_{\nu}\left((z^{2}-\zeta% ^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{\zeta\cos\theta}\cos% \left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta-\frac{\sin\left(\nu\pi% \right)}{\pi}\int_{0}^{\infty}e^{-\zeta\cosh t-z\sinh t-\nu t}\mathrm{d}t,$ $\Re\left(z+\zeta\right)>0$,
 10.9.14 $\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}Y_{\nu}\left((z^{2}-\zeta% ^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{\zeta\cos\theta}\sin% \left(z\sin\theta-\nu\theta\right)\mathrm{d}\theta-\frac{1}{\pi}\int_{0}^{% \infty}\left(e^{\nu t+\zeta\cosh t}+e^{-\nu t-\zeta\cosh t}\cos\left(\nu\pi% \right)\right)\*e^{-z\sinh t}\mathrm{d}t,$ $\Re\left(z\pm\zeta\right)>0$.
 10.9.15 $\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}{H^{(1)}_{\nu}}\left((z^{% 2}-\zeta^{2})^{\frac{1}{2}}\right)=\frac{1}{\pi i}e^{-\frac{1}{2}\nu\pi i}\int% _{-\infty}^{\infty}e^{iz\cosh t+i\zeta\sinh t-\nu t}\mathrm{d}t,$ $\Im\left(z\pm\zeta\right)>0$,
 10.9.16 $\left(\frac{z+\zeta}{z-\zeta}\right)^{\frac{1}{2}\nu}{H^{(2)}_{\nu}}\left((z^{% 2}-\zeta^{2})^{\frac{1}{2}}\right)=-\frac{1}{\pi i}e^{\frac{1}{2}\nu\pi i}\int% _{-\infty}^{\infty}e^{-iz\cosh t-i\zeta\sinh t-\nu t}\mathrm{d}t,$ $\Im\left(z\pm\zeta\right)<0$.

§10.9(ii) Contour Integrals

Schläfli–Sommerfeld Integrals

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

 10.9.17 $J_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{\infty-\pi i}^{\infty+\pi i}e^{z% \sinh t-\nu t}\mathrm{d}t,$

and

 10.9.18 $\displaystyle{H^{(1)}_{\nu}}\left(z\right)$ $\displaystyle=\frac{1}{\pi i}\int_{-\infty}^{\infty+\pi i}e^{z\sinh t-\nu t}% \mathrm{d}t,$ $\displaystyle{H^{(2)}_{\nu}}\left(z\right)$ $\displaystyle=-\frac{1}{\pi i}\int_{-\infty}^{\infty-\pi i}e^{z\sinh t-\nu t}% \mathrm{d}t.$

Schläfli’s Integral

 10.9.19 $J_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}}{2\pi i}\int_{-\infty}^{(0+)% }\exp\left(t-\frac{z^{2}}{4t}\right)\frac{\mathrm{d}t}{t^{\nu+1}},$

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 $J_{\nu}\left(z\right)=\frac{\Gamma\left(\frac{1}{2}-\nu\right)(\frac{1}{2}z)^{% \nu}}{\pi^{\frac{3}{2}}i}\int_{0}^{(1+)}\cos\left(zt\right)(t^{2}-1)^{\nu-% \frac{1}{2}}\mathrm{d}t,$ $\nu\neq\tfrac{1}{2},\tfrac{3}{2},\dotsc$.
 10.9.21 $\displaystyle{H^{(1)}_{\nu}}\left(z\right)$ $\displaystyle=\frac{\Gamma\left(\tfrac{1}{2}-\nu\right)(\tfrac{1}{2}z)^{\nu}}{% \pi^{\frac{3}{2}}i}\int_{1+i\infty}^{(1+)}e^{izt}(t^{2}-1)^{\nu-\frac{1}{2}}% \mathrm{d}t,$ $\displaystyle{H^{(2)}_{\nu}}\left(z\right)$ $\displaystyle=\frac{\Gamma\left(\tfrac{1}{2}-\nu\right)(\tfrac{1}{2}z)^{\nu}}{% \pi^{\frac{3}{2}}i}\int_{1-i\infty}^{(1+)}e^{-izt}(t^{2}-1)^{\nu-\frac{1}{2}}% \mathrm{d}t,$ $\nu\neq\tfrac{1}{2},\tfrac{3}{2},\dotsc,|\operatorname{ph}z|<\tfrac{1}{2}\pi$.

Mellin–Barnes Type Integrals

 10.9.22 $J_{\nu}\left(x\right)=\frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma% \left(-t\right)(\tfrac{1}{2}x)^{\nu+2t}}{\Gamma\left(\nu+t+1\right)}\mathrm{d}t,$ $\Re\nu>0$, $x>0$,

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

 10.9.23 $J_{\nu}\left(z\right)=\frac{1}{2\pi i}\int_{-\infty-ic}^{-\infty+ic}\frac{% \Gamma\left(t\right)}{\Gamma\left(\nu-t+1\right)}(\tfrac{1}{2}z)^{\nu-2t}% \mathrm{d}t,$

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 $\displaystyle{H^{(1)}_{\nu}}\left(z\right)$ $\displaystyle=-\frac{e^{-\frac{1}{2}\nu\pi i}}{2\pi^{2}}\*\int_{c-i\infty}^{c+% i\infty}\Gamma\left(t\right)\Gamma\left(t-\nu\right)(-\tfrac{1}{2}iz)^{\nu-2t}% \mathrm{d}t,$ $0<\operatorname{ph}z<\pi$, 10.9.25 $\displaystyle{H^{(2)}_{\nu}}\left(z\right)$ $\displaystyle=\frac{e^{\frac{1}{2}\nu\pi i}}{2\pi^{2}}\int_{c-i\infty}^{c+i% \infty}\Gamma\left(t\right)\Gamma\left(t-\nu\right)(\tfrac{1}{2}iz)^{\nu-2t}% \mathrm{d}t,$ $-\pi<\operatorname{ph}z<0$.

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

 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, $\int$: 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 $J_{\nu}\left(z\right)J_{\nu}\left(\zeta\right)=\frac{2}{\pi}\int_{0}^{\pi/2}J_% {2\nu}\left(2(z\zeta)^{\frac{1}{2}}\sin\theta\right)\cos\left((z-\zeta)\cos% \theta\right)\mathrm{d}\theta,$ $\Re\nu>-\tfrac{1}{2}$,

where the square root has its principal value.

 10.9.28 $J_{\nu}\left(z\right)J_{\nu}\left(\zeta\right)=\frac{1}{2\pi i}\int_{c-i\infty% }^{c+i\infty}\*\exp\left(\frac{1}{2}t-\frac{z^{2}+\zeta^{2}}{2t}\right)I_{\nu}% \left(\frac{z\zeta}{t}\right)\frac{\mathrm{d}t}{t},$ $\Re\nu>-1$,

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

Mellin–Barnes Type

 10.9.29 $J_{\mu}\left(x\right)J_{\nu}\left(x\right)=\frac{1}{2\pi i}\int_{-i\infty}^{i% \infty}\frac{\Gamma\left(-t\right)\Gamma\left(2t+\mu+\nu+1\right)(\tfrac{1}{2}% x)^{\mu+\nu+2t}}{\Gamma\left(t+\mu+1\right)\Gamma\left(t+\nu+1\right)\Gamma% \left(t+\mu+\nu+1\right)}\mathrm{d}t,$ $x>0$,

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

 10.9.30 ${J_{\nu}}^{2}\left(z\right)+{Y_{\nu}}^{2}\left(z\right)=\frac{8}{\pi^{2}}\int_% {0}^{\infty}\cosh\left(2\nu t\right)K_{0}\left(2z\sinh t\right)\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$.

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

§10.9(iv) Compendia

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).