# §10.9 Integral Representations

## §10.9(i) Integrals along the Real Line

### Bessel’s Integral

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

### Neumann’s Integral

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

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

### Poisson’s and Related Integrals

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

### Schläfli’s and Related Integrals

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

### Mehler–Sonine and Related Integrals

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

In particular,

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

## §10.9(ii) Contour Integrals

### Schläfli–Sommerfeld Integrals

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

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

and

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

### Schläfli’s Integral

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

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

### Mellin–Barnes Type Integrals

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

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

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

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 $\max(\Re{\nu},0)$.

 10.9.24 $\displaystyle\mathop{{H^{(1)}_{\nu}}\/}\nolimits\!\left(z\right)$ $\displaystyle=-\frac{e^{-\frac{1}{2}\nu\pi i}}{2\pi^{2}}\*\int_{c-i\infty}^{c+% i\infty}\mathop{\Gamma\/}\nolimits\!\left(t\right)\mathop{\Gamma\/}\nolimits\!% \left(t-\nu\right)(-\tfrac{1}{2}iz)^{\nu-2t}\mathrm{d}t,$ $0<\mathop{\mathrm{ph}\/}\nolimits z<\pi$, 10.9.25 $\displaystyle\mathop{{H^{(2)}_{\nu}}\/}\nolimits\!\left(z\right)$ $\displaystyle=\frac{e^{\frac{1}{2}\nu\pi i}}{2\pi^{2}}\int_{c-i\infty}^{c+i% \infty}\mathop{\Gamma\/}\nolimits\!\left(t\right)\mathop{\Gamma\/}\nolimits\!% \left(t-\nu\right)(\tfrac{1}{2}iz)^{\nu-2t}\mathrm{d}t,$ $-\pi<\mathop{\mathrm{ph}\/}\nolimits z<0$.

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 $\mathop{J_{\mu}\/}\nolimits\!\left(z\right)\mathop{J_{\nu}\/}\nolimits\!\left(% z\right)=\frac{2}{\pi}\int_{0}^{\pi/2}\mathop{J_{\mu+\nu}\/}\nolimits\!\left(2% z\mathop{\cos\/}\nolimits\theta\right)\mathop{\cos\/}\nolimits(\mu-\nu)\theta% \mathrm{d}\theta,$ $\Re{(\mu+\nu)}>-1$.
 10.9.27 $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)\mathop{J_{\nu}\/}\nolimits\!\left(% \zeta\right)=\frac{2}{\pi}\int_{0}^{\pi/2}\mathop{J_{2\nu}\/}\nolimits\!\left(% 2(z\zeta)^{\frac{1}{2}}\mathop{\sin\/}\nolimits\theta\right)\mathop{\cos\/}% \nolimits\left((z-\zeta)\mathop{\cos\/}\nolimits\theta\right)\mathrm{d}\theta,$ $\Re{\nu}>-\tfrac{1}{2}$,

where the square root has its principal value.

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

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

### Mellin–Barnes Type

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

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

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

For the function $\mathop{K_{0}\/}\nolimits$ 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).