# §10.32 Integral Representations

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

 10.32.1 $\mathop{I_{0}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{\pm z% \mathop{\cos\/}\nolimits\theta}\mathrm{d}\theta=\frac{1}{\pi}\int_{0}^{\pi}% \mathop{\cosh\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta\right)\mathrm% {d}\theta.$
 10.32.2 $\mathop{I_{\nu}\/}\nolimits\!\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{\pi^{% \frac{1}{2}}\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right)}\int_{0}^% {\pi}e^{\pm z\mathop{\cos\/}\nolimits\theta}(\mathop{\sin\/}\nolimits\theta)^{% 2\nu}\mathrm{d}\theta=\frac{(\frac{1}{2}z)^{\nu}}{\pi^{\frac{1}{2}}\mathop{% \Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right)}\int_{-1}^{1}(1-t^{2})^{\nu-% \frac{1}{2}}e^{\pm zt}\mathrm{d}t,$ $\Re{\nu}>-\tfrac{1}{2}$.
 10.32.3 $\mathop{I_{n}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{z% \mathop{\cos\/}\nolimits\theta}\mathop{\cos\/}\nolimits\!\left(n\theta\right)% \mathrm{d}\theta.$
 10.32.4 $\mathop{I_{\nu}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{z% \mathop{\cos\/}\nolimits\theta}\mathop{\cos\/}\nolimits\!\left(\nu\theta\right% )\mathrm{d}\theta-\frac{\mathop{\sin\/}\nolimits\!\left(\nu\pi\right)}{\pi}% \int_{0}^{\infty}e^{-z\mathop{\cosh\/}\nolimits t-\nu t}\mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$.
 10.32.5 $\mathop{K_{0}\/}\nolimits\!\left(z\right)=-\frac{1}{\pi}\int_{0}^{\pi}e^{\pm z% \mathop{\cos\/}\nolimits\theta}\left(\gamma+\mathop{\ln\/}\nolimits\!\left(2z(% \mathop{\sin\/}\nolimits\theta)^{2}\right)\right)\mathrm{d}\theta.$
 10.32.6 $\mathop{K_{0}\/}\nolimits\!\left(x\right)=\int_{0}^{\infty}\mathop{\cos\/}% \nolimits\!\left(x\mathop{\sinh\/}\nolimits t\right)\mathrm{d}t=\int_{0}^{% \infty}\frac{\mathop{\cos\/}\nolimits\!\left(xt\right)}{\sqrt{t^{2}+1}}\mathrm% {d}t,$ $x>0$.
 10.32.7 $\mathop{K_{\nu}\/}\nolimits\!\left(x\right)=\mathop{\sec\/}\nolimits\!\left(% \tfrac{1}{2}\nu\pi\right)\int_{0}^{\infty}\mathop{\cos\/}\nolimits\!\left(x% \mathop{\sinh\/}\nolimits t\right)\mathop{\cosh\/}\nolimits\!\left(\nu t\right% )\mathrm{d}t=\mathop{\csc\/}\nolimits\!\left(\tfrac{1}{2}\nu\pi\right)\int_{0}% ^{\infty}\mathop{\sin\/}\nolimits\!\left(x\mathop{\sinh\/}\nolimits t\right)% \mathop{\sinh\/}\nolimits\!\left(\nu t\right)\mathrm{d}t,$ $|\Re{\nu}|<1$, $x>0$.
 10.32.8 $\mathop{K_{\nu}\/}\nolimits\!\left(z\right)=\frac{\pi^{\frac{1}{2}}(\frac{1}{2% }z)^{\nu}}{\mathop{\Gamma\/}\nolimits\!\left(\nu+\frac{1}{2}\right)}\int_{0}^{% \infty}e^{-z\mathop{\cosh\/}\nolimits t}(\mathop{\sinh\/}\nolimits t)^{2\nu}% \mathrm{d}t=\frac{\pi^{\frac{1}{2}}(\frac{1}{2}z)^{\nu}}{\mathop{\Gamma\/}% \nolimits\!\left(\nu+\frac{1}{2}\right)}\int_{1}^{\infty}e^{-zt}(t^{2}-1)^{\nu% -\frac{1}{2}}\mathrm{d}t,$ $\Re{\nu}>-\tfrac{1}{2}$, $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$.
 10.32.9 $\mathop{K_{\nu}\/}\nolimits\!\left(z\right)=\int_{0}^{\infty}e^{-z\mathop{% \cosh\/}\nolimits t}\mathop{\cosh\/}\nolimits\!\left(\nu t\right)\mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$.
 10.32.10 $\mathop{K_{\nu}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}(\tfrac{1}{2}z)^{\nu}% \int_{0}^{\infty}\mathop{\exp\/}\nolimits\left(-t-\frac{z^{2}}{4t}\right)\frac% {\mathrm{d}t}{t^{\nu+1}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{4}\pi$.

### Basset’s Integral

 10.32.11 $\mathop{K_{\nu}\/}\nolimits\!\left(xz\right)=\frac{\mathop{\Gamma\/}\nolimits% \!\left(\nu+\frac{1}{2}\right)(2z)^{\nu}}{\pi^{\frac{1}{2}}x^{\nu}}\int_{0}^{% \infty}\frac{\mathop{\cos\/}\nolimits\!\left(xt\right)\mathrm{d}t}{(t^{2}+z^{2% })^{\nu+\frac{1}{2}}},$ $\Re{\nu}>-\tfrac{1}{2}$, $x>0$, $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$.

## §10.32(ii) Contour Integrals

 10.32.12 $\mathop{I_{\nu}\/}\nolimits\!\left(z\right)=\frac{1}{2\pi i}\int_{\infty-i\pi}% ^{\infty+i\pi}e^{z\mathop{\cosh\/}\nolimits t-\nu t}\mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$.

### Mellin–Barnes Type

 10.32.13 $\mathop{K_{\nu}\/}\nolimits\!\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{4\pi i% }\int_{c-i\infty}^{c+i\infty}\mathop{\Gamma\/}\nolimits\!\left(t\right)\mathop% {\Gamma\/}\nolimits\!\left(t-\nu\right)(\tfrac{1}{2}z)^{-2t}\mathrm{d}t,$ $c>\max(\Re{\nu},0),|\mathop{\mathrm{ph}\/}\nolimits z|<\frac{1}{2}\pi$. Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{K_{\NVar{\nu}}\/}\nolimits\!\left(\NVar{z}\right)$: modified Bessel function of the second kind, $\mathop{\mathrm{ph}\/}\nolimits$: phase, $\Re{}$: real part, $z$: complex variable and $\nu$: complex parameter Referenced by: Equation (10.32.13) Permalink: http://dlmf.nist.gov/10.32.E13 Encodings: TeX, pMML, png Errata (effective with 1.0.11): Originally the constraint $|\mathop{\mathrm{ph}\/}\nolimits z|<\frac{1}{2}\pi$ was written incorrectly as $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$. Reported 2015-05-20 by Richard Paris See also: Annotations for 10.32(ii)
 10.32.14 $\mathop{K_{\nu}\/}\nolimits\!\left(z\right)=\frac{1}{2\pi^{2}i}\left(\frac{\pi% }{2z}\right)^{\frac{1}{2}}e^{-z}\mathop{\cos\/}\nolimits(\nu\pi)\*\int_{-i% \infty}^{i\infty}\mathop{\Gamma\/}\nolimits\!\left(t\right)\mathop{\Gamma\/}% \nolimits\!\left(\tfrac{1}{2}-t-\nu\right)\mathop{\Gamma\/}\nolimits\!\left(% \tfrac{1}{2}-t+\nu\right)(2z)^{t}\mathrm{d}t,$ $\nu-\tfrac{1}{2}\notin\mathbb{Z},|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{3}% {2}\pi$.

In (10.32.14) the integration contour separates the poles of $\mathop{\Gamma\/}\nolimits\!\left(t\right)$ from the poles of $\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-t-\nu\right)\mathop{\Gamma\/}% \nolimits\!\left(\frac{1}{2}-t+\nu\right)$.

## §10.32(iii) Products

 10.32.15 $\mathop{I_{\mu}\/}\nolimits\!\left(z\right)\mathop{I_{\nu}\/}\nolimits\!\left(% z\right)=\frac{2}{\pi}\int_{0}^{\frac{1}{2}\pi}\mathop{I_{\mu+\nu}\/}\nolimits% \!\left(2z\mathop{\cos\/}\nolimits\theta\right)\mathop{\cos\/}\nolimits((\mu-% \nu)\theta)\mathrm{d}\theta,$ $\Re{(\mu+\nu)}>-1$.
 10.32.16 $\mathop{I_{\mu}\/}\nolimits\!\left(x\right)\mathop{K_{\nu}\/}\nolimits\!\left(% x\right)=\int_{0}^{\infty}\mathop{J_{\mu\pm\nu}\/}\nolimits\!\left(2x\mathop{% \sinh\/}\nolimits t\right)e^{(-\mu\pm\nu)t}\mathrm{d}t,$ $\Re{(\mu\mp\nu)}>-\tfrac{1}{2}$, $\Re{(\mu\pm\nu)}>-1$, $x>0$.
 10.32.17 $\mathop{K_{\mu}\/}\nolimits\!\left(z\right)\mathop{K_{\nu}\/}\nolimits\!\left(% z\right)=2\int_{0}^{\infty}\mathop{K_{\mu\pm\nu}\/}\nolimits\!\left(2z\mathop{% \cosh\/}\nolimits t\right)\mathop{\cosh\/}\nolimits((\mu\mp\nu)t)\mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$.
 10.32.18 $\mathop{K_{\nu}\/}\nolimits\!\left(z\right)\mathop{K_{\nu}\/}\nolimits\!\left(% \zeta\right)=\frac{1}{2}\int_{0}^{\infty}\mathop{\exp\/}\nolimits\left(-\frac{% t}{2}-\frac{z^{2}+\zeta^{2}}{2t}\right)\mathop{K_{\nu}\/}\nolimits\left(\frac{% z\zeta}{t}\right)\frac{\mathrm{d}t}{t},$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$, $|\mathop{\mathrm{ph}\/}\nolimits\zeta|<\pi$, $|\mathop{\mathrm{ph}\/}\nolimits(z+\zeta)|<\tfrac{1}{4}\pi$.

### Mellin–Barnes Type

 10.32.19 $\mathop{K_{\mu}\/}\nolimits\!\left(z\right)\mathop{K_{\nu}\/}\nolimits\!\left(% z\right)=\frac{1}{8\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\mathop{\Gamma\/}% \nolimits\!\left(t+\frac{1}{2}\mu+\frac{1}{2}\nu\right)\mathop{\Gamma\/}% \nolimits\!\left(t+\frac{1}{2}\mu-\frac{1}{2}\nu\right)\mathop{\Gamma\/}% \nolimits\!\left(t-\frac{1}{2}\mu+\frac{1}{2}\nu\right)\mathop{\Gamma\/}% \nolimits\!\left(t-\frac{1}{2}\mu-\frac{1}{2}\nu\right)}{\mathop{\Gamma\/}% \nolimits\!\left(2t\right)}(\tfrac{1}{2}z)^{-2t}\mathrm{d}t,$ $c>\tfrac{1}{2}(|\Re{\mu}|+|\Re{\nu}|),|\mathop{\mathrm{ph}\/}\nolimits z|<% \tfrac{1}{2}\pi$.

For similar integrals for $\mathop{J_{\nu}\/}\nolimits\!\left(z\right)\mathop{K_{\nu}\/}\nolimits\!\left(% z\right)$ and $\mathop{I_{\nu}\/}\nolimits\!\left(z\right)\mathop{K_{\nu}\/}\nolimits\!\left(% z\right)$ see Paris and Kaminski (2001, p. 116).

## §10.32(iv) Compendia

For collections of integral representations of modified Bessel functions, or products of modified Bessel functions, see Erdélyi et al. (1953b, §§7.3, 7.12, and 7.14.2), Erdélyi et al. (1954a, pp. 48–60, 105–115, 276–285, and 357–359), Gröbner and Hofreiter (1950, pp. 193–194), Magnus et al. (1966, §3.7), Marichev (1983, pp. 191–216), and Watson (1944, Chapters 6, 12, and 13).