# §10.32 Integral Representations

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

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

### Basset’s Integral

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

## §10.32(ii) Contour Integrals

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

### Mellin–Barnes Type

 10.32.13 $K_{\nu}\left(z\right)=\frac{(\frac{1}{2}z)^{\nu}}{4\pi i}\int_{c-i\infty}^{c+i% \infty}\Gamma\left(t\right)\Gamma\left(t-\nu\right)(\tfrac{1}{2}z)^{-2t}% \mathrm{d}t,$ $c>\max(\Re\nu,0),|\operatorname{ph}z|<\frac{1}{2}\pi$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: 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, $K_{\NVar{\nu}}\left(\NVar{z}\right)$: modified Bessel function of the second kind, $\operatorname{ph}$: phase, $\Re$: real part, $z$: complex variable and $\nu$: complex parameter Keywords: Mellin transform Referenced by: Erratum (V1.0.11) for 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 $|\operatorname{ph}z|<\frac{1}{2}\pi$ was written incorrectly as $|\operatorname{ph}z|<\pi$. Reported 2015-05-20 by Richard Paris See also: Annotations for §10.32(ii), §10.32(ii), §10.32 and Ch.10
 10.32.14 $K_{\nu}\left(z\right)=\frac{1}{2\pi^{2}i}\left(\frac{\pi}{2z}\right)^{\frac{1}% {2}}e^{-z}\cos\left(\nu\pi\right)\*\int_{-i\infty}^{i\infty}\Gamma\left(t% \right)\Gamma\left(\tfrac{1}{2}-t-\nu\right)\Gamma\left(\tfrac{1}{2}-t+\nu% \right)(2z)^{t}\mathrm{d}t,$ $\nu-\tfrac{1}{2}\notin\mathbb{Z},|\operatorname{ph}z|<\tfrac{3}{2}\pi$.

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

## §10.32(iii) Products

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

### Mellin–Barnes Type

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

For similar integrals for $J_{\nu}\left(z\right)K_{\nu}\left(z\right)$ and $I_{\nu}\left(z\right)K_{\nu}\left(z\right)$ 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).