# §11.5 Integral Representations

## §11.5(i) Integrals Along the Real Line

 11.5.1 $\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(z\right)=\frac{2(\tfrac{1}{2}z)^{% \nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int% _{0}^{1}(1-t^{2})^{\nu-\frac{1}{2}}\mathop{\sin\/}\nolimits\!\left(zt\right)% \mathrm{d}t=\frac{2(\tfrac{1}{2}z)^{\nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits% \!\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\pi/2}\mathop{\sin\/}\nolimits\!% \left(z\mathop{\cos\/}\nolimits\theta\right)(\mathop{\sin\/}\nolimits\theta)^{% 2\nu}\mathrm{d}\theta,$ $\Re{\nu}>-\tfrac{1}{2}$,
 11.5.2 $\mathop{\mathbf{K}_{\nu}\/}\nolimits\!\left(z\right)=\frac{2(\tfrac{1}{2}z)^{% \nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int% _{0}^{\infty}e^{-zt}(1+t^{2})^{\nu-\frac{1}{2}}\mathrm{d}t,$ $\Re{z}>0$,
 11.5.3 $\mathop{\mathbf{K}_{0}\/}\nolimits\!\left(z\right)=\frac{2}{\pi}\int_{0}^{% \infty}e^{-z\mathop{\sinh\/}\nolimits t}\mathrm{d}t,$ $\Re{z}>0$,
 11.5.4 $\mathop{\mathbf{M}_{\nu}\/}\nolimits\!\left(z\right)=-\frac{2(\tfrac{1}{2}z)^{% \nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int% _{0}^{1}e^{-zt}(1-t^{2})^{\nu-\frac{1}{2}}\mathrm{d}t,$ $\Re{\nu}>-\tfrac{1}{2}$,
 11.5.5 $\mathop{\mathbf{M}_{0}\/}\nolimits\!\left(z\right)=-\frac{2}{\pi}\int_{0}^{\pi% /2}e^{-z\mathop{\cos\/}\nolimits\theta}\mathrm{d}\theta,$
 11.5.6 $\mathop{\mathbf{L}_{\nu}\/}\nolimits\!\left(z\right)=\frac{2(\tfrac{1}{2}z)^{% \nu}}{\sqrt{\pi}\mathop{\Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int% _{0}^{\pi/2}\mathop{\sinh\/}\nolimits\!\left(z\mathop{\cos\/}\nolimits\theta% \right)(\mathop{\sin\/}\nolimits\theta)^{2\nu}\mathrm{d}\theta,$ $\Re{\nu}>-\tfrac{1}{2}$,
 11.5.7 $\mathop{I_{-\nu}\/}\nolimits\!\left(x\right)-\mathop{\mathbf{L}_{\nu}\/}% \nolimits\!\left(x\right)=\frac{2(\tfrac{1}{2}x)^{\nu}}{\sqrt{\pi}\mathop{% \Gamma\/}\nolimits\!\left(\nu+\tfrac{1}{2}\right)}\int_{0}^{\infty}(1+t^{2})^{% \nu-\frac{1}{2}}\mathop{\sin\/}\nolimits\!\left(xt\right)\mathrm{d}t,$ $x>0$, $\Re{\nu}<\tfrac{1}{2}$.

## §11.5(ii) Contour Integrals

For loop-integral versions of (11.5.1), (11.5.2), (11.5.4), and (11.5.7) see Babister (1967, §§3.3 and 3.14).

### Mellin–Barnes Integrals

 11.5.8 $(\tfrac{1}{2}x)^{-\nu-1}\mathop{\mathbf{H}_{\nu}\/}\nolimits\!\left(x\right)=-% \frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\pi\mathop{\csc\/}\nolimits\!% \left(\pi s\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{2}+s\right)% \mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{2}+\nu+s\right)}(\tfrac{1}{4}x^{2}% )^{s}\mathrm{d}s,$ $x>0$, $\Re{\nu}>-1$,
 11.5.9 $(\tfrac{1}{2}z)^{-\nu-1}\mathop{\mathbf{L}_{\nu}\/}\nolimits\!\left(z\right)=% \frac{1}{2\pi i}\int_{\infty}^{(0+)}\frac{\pi\mathop{\csc\/}\nolimits\!\left(% \pi s\right)}{\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{2}+s\right)\mathop{% \Gamma\/}\nolimits\!\left(\tfrac{3}{2}+\nu+s\right)}(-\tfrac{1}{4}z^{2})^{s}% \mathrm{d}s.$

In (11.5.8) and (11.5.9) the path of integration separates the poles of the integrand at $s=0,1,2,\dots$ from those at $s=-1,-2,-3,\dots$.

## §11.5(iii) Compendia

For further integral representations see Babister (1967, §§3.3, 3.14), Erdélyi et al. (1954a, §§5.17, 15.3), Magnus et al. (1966, p. 114), Oberhettinger (1972), Oberhettinger (1974, §2.7), Oberhettinger and Badii (1973, §2.14), and Watson (1944, pp. 330, 374, and 426).
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