text: # §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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