\[(\tfrac{1}{2}x)^{-\nu-1}\mathbf{H}_{\nu}\left(x\right)=-\frac{1}{2\pi i}\int_{%
-i\infty}^{i\infty}\frac{\pi\csc\left(\pi s\right)}{\Gamma\left(\tfrac{3}{2}+s%
\right)\Gamma\left(\tfrac{3}{2}+\nu+s\right)}(\tfrac{1}{4}x^{2})^{s}\,\mathrm{%
d}s,\]