${\tilde{J}}_{\nu }\left(x\right)={\left(\frac{2\tanh \left(\frac{1}{2}\pi \nu \right)}{\pi \nu }\right)}^{\frac{1}{2}}\cos \left(\nu \ln \left(12x\right)-{\gamma }_{\nu }\right)+O\left(x^2\right)$