J ~ ν ⁡ ( x ) = ( 2 ⁢ tanh ⁡ ( 1 2 ⁢ π ⁢ ν ) π ⁢ ν ) 1 2 ⁢ cos ⁡ ( ν ⁢ ln ⁡ ( 1 2 ⁢ x ) − γ ν ) + O ⁡ ( x 2 ) {\widetilde {J}}_{\nu }\left(x\right)=\left({\frac {2\tanh({\frac {1}{2}}\pi \nu )}{\pi \nu }}\right)^{\frac {1}{2}}\cos \left(\nu \ln({\tfrac {1}{2}}x)-\gamma _{\nu }\right)+O(x^{2})