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})