Question Number 222095 by wewji12 last updated on 17/Jun/25
$$\mathrm{could}\:\:\mathrm{I}\:\mathrm{consider}\:\:{Y}_{\nu} \left({z}\right)=\mathrm{cot}\left(\nu\pi\right){J}_{\nu} \left({z}\right)−\mathrm{csc}\left(\nu\pi\right){J}_{−\nu} \left({z}\right) \\ $$$$\mathrm{as}\:\infty−\infty\:\mathrm{form}\:\mathrm{limit}\:\mathrm{when}\:\nu\in\mathbb{Z} \\ $$$$\mathrm{and}\:\mathrm{How}\:\mathrm{can}\:\mathrm{i}\:\mathrm{calculate} \\ $$$${Y}_{\nu} \left({z}\right)=\mathrm{cot}\left(\nu\pi\right){J}_{\nu} \left({z}\right)−\mathrm{csc}\left(\nu\pi\right){J}_{−\nu} \left({z}\right)…?? \\ $$$$\underset{\alpha\rightarrow\nu} {\mathrm{lim}}\:\frac{\mathrm{cot}^{\mathrm{2}} \left(\alpha\pi\right){J}_{\alpha} ^{\mathrm{2}} \left({z}\right)−\mathrm{csc}^{\mathrm{2}} \left(\alpha\pi\right){J}_{−\alpha} ^{\:\mathrm{2}} \left({z}\right)}{\mathrm{cot}\left(\alpha\pi\right){J}_{\alpha} \left({z}\right)+\mathrm{csc}\left(\alpha\pi\right){J}_{−\alpha} \left({z}\right)}….. \\ $$$$\underset{\alpha\rightarrow\nu} {\mathrm{lim}}\frac{\frac{\partial\:\:}{\partial\alpha}\left(\mathrm{cot}^{\mathrm{2}} \left(\alpha\pi\right){J}_{\alpha} ^{\mathrm{2}} \left({z}\right)−\mathrm{csc}^{\mathrm{2}} \left(\alpha\pi\right){J}_{−\alpha} ^{\mathrm{2}} \left({z}\right)\right)}{\frac{\partial\:\:}{\partial\alpha}\left(\mathrm{cot}\left(\alpha\pi\right){J}_{\alpha} ^{\:} \left({z}\right)+\mathrm{csc}\left(\alpha\pi\right){J}_{−\alpha} \left({z}\right)\right)}….??…. \\ $$$$:\left(\right. \\ $$