Bessel functions are the solutions $y(x)$ of Bessel's differential equation $$ x^2 \frac{\mathrm{d}^2y}{\mathrm{d}x^2}+x\frac{\mathrm{d}y}{\mathrm{d}x}+(x^2-\alpha^2)y = 0 $$
The Bessel functions of the first kind, $J_\alpha$ can be defined as $$ J_\alpha(x)=\sum_{m=1}^{\infty}\frac{(-1)^m}{m!\Gamma(m+\alpha+1)}\left(\frac{x}{2}\right)^{2m+\alpha} $$
On the other hand, there is a Laurent series about $J_n$, $$ \exp\left(\left(\frac{x}{2}\right)\left(t-\frac{1}{t}\right)\right) = \sum_{m=-\infty}^{\infty}J_n(x)t^n $$
If I have to prove the Laurent series, I need to compute line integrals by definition of the series, which bothers me.
Could someone please show me how to calculate the line integral?