Set $n=1+\epsilon$ and let $\epsilon$ tend to zero. $$ \begin{align} c_1 &= \frac1{2\pi} \left[\frac{e^{i\pi (1+\epsilon)}-e^{-i\pi(1+\epsilon)}}{(1+\epsilon)^2-1}\right]\\ &= \frac1{2\pi} \left[\frac{-e^{i\pi\epsilon}+e^{-i\pi\epsilon}}{(1+\epsilon)^2-1}\right]\\ &\approx \frac1{2\pi} \left[\frac{-1-i\pi\epsilon+1-i\pi\epsilon}{1+2\epsilon-1}\right]\\ &\approx \frac1{2\pi} \left[\frac{-2i\pi\epsilon}{2\epsilon}\right]\\ &\approx \frac{-i}2 \\ &\approx \frac1{2i} \end{align} $$
Can someone please explain this to me, I get the first 2 steps but get confused on what happens to the numerator in the 3rd step. thanks in advance
$$e^x=1+\frac x{1!}+\frac {x^2}{2!}+\cdots$$
For $x\to0,$ $$e^x=1+x+O(x^2)$$