I am trying to find the residues of the simple poles in $$f(w)= \frac{e^{iwx}}{kw^2+s}.$$
The simples poles are at points $$w=\pm i\sqrt{\frac{s}{k}}.$$ In general, I have used the formula $$\text{Res}(f(w),w_0)=\frac{1}{(n-1)!}\lim_{w\to w_0}\frac{d^{n-1}}{dw^{n-1}}\left((w-w_0)^n f(w)\right),$$ where $n$ denotes the order of the pole. However, in my particular case, I can not simplify $$\left(w-i\sqrt{\frac{s}{k}}\right)\frac{e^{iwx}}{kw^2+s}$$ to solve for the residues at each singular point.
The denominator is $k(w-i\sqrt {\frac s k})(w+i\sqrt {\frac s k})$. You should be able to write down the residue using this.