For the function $$\alpha(\omega)=\frac{R}{i\omega-\lambda}+\frac{R^*}{i\omega-\lambda^*},$$ where $R$ and $\lambda$ are both complex numbers, What is the simplest way to obtain the inverse Fourier transform (not going into Laplace transform): $$h(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} \alpha(\omega)\exp(i\omega t)\,\mathrm{d}\omega.$$ (The solution should be $h(t)=R\exp(\lambda t)+R^*\exp(\lambda^*t)$.)
and then go back to the frequency domain with $$\alpha(\omega)=\int_{-\infty}^{+\infty} h(t)\exp(-i\omega t)\,\mathrm{d}t.$$