Restrictions on the inverse Laplace transform for fractional-order systems

The inverse bilateral Laplace transforms of some prototype fractional-order transfer functions are studied. For each prototype transfer function, the inverse transform is attempted for different positions of the branch cut and various regions of convergence. It is seen that certain choices of the branch cut are required in order for the Bromwich integral for the inverse transform to be evaluated. As a result, the variety of inverse transforms that can be found for these fractional-order transfer functions is restricted by comparison with those that can be found for rational transfer functions. Specifically, it is seen that each of the prototype fractional-order transfer functions is excluded from representing either a causal or an anticausal system. It is postulated that such a restriction will apply to any fractional-order transfer function.