Recursive Generation of The Semi-Classical Expansion in Arbitrary Dimension

We present a recursive procedure, which is based on the small time expansion of the propagator, in order to generate a semi-classical expansion of the $\mathit{\text{quantum}}\mathit{\text{action}}$ for a quantum mechanical potential in arbitrary dimensions. In the method we use the spectral information emerges from the singularities of the propagator on the complex $t$ plane, which are handled by the $i\text{ve}$ prescription and basic complex analysis. This feature allows for generalization to higher dimensions. We illustrate the procedure by providing simple examples in non-relativistic quantum mechanics.